Average Error: 32.3 → 0.4
Time: 4.9s
Precision: 64
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
↓
\[\frac{1}{\sqrt{\log 10}} \cdot \left(2 \cdot \log \left(\sqrt[3]{{\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}}\right) + \log \left(\sqrt[3]{{\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}}\right)\right)\]
\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}
↓
\frac{1}{\sqrt{\log 10}} \cdot \left(2 \cdot \log \left(\sqrt[3]{{\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}}\right) + \log \left(\sqrt[3]{{\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}}\right)\right)
double f(double re, double im) {
        double r86104 = re;
        double r86105 = r86104 * r86104;
        double r86106 = im;
        double r86107 = r86106 * r86106;
        double r86108 = r86105 + r86107;
        double r86109 = sqrt(r86108);
        double r86110 = log(r86109);
        double r86111 = 10.0;
        double r86112 = log(r86111);
        double r86113 = r86110 / r86112;
        return r86113;
}
↓
double f(double re, double im) {
        double r86114 = 1.0;
        double r86115 = 10.0;
        double r86116 = log(r86115);
        double r86117 = sqrt(r86116);
        double r86118 = r86114 / r86117;
        double r86119 = 2.0;
        double r86120 = re;
        double r86121 = im;
        double r86122 = hypot(r86120, r86121);
        double r86123 = pow(r86122, r86118);
        double r86124 = cbrt(r86123);
        double r86125 = log(r86124);
        double r86126 = r86119 * r86125;
        double r86127 = r86126 + r86125;
        double r86128 = r86118 * r86127;
        return r86128;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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 hypot-def0.6

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

  5. Applied add-sqr-sqrt0.6

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

  6. Applied pow10.6

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

  7. Applied log-pow0.6

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

  8. Applied times-frac0.6

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

  10. Applied add-log-exp0.6

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

  11. Simplified0.3

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

  13. Applied add-cube-cbrt0.3

    \[\leadsto \frac{1}{\sqrt{\log 10}} \cdot \log \color{blue}{\left(\left(\sqrt[3]{{\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}} \cdot \sqrt[3]{{\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}}\right) \cdot \sqrt[3]{{\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}}\right)}\]

  14. Applied log-prod0.4

    \[\leadsto \frac{1}{\sqrt{\log 10}} \cdot \color{blue}{\left(\log \left(\sqrt[3]{{\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}} \cdot \sqrt[3]{{\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}}\right) + \log \left(\sqrt[3]{{\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}}\right)\right)}\]

  15. Simplified0.4

    \[\leadsto \frac{1}{\sqrt{\log 10}} \cdot \left(\color{blue}{2 \cdot \log \left(\sqrt[3]{{\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}}\right)} + \log \left(\sqrt[3]{{\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}}\right)\right)\]

  16. Final simplification0.4

    \[\leadsto \frac{1}{\sqrt{\log 10}} \cdot \left(2 \cdot \log \left(\sqrt[3]{{\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}}\right) + \log \left(\sqrt[3]{{\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}}\right)\right)\]

Reproduce

herbie shell --seed 2020036 +o rules:numerics
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10)))