math.log/2 on complex, imaginary part

Average Error: 31.7 → 0.4

Time: 4.5s

Precision: 64

Math

\[\frac{\tan^{-1}_* \frac{im}{re} \cdot \log base - \log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\]

↓

\[-1 \cdot \left(\tan^{-1}_* \frac{im}{re} \cdot \frac{-1}{\log base}\right)\]

C

double code(double re, double im, double base) {
	return (((atan2(im, re) * log(base)) - (log(sqrt(((re * re) + (im * im)))) * 0.0)) / ((log(base) * log(base)) + (0.0 * 0.0)));
}

↓

double code(double re, double im, double base) {
	return (-1.0 * (atan2(im, re) * (-1.0 / log(base))));
}

Error

Bits error versus re

Bits error versus im

Bits error versus base

Derivation

1. Initial program 31.7

   \[\frac{\tan^{-1}_* \frac{im}{re} \cdot \log base - \log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\]

2. Taylor expanded around inf 0.3

   \[\leadsto \color{blue}{-1 \cdot \frac{\tan^{-1}_* \frac{im}{re}}{\log \left(\frac{1}{base}\right)}}\]
3. Using strategy rm

4. Applied div-inv 0.4

   \[\leadsto -1 \cdot \color{blue}{\left(\tan^{-1}_* \frac{im}{re} \cdot \frac{1}{\log \left(\frac{1}{base}\right)}\right)}\]

5. Simplified 0.4

   \[\leadsto -1 \cdot \left(\tan^{-1}_* \frac{im}{re} \cdot \color{blue}{\frac{-1}{\log base}}\right)\]

6. Final simplification 0.4

   \[\leadsto -1 \cdot \left(\tan^{-1}_* \frac{im}{re} \cdot \frac{-1}{\log base}\right)\]

Reproduce

herbie shell --seed 2020056 +o rules:numerics
(FPCore (re im base)
  :name "math.log/2 on complex, imaginary part"
  :precision binary64
  (/ (- (* (atan2 im re) (log base)) (* (log (sqrt (+ (* re re) (* im im)))) 0.0)) (+ (* (log base) (log base)) (* 0.0 0.0))))