Average Error: 38.4 → 13.2
Time: 20.5s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
↓
\[0.5 \cdot \sqrt{\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot 2}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}
↓
0.5 \cdot \sqrt{\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot 2}
double f(double re, double im) {
        double r26178 = 0.5;
        double r26179 = 2.0;
        double r26180 = re;
        double r26181 = r26180 * r26180;
        double r26182 = im;
        double r26183 = r26182 * r26182;
        double r26184 = r26181 + r26183;
        double r26185 = sqrt(r26184);
        double r26186 = r26185 - r26180;
        double r26187 = r26179 * r26186;
        double r26188 = sqrt(r26187);
        double r26189 = r26178 * r26188;
        return r26189;
}
↓
double f(double re, double im) {
        double r26190 = 0.5;
        double r26191 = re;
        double r26192 = im;
        double r26193 = hypot(r26191, r26192);
        double r26194 = r26193 - r26191;
        double r26195 = 2.0;
        double r26196 = r26194 * r26195;
        double r26197 = sqrt(r26196);
        double r26198 = r26190 * r26197;
        return r26198;
}

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 38.4

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

  2. Simplified13.2

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

  3. Final simplification13.2

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

Reproduce

herbie shell --seed 2019326 +o rules:numerics
(FPCore (re im)
  :name "math.sqrt on complex, imaginary part, im greater than 0 branch"
  :precision binary64
  (* 0.5 (sqrt (* 2 (- (sqrt (+ (* re re) (* im im))) re)))))