Average Error: 30.0 → 0.0
Time: 8.9s
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 r1752056 = re;
        double r1752057 = r1752056 * r1752056;
        double r1752058 = im;
        double r1752059 = r1752058 * r1752058;
        double r1752060 = r1752057 + r1752059;
        double r1752061 = sqrt(r1752060);
        double r1752062 = log(r1752061);
        return r1752062;
}
↓
double f(double re, double im) {
        double r1752063 = re;
        double r1752064 = im;
        double r1752065 = hypot(r1752063, r1752064);
        double r1752066 = sqrt(r1752065);
        double r1752067 = log(r1752066);
        double r1752068 = r1752067 + r1752067;
        return r1752068;
}

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)))))