math.log/2 on complex, imaginary part

Average Error: 31.9 → 0.3

Time: 16.3s

Precision: 64

Math

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

↓

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

C

double f(double re, double im, double base) {
        double r49720 = im;
        double r49721 = re;
        double r49722 = atan2(r49720, r49721);
        double r49723 = base;
        double r49724 = log(r49723);
        double r49725 = r49722 * r49724;
        double r49726 = r49721 * r49721;
        double r49727 = r49720 * r49720;
        double r49728 = r49726 + r49727;
        double r49729 = sqrt(r49728);
        double r49730 = log(r49729);
        double r49731 = 0.0;
        double r49732 = r49730 * r49731;
        double r49733 = r49725 - r49732;
        double r49734 = r49724 * r49724;
        double r49735 = r49731 * r49731;
        double r49736 = r49734 + r49735;
        double r49737 = r49733 / r49736;
        return r49737;
}

↓

double f(double re, double im, double base) {
        double r49738 = im;
        double r49739 = re;
        double r49740 = atan2(r49738, r49739);
        double r49741 = base;
        double r49742 = log(r49741);
        double r49743 = r49740 / r49742;
        return r49743;
}

Error

Bits error versus re

Bits error versus im

Bits error versus base

Derivation

1. Initial program 31.9

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

2. Simplified 0.4

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.0, \log \left(\mathsf{hypot}\left(im, re\right)\right), \log base \cdot \tan^{-1}_* \frac{im}{re}\right)}{\mathsf{fma}\left(\log base, \log base, 0.0 \cdot 0.0\right)}}\]
3. Using strategy rm

4. Applied add-sqr-sqrt 0.4

   \[\leadsto \frac{\mathsf{fma}\left(-0.0, \log \left(\mathsf{hypot}\left(im, re\right)\right), \log base \cdot \tan^{-1}_* \frac{im}{re}\right)}{\color{blue}{\sqrt{\mathsf{fma}\left(\log base, \log base, 0.0 \cdot 0.0\right)} \cdot \sqrt{\mathsf{fma}\left(\log base, \log base, 0.0 \cdot 0.0\right)}}}\]

5. Applied *-un-lft-identity 0.4

   \[\leadsto \frac{\color{blue}{1 \cdot \mathsf{fma}\left(-0.0, \log \left(\mathsf{hypot}\left(im, re\right)\right), \log base \cdot \tan^{-1}_* \frac{im}{re}\right)}}{\sqrt{\mathsf{fma}\left(\log base, \log base, 0.0 \cdot 0.0\right)} \cdot \sqrt{\mathsf{fma}\left(\log base, \log base, 0.0 \cdot 0.0\right)}}\]

6. Applied times-frac 0.4

   \[\leadsto \color{blue}{\frac{1}{\sqrt{\mathsf{fma}\left(\log base, \log base, 0.0 \cdot 0.0\right)}} \cdot \frac{\mathsf{fma}\left(-0.0, \log \left(\mathsf{hypot}\left(im, re\right)\right), \log base \cdot \tan^{-1}_* \frac{im}{re}\right)}{\sqrt{\mathsf{fma}\left(\log base, \log base, 0.0 \cdot 0.0\right)}}}\]

7. Simplified 0.4

   \[\leadsto \color{blue}{\frac{1}{\mathsf{hypot}\left(0.0, \log base\right)}} \cdot \frac{\mathsf{fma}\left(-0.0, \log \left(\mathsf{hypot}\left(im, re\right)\right), \log base \cdot \tan^{-1}_* \frac{im}{re}\right)}{\sqrt{\mathsf{fma}\left(\log base, \log base, 0.0 \cdot 0.0\right)}}\]

8. Simplified 0.4

   \[\leadsto \frac{1}{\mathsf{hypot}\left(0.0, \log base\right)} \cdot \color{blue}{\frac{\mathsf{fma}\left(\log base, \tan^{-1}_* \frac{im}{re}, \log \left(\mathsf{hypot}\left(im, re\right)\right) \cdot \left(-0.0\right)\right)}{\mathsf{hypot}\left(0.0, \log base\right)}}\]

9. Taylor expanded around 0 0.3

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

10. Final simplification 0.3

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

Reproduce

herbie shell --seed 2019194 +o rules:numerics
(FPCore (re im base)
  :name "math.log/2 on complex, imaginary part"
  (/ (- (* (atan2 im re) (log base)) (* (log (sqrt (+ (* re re) (* im im)))) 0.0)) (+ (* (log base) (log base)) (* 0.0 0.0))))