math.log10 on complex, real part

Average Error: 32.3 → 0.3

Time: 4.9s

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 r83404 = re;
        double r83405 = r83404 * r83404;
        double r83406 = im;
        double r83407 = r83406 * r83406;
        double r83408 = r83405 + r83407;
        double r83409 = sqrt(r83408);
        double r83410 = log(r83409);
        double r83411 = 10.0;
        double r83412 = log(r83411);
        double r83413 = r83410 / r83412;
        return r83413;
}

↓

double f(double re, double im) {
        double r83414 = 1.0;
        double r83415 = 10.0;
        double r83416 = log(r83415);
        double r83417 = sqrt(r83416);
        double r83418 = r83414 / r83417;
        double r83419 = re;
        double r83420 = im;
        double r83421 = hypot(r83419, r83420);
        double r83422 = pow(r83421, r83418);
        double r83423 = log(r83422);
        double r83424 = r83418 * r83423;
        return r83424;
}

Error

Bits error versus re

Bits error versus im

Derivation

1. Initial program 32.3

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

3. Applied *-un-lft-identity 32.3

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

4. Applied sqrt-prod 32.3

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

5. Simplified 32.3

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

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 0.5

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

15. Applied add-log-exp 0.5

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

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 2020025 +o rules:numerics
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10)))