Average Error: 0.8 → 0.8
Time: 2.8s
Precision: 64
\[\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}\]
↓
\[\frac{1}{\sqrt{\log 10}} \cdot \left(\tan^{-1}_* \frac{im}{re} \cdot \sqrt{\frac{1}{\log 10}}\right)\]
\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}
↓
\frac{1}{\sqrt{\log 10}} \cdot \left(\tan^{-1}_* \frac{im}{re} \cdot \sqrt{\frac{1}{\log 10}}\right)
double f(double re, double im) {
        double r76483 = im;
        double r76484 = re;
        double r76485 = atan2(r76483, r76484);
        double r76486 = 10.0;
        double r76487 = log(r76486);
        double r76488 = r76485 / r76487;
        return r76488;
}
↓
double f(double re, double im) {
        double r76489 = 1.0;
        double r76490 = 10.0;
        double r76491 = log(r76490);
        double r76492 = sqrt(r76491);
        double r76493 = r76489 / r76492;
        double r76494 = im;
        double r76495 = re;
        double r76496 = atan2(r76494, r76495);
        double r76497 = r76489 / r76491;
        double r76498 = sqrt(r76497);
        double r76499 = r76496 * r76498;
        double r76500 = r76493 * r76499;
        return r76500;
}

Error

Bits error versus re

Bits error versus im

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Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.8

    \[\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}\]
  2. Using strategy rm

  3. Applied add-sqr-sqrt0.8

    \[\leadsto \frac{\tan^{-1}_* \frac{im}{re}}{\color{blue}{\sqrt{\log 10} \cdot \sqrt{\log 10}}}\]

  4. Applied *-un-lft-identity0.8

    \[\leadsto \frac{\color{blue}{1 \cdot \tan^{-1}_* \frac{im}{re}}}{\sqrt{\log 10} \cdot \sqrt{\log 10}}\]

  5. Applied times-frac0.8

    \[\leadsto \color{blue}{\frac{1}{\sqrt{\log 10}} \cdot \frac{\tan^{-1}_* \frac{im}{re}}{\sqrt{\log 10}}}\]

  6. Taylor expanded around 0 0.8

    \[\leadsto \frac{1}{\sqrt{\log 10}} \cdot \color{blue}{\left(\tan^{-1}_* \frac{im}{re} \cdot \sqrt{\frac{1}{\log 10}}\right)}\]

  7. Final simplification0.8

    \[\leadsto \frac{1}{\sqrt{\log 10}} \cdot \left(\tan^{-1}_* \frac{im}{re} \cdot \sqrt{\frac{1}{\log 10}}\right)\]

Reproduce

herbie shell --seed 2020034 
(FPCore (re im)
  :name "math.log10 on complex, imaginary part"
  :precision binary64
  (/ (atan2 im re) (log 10)))