Average Error: 30.0 → 0.0
Time: 9.0s
Precision: 64
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\log \left(\sqrt{\mathsf{hypot}\left(re, im\right)}\right) + \log \left(\sqrt{\mathsf{hypot}\left(re, im\right)}\right)\]
\log \left(\sqrt{re \cdot re + im \cdot im}\right)
↓
\log \left(\sqrt{\mathsf{hypot}\left(re, im\right)}\right) + \log \left(\sqrt{\mathsf{hypot}\left(re, im\right)}\right)
double f(double re, double im) {
        double r1611818 = re;
        double r1611819 = r1611818 * r1611818;
        double r1611820 = im;
        double r1611821 = r1611820 * r1611820;
        double r1611822 = r1611819 + r1611821;
        double r1611823 = sqrt(r1611822);
        double r1611824 = log(r1611823);
        return r1611824;
}
↓
double f(double re, double im) {
        double r1611825 = re;
        double r1611826 = im;
        double r1611827 = hypot(r1611825, r1611826);
        double r1611828 = sqrt(r1611827);
        double r1611829 = log(r1611828);
        double r1611830 = r1611829 + r1611829;
        return r1611830;
}

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 30.0

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

  2. Simplified0.0

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

  4. Applied add-sqr-sqrt0.0

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

  5. Applied log-prod0.0

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

  6. Final simplification0.0

    \[\leadsto \log \left(\sqrt{\mathsf{hypot}\left(re, im\right)}\right) + \log \left(\sqrt{\mathsf{hypot}\left(re, im\right)}\right)\]

Reproduce

herbie shell --seed 2019158 +o rules:numerics
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  (log (sqrt (+ (* re re) (* im im)))))