Average Error: 37.4 → 12.9
Time: 17.2s
Precision: 64
\[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
↓
\[\sqrt{\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot 2.0} \cdot 0.5\]
0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}
↓
\sqrt{\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot 2.0} \cdot 0.5
double f(double re, double im) {
        double r653084 = 0.5;
        double r653085 = 2.0;
        double r653086 = re;
        double r653087 = r653086 * r653086;
        double r653088 = im;
        double r653089 = r653088 * r653088;
        double r653090 = r653087 + r653089;
        double r653091 = sqrt(r653090);
        double r653092 = r653091 - r653086;
        double r653093 = r653085 * r653092;
        double r653094 = sqrt(r653093);
        double r653095 = r653084 * r653094;
        return r653095;
}
↓
double f(double re, double im) {
        double r653096 = re;
        double r653097 = im;
        double r653098 = hypot(r653096, r653097);
        double r653099 = r653098 - r653096;
        double r653100 = 2.0;
        double r653101 = r653099 * r653100;
        double r653102 = sqrt(r653101);
        double r653103 = 0.5;
        double r653104 = r653102 * r653103;
        return r653104;
}

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 37.4

    \[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]

  2. Simplified12.9

    \[\leadsto \color{blue}{0.5 \cdot \sqrt{\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot 2.0}}\]

  3. Final simplification12.9

    \[\leadsto \sqrt{\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot 2.0} \cdot 0.5\]

Reproduce

herbie shell --seed 2019146 +o rules:numerics
(FPCore (re im)
  :name "math.sqrt on complex, imaginary part, im greater than 0 branch"
  (* 0.5 (sqrt (* 2.0 (- (sqrt (+ (* re re) (* im im))) re)))))