Average Error: 37.5 → 13.6
Time: 20.7s
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 r622203 = 0.5;
        double r622204 = 2.0;
        double r622205 = re;
        double r622206 = r622205 * r622205;
        double r622207 = im;
        double r622208 = r622207 * r622207;
        double r622209 = r622206 + r622208;
        double r622210 = sqrt(r622209);
        double r622211 = r622210 - r622205;
        double r622212 = r622204 * r622211;
        double r622213 = sqrt(r622212);
        double r622214 = r622203 * r622213;
        return r622214;
}
↓
double f(double re, double im) {
        double r622215 = re;
        double r622216 = im;
        double r622217 = hypot(r622215, r622216);
        double r622218 = r622217 - r622215;
        double r622219 = 2.0;
        double r622220 = r622218 * r622219;
        double r622221 = sqrt(r622220);
        double r622222 = 0.5;
        double r622223 = r622221 * r622222;
        return r622223;
}

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 37.5

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

  2. Simplified13.6

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

  3. Final simplification13.6

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

Reproduce

herbie shell --seed 2019151 +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)))))