Average Error: 38.5 → 13.0
Time: 23.9s
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 r30951 = 0.5;
        double r30952 = 2.0;
        double r30953 = re;
        double r30954 = r30953 * r30953;
        double r30955 = im;
        double r30956 = r30955 * r30955;
        double r30957 = r30954 + r30956;
        double r30958 = sqrt(r30957);
        double r30959 = r30958 - r30953;
        double r30960 = r30952 * r30959;
        double r30961 = sqrt(r30960);
        double r30962 = r30951 * r30961;
        return r30962;
}
↓
double f(double re, double im) {
        double r30963 = 0.5;
        double r30964 = re;
        double r30965 = im;
        double r30966 = hypot(r30964, r30965);
        double r30967 = r30966 - r30964;
        double r30968 = 2.0;
        double r30969 = r30967 * r30968;
        double r30970 = sqrt(r30969);
        double r30971 = r30963 * r30970;
        return r30971;
}

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 38.5

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

  2. Simplified13.0

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

  3. Final simplification13.0

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

Reproduce

herbie shell --seed 2019325 +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)))))