math.log/2 on complex, imaginary part

Average Error: 32.1 → 0.3

Time: 5.1s

Precision: binary64

Cost: 13056

Math

\[\frac{\tan^{-1}_* \frac{im}{re} \cdot \log base - \log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]

↓

\[\frac{\tan^{-1}_* \frac{im}{re}}{\log base}\]

FPCore

(FPCore (re im base)
 :precision binary64
 (/
  (- (* (atan2 im re) (log base)) (* (log (sqrt (+ (* re re) (* im im)))) 0.0))
  (+ (* (log base) (log base)) (* 0.0 0.0))))

↓

(FPCore (re im base) :precision binary64 (/ (atan2 im re) (log base)))

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 atan2(im, re) / log(base);
}

Error

Bits error versus re

Bits error versus im

Bits error versus base

Alternatives

Alternative 1
Error	56.2
Cost	385

\[\begin{array}{l} \mathbf{if}\;re \leq 1.5180857450401232 \cdot 10^{+80}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Alternative 2
Error	59.5
Cost	64

\[1\]

Derivation

1. Initial program 32.1

   \[\frac{\tan^{-1}_* \frac{im}{re} \cdot \log base - \log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]

2. Simplified 0.3

   \[\leadsto \color{blue}{\frac{\tan^{-1}_* \frac{im}{re}}{\log base}}\]

3. Simplified 0.3

   \[\leadsto \color{blue}{\frac{\tan^{-1}_* \frac{im}{re}}{\log base}}\]

4. Final simplification 0.3

   \[\leadsto \frac{\tan^{-1}_* \frac{im}{re}}{\log base}\]

Reproduce

herbie shell --seed 2021044 
(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))))