math.log/2 on complex, real part

Average Error: 31.8 → 0.6

Time: 17.3s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

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

C

double f(double re, double im, double base) {
        double r57141 = re;
        double r57142 = r57141 * r57141;
        double r57143 = im;
        double r57144 = r57143 * r57143;
        double r57145 = r57142 + r57144;
        double r57146 = sqrt(r57145);
        double r57147 = log(r57146);
        double r57148 = base;
        double r57149 = log(r57148);
        double r57150 = r57147 * r57149;
        double r57151 = atan2(r57143, r57141);
        double r57152 = 0.0;
        double r57153 = r57151 * r57152;
        double r57154 = r57150 + r57153;
        double r57155 = r57149 * r57149;
        double r57156 = r57152 * r57152;
        double r57157 = r57155 + r57156;
        double r57158 = r57154 / r57157;
        return r57158;
}

↓

double f(double re, double im, double base) {
        double r57159 = re;
        double r57160 = im;
        double r57161 = hypot(r57159, r57160);
        double r57162 = log(r57161);
        double r57163 = base;
        double r57164 = log(r57163);
        double r57165 = atan2(r57160, r57159);
        double r57166 = 0.0;
        double r57167 = r57165 * r57166;
        double r57168 = fma(r57162, r57164, r57167);
        double r57169 = hypot(r57164, r57166);
        double r57170 = r57168 / r57169;
        double r57171 = sqrt(r57169);
        double r57172 = r57170 / r57171;
        double r57173 = r57166 * r57166;
        double r57174 = fma(r57164, r57164, r57173);
        double r57175 = sqrt(r57174);
        double r57176 = sqrt(r57175);
        double r57177 = r57172 / r57176;
        return r57177;
}

Error

Bits error versus re

Bits error versus im

Bits error versus base

Derivation

1. Initial program 31.8

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

2. Simplified 0.5

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

4. Applied add-sqr-sqrt 0.5

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

5. Applied associate-/r* 0.4

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

6. Simplified 0.4

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

8. Applied *-un-lft-identity 0.4

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

9. Applied *-un-lft-identity 0.4

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

10. Applied times-frac 0.4

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

11. Simplified 0.4

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

13. Applied add-sqr-sqrt 0.4

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

14. Applied sqrt-prod 0.7

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

15. Applied associate-/r* 0.6

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

16. Simplified 0.6

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

17. Final simplification 0.6

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

Reproduce

herbie shell --seed 2020042 +o rules:numerics
(FPCore (re im base)
  :name "math.log/2 on complex, real part"
  :precision binary64
  (/ (+ (* (log (sqrt (+ (* re re) (* im im)))) (log base)) (* (atan2 im re) 0.0)) (+ (* (log base) (log base)) (* 0.0 0.0))))