Average Error: 37.0 → 13.0
Time: 30.0s
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 r908195 = 0.5;
        double r908196 = 2.0;
        double r908197 = re;
        double r908198 = r908197 * r908197;
        double r908199 = im;
        double r908200 = r908199 * r908199;
        double r908201 = r908198 + r908200;
        double r908202 = sqrt(r908201);
        double r908203 = r908202 - r908197;
        double r908204 = r908196 * r908203;
        double r908205 = sqrt(r908204);
        double r908206 = r908195 * r908205;
        return r908206;
}
↓
double f(double re, double im) {
        double r908207 = re;
        double r908208 = im;
        double r908209 = hypot(r908207, r908208);
        double r908210 = r908209 - r908207;
        double r908211 = 2.0;
        double r908212 = r908210 * r908211;
        double r908213 = sqrt(r908212);
        double r908214 = 0.5;
        double r908215 = r908213 * r908214;
        return r908215;
}

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 37.0

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

  2. Simplified13.0

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

  3. Final simplification13.0

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

Reproduce

herbie shell --seed 2019125 +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)))))