Average Error: 37.5 → 13.2
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 r533196 = 0.5;
        double r533197 = 2.0;
        double r533198 = re;
        double r533199 = r533198 * r533198;
        double r533200 = im;
        double r533201 = r533200 * r533200;
        double r533202 = r533199 + r533201;
        double r533203 = sqrt(r533202);
        double r533204 = r533203 - r533198;
        double r533205 = r533197 * r533204;
        double r533206 = sqrt(r533205);
        double r533207 = r533196 * r533206;
        return r533207;
}
↓
double f(double re, double im) {
        double r533208 = re;
        double r533209 = im;
        double r533210 = hypot(r533208, r533209);
        double r533211 = r533210 - r533208;
        double r533212 = 2.0;
        double r533213 = r533211 * r533212;
        double r533214 = sqrt(r533213);
        double r533215 = 0.5;
        double r533216 = r533214 * r533215;
        return r533216;
}

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. Simplified13.2

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

  3. Final simplification13.2

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

Reproduce

herbie shell --seed 2019144 +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)))))