math.log/2 on complex, real part

Average Error: 32.0 → 0.5

Time: 12.6s

Precision: 64

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

Math

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

↓

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

C

double f(double re, double im, double base) {
        double r53221 = re;
        double r53222 = r53221 * r53221;
        double r53223 = im;
        double r53224 = r53223 * r53223;
        double r53225 = r53222 + r53224;
        double r53226 = sqrt(r53225);
        double r53227 = log(r53226);
        double r53228 = base;
        double r53229 = log(r53228);
        double r53230 = r53227 * r53229;
        double r53231 = atan2(r53223, r53221);
        double r53232 = 0.0;
        double r53233 = r53231 * r53232;
        double r53234 = r53230 + r53233;
        double r53235 = r53229 * r53229;
        double r53236 = r53232 * r53232;
        double r53237 = r53235 + r53236;
        double r53238 = r53234 / r53237;
        return r53238;
}

↓

double f(double re, double im, double base) {
        double r53239 = 1.0;
        double r53240 = base;
        double r53241 = log(r53240);
        double r53242 = 0.0;
        double r53243 = hypot(r53241, r53242);
        double r53244 = re;
        double r53245 = im;
        double r53246 = hypot(r53244, r53245);
        double r53247 = log(r53246);
        double r53248 = atan2(r53245, r53244);
        double r53249 = r53248 * r53242;
        double r53250 = fma(r53247, r53241, r53249);
        double r53251 = r53243 / r53250;
        double r53252 = r53239 / r53251;
        double r53253 = r53252 / r53243;
        return r53253;
}

Error

Bits error versus re

Bits error versus im

Bits error versus base

Derivation

1. Initial program 32.0

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

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 *-un-lft-identity 0.5

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

6. Applied times-frac 0.5

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

7. Simplified 0.5

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

8. Simplified 0.5

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

10. Applied clear-num 0.5

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

12. Applied associate-*l/ 0.5

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

13. Simplified 0.4

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

15. Applied clear-num 0.5

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

16. Final simplification 0.5

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

Reproduce

herbie shell --seed 2020047 +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))))