Average Error: 38.0 → 13.4
Time: 20.5s
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 r624187 = 0.5;
        double r624188 = 2.0;
        double r624189 = re;
        double r624190 = r624189 * r624189;
        double r624191 = im;
        double r624192 = r624191 * r624191;
        double r624193 = r624190 + r624192;
        double r624194 = sqrt(r624193);
        double r624195 = r624194 - r624189;
        double r624196 = r624188 * r624195;
        double r624197 = sqrt(r624196);
        double r624198 = r624187 * r624197;
        return r624198;
}
↓
double f(double re, double im) {
        double r624199 = re;
        double r624200 = im;
        double r624201 = hypot(r624199, r624200);
        double r624202 = r624201 - r624199;
        double r624203 = 2.0;
        double r624204 = r624202 * r624203;
        double r624205 = sqrt(r624204);
        double r624206 = 0.5;
        double r624207 = r624205 * r624206;
        return r624207;
}

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 38.0

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

  2. Simplified13.4

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

  3. Final simplification13.4

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

Reproduce

herbie shell --seed 2019133 +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)))))