math.log10 on complex, real part

Average Error: 32.4 → 0.3

Time: 5.1s

Precision: 64

Math

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

↓

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

C

double f(double re, double im) {
        double r90037 = re;
        double r90038 = r90037 * r90037;
        double r90039 = im;
        double r90040 = r90039 * r90039;
        double r90041 = r90038 + r90040;
        double r90042 = sqrt(r90041);
        double r90043 = log(r90042);
        double r90044 = 10.0;
        double r90045 = log(r90044);
        double r90046 = r90043 / r90045;
        return r90046;
}

↓

double f(double re, double im) {
        double r90047 = 1.0;
        double r90048 = 10.0;
        double r90049 = log(r90048);
        double r90050 = sqrt(r90049);
        double r90051 = r90047 / r90050;
        double r90052 = re;
        double r90053 = im;
        double r90054 = hypot(r90052, r90053);
        double r90055 = pow(r90054, r90051);
        double r90056 = log(r90055);
        double r90057 = r90051 * r90056;
        return r90057;
}

Error

Bits error versus re

Bits error versus im

Derivation

1. Initial program 32.4

   \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
2. Using strategy rm

3. Applied *-un-lft-identity 32.4

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

4. Applied sqrt-prod 32.4

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

5. Simplified 32.4

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

6. Simplified 0.6

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

8. Applied add-sqr-sqrt 0.6

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

9. Applied pow1 0.6

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

10. Applied pow1 0.6

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

11. Applied pow-prod-down 0.6

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

12. Applied log-pow 0.6

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

13. Applied times-frac 0.6

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

15. Applied add-log-exp 0.6

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

16. Simplified 0.3

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

17. Final simplification 0.3

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

Reproduce

herbie shell --seed 2020049 +o rules:numerics
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10)))