Average Error: 38.8 → 13.5
Time: 31.2s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
↓
\[\sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)} \cdot 0.5\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}
↓
\sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)} \cdot 0.5
double f(double re, double im) {
        double r829120 = 0.5;
        double r829121 = 2.0;
        double r829122 = re;
        double r829123 = r829122 * r829122;
        double r829124 = im;
        double r829125 = r829124 * r829124;
        double r829126 = r829123 + r829125;
        double r829127 = sqrt(r829126);
        double r829128 = r829127 - r829122;
        double r829129 = r829121 * r829128;
        double r829130 = sqrt(r829129);
        double r829131 = r829120 * r829130;
        return r829131;
}
↓
double f(double re, double im) {
        double r829132 = 2.0;
        double r829133 = re;
        double r829134 = im;
        double r829135 = hypot(r829133, r829134);
        double r829136 = r829135 - r829133;
        double r829137 = r829132 * r829136;
        double r829138 = sqrt(r829137);
        double r829139 = 0.5;
        double r829140 = r829138 * r829139;
        return r829140;
}

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 38.8

    \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]

  2. Simplified13.5

    \[\leadsto \color{blue}{0.5 \cdot \sqrt{\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot 2}}\]

  3. Final simplification13.5

    \[\leadsto \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) - re\right)} \cdot 0.5\]

Reproduce

herbie shell --seed 2019200 +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)))))