math.log10 on complex, real part

Average Error: 32.7 → 0.6

Time: 11.0s

Precision: 64

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

Math

\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]

↓

\[\frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\log 10}\]

C

double f(double re, double im) {
        double r92299 = re;
        double r92300 = r92299 * r92299;
        double r92301 = im;
        double r92302 = r92301 * r92301;
        double r92303 = r92300 + r92302;
        double r92304 = sqrt(r92303);
        double r92305 = log(r92304);
        double r92306 = 10.0;
        double r92307 = log(r92306);
        double r92308 = r92305 / r92307;
        return r92308;
}

↓

double f(double re, double im) {
        double r92309 = re;
        double r92310 = im;
        double r92311 = hypot(r92309, r92310);
        double r92312 = log(r92311);
        double r92313 = 10.0;
        double r92314 = log(r92313);
        double r92315 = r92312 / r92314;
        return r92315;
}

Error

Bits error versus re

Bits error versus im

Derivation

1. Initial program 32.7

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]

2. Simplified 0.6

   \[\leadsto \color{blue}{\frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\log 10}}\]
3. Using strategy rm

4. Applied add-sqr-sqrt 0.6

   \[\leadsto \frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\color{blue}{\sqrt{\log 10} \cdot \sqrt{\log 10}}}\]

5. Applied pow1 0.6

   \[\leadsto \frac{\log \color{blue}{\left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{1}\right)}}{\sqrt{\log 10} \cdot \sqrt{\log 10}}\]

6. Applied log-pow 0.6

   \[\leadsto \frac{\color{blue}{1 \cdot \log \left(\mathsf{hypot}\left(re, im\right)\right)}}{\sqrt{\log 10} \cdot \sqrt{\log 10}}\]

7. Applied times-frac 0.5

   \[\leadsto \color{blue}{\frac{1}{\sqrt{\log 10}} \cdot \frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\sqrt{\log 10}}}\]
8. Using strategy rm

9. Applied *-un-lft-identity 0.5

   \[\leadsto \frac{1}{\color{blue}{1 \cdot \sqrt{\log 10}}} \cdot \frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\sqrt{\log 10}}\]

10. Applied *-un-lft-identity 0.5

    \[\leadsto \frac{\color{blue}{1 \cdot 1}}{1 \cdot \sqrt{\log 10}} \cdot \frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\sqrt{\log 10}}\]

11. Applied times-frac 0.5

    \[\leadsto \color{blue}{\left(\frac{1}{1} \cdot \frac{1}{\sqrt{\log 10}}\right)} \cdot \frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\sqrt{\log 10}}\]

12. Applied associate-*l* 0.5

    \[\leadsto \color{blue}{\frac{1}{1} \cdot \left(\frac{1}{\sqrt{\log 10}} \cdot \frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\sqrt{\log 10}}\right)}\]

13. Simplified 0.6

    \[\leadsto \frac{1}{1} \cdot \color{blue}{\frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\log 10}}\]

14. Final simplification 0.6

    \[\leadsto \frac{\log \left(\mathsf{hypot}\left(re, im\right)\right)}{\log 10}\]

Reproduce

herbie shell --seed 2020047 +o rules:numerics
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10)))