calculator

name	calculator
description	Exact symbolic + numeric math with sympy/mpmath — derive formulas, evaluate constants, propagate errors, convert units. Use for ANY multi-step arithmetic or algebra instead of mental math.
version	1.0.0
author	Hermes / AIP
license	MIT
metadata	{"hermes":{"tags":["math","sympy","mpmath","units","derivation","rigor"],"category":"astronomy","related_skills":["cmd-plotting"]}}

Calculator — sympy/mpmath for honest math

When to Use

EVERY time a task involves a formula, a derivation, a unit conversion, or any arithmetic beyond a single trivial operation. The rule from your SOUL: never hand-evaluate formulas — mental algebra and mental arithmetic both fabricate precision. Derive symbolically, evaluate numerically, and quote exactly what the code printed.

sympy (1.14) and mpmath are preinstalled in the analysis environment — plain python3 imports them directly. astropy.units and astropy.constants are there too. Never pip install any of these.

Core recipe

import sympy as sp

# 1. Declare symbols and the formula SYMBOLICALLY
M, t = sp.symbols("M t", positive=True)
t_ms = 10 * (M) ** sp.Rational(-5, 2)   # main-sequence lifetime in Gyr, M in Msun

# 2. Manipulate symbolically if needed (solve, diff, integrate, simplify)
M_of_t = sp.solve(sp.Eq(t, t_ms), M)[0]

# 3. Substitute numbers LAST, evaluate with evalf — quote THIS output
print(t_ms.subs(M, 1.1).evalf())        # -> 7.877... Gyr (NOT a guessed 3.6!)
print(M_of_t.subs(t, 4.0).evalf())      # mass whose t_MS = 4 Gyr

Error propagation (do this instead of hand-waving uncertainties)

import sympy as sp
x, y, sx, sy = sp.symbols("x y sigma_x sigma_y", positive=True)
f = x * y**2
sigma_f = sp.sqrt((sp.diff(f, x) * sx) ** 2 + (sp.diff(f, y) * sy) ** 2)
print(sp.simplify(sigma_f / f))          # relative error, exact
print(sigma_f.subs({x: 3.2, y: 1.7, sx: 0.1, sy: 0.05}).evalf())

Units & constants — astropy, not memory

from astropy import units as u, constants as c
E = (c.G * u.Msun**2 / u.Rsun).to(u.erg)       # gravitational self-energy scale
v = (500 * u.km / u.s).to(u.pc / u.Myr)        # unit conversion
print(E, v)

Never recall constants from memory when astropy.constants has them.

High precision / special functions — mpmath

import mpmath as mp
mp.mp.dps = 50                       # 50 significant digits
print(mp.quad(lambda x: mp.exp(-x**2), [0, mp.inf]))   # sqrt(pi)/2 to 50 digits

Honesty contract

• The number you report MUST be a number your code printed in this session.
• If symbolic and numeric paths disagree, or a result surprises you, show both and say so — do not silently pick the one that matches expectations.
• Keep the derivation script in the task's folder (e.g. analysis/derive_age.py) so the user can rerun it.