math.log10 on complex, real part

Average Error: 32.8 → 0.3

Time: 5.0s

Precision: 64

Math

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

↓

\[\frac{1}{\sqrt{\log 10}} \cdot \log \left({\left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{\frac{1}{2}} \cdot {\left(\mathsf{hypot}\left(re, im\right)\right)}^{\frac{1}{2}}\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)\]

C

double f(double re, double im) {
        double r45148 = re;
        double r45149 = r45148 * r45148;
        double r45150 = im;
        double r45151 = r45150 * r45150;
        double r45152 = r45149 + r45151;
        double r45153 = sqrt(r45152);
        double r45154 = log(r45153);
        double r45155 = 10.0;
        double r45156 = log(r45155);
        double r45157 = r45154 / r45156;
        return r45157;
}

↓

double f(double re, double im) {
        double r45158 = 1.0;
        double r45159 = 10.0;
        double r45160 = log(r45159);
        double r45161 = sqrt(r45160);
        double r45162 = r45158 / r45161;
        double r45163 = re;
        double r45164 = im;
        double r45165 = hypot(r45163, r45164);
        double r45166 = 0.5;
        double r45167 = pow(r45165, r45166);
        double r45168 = r45167 * r45167;
        double r45169 = pow(r45168, r45162);
        double r45170 = log(r45169);
        double r45171 = r45162 * r45170;
        return r45171;
}

Error

Bits error versus re

Bits error versus im

Derivation

1. Initial program 32.8

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

3. Applied *-un-lft-identity 32.8

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

4. Applied sqrt-prod 32.8

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

5. Simplified 32.8

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

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. Using strategy rm

18. Applied add-sqr-sqrt 0.3

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

19. Simplified 0.3

    \[\leadsto \frac{1}{\sqrt{\log 10}} \cdot \log \left({\left(\color{blue}{{\left(\mathsf{hypot}\left(re, im\right)\right)}^{\frac{1}{2}}} \cdot \sqrt{\mathsf{hypot}\left(re, im\right)}\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)\]

20. Simplified 0.3

    \[\leadsto \frac{1}{\sqrt{\log 10}} \cdot \log \left({\left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{\frac{1}{2}} \cdot \color{blue}{{\left(\mathsf{hypot}\left(re, im\right)\right)}^{\frac{1}{2}}}\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)\]

21. Final simplification 0.3

    \[\leadsto \frac{1}{\sqrt{\log 10}} \cdot \log \left({\left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{\frac{1}{2}} \cdot {\left(\mathsf{hypot}\left(re, im\right)\right)}^{\frac{1}{2}}\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)\]

Reproduce

herbie shell --seed 2020046 +o rules:numerics
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10)))