Average Error: 38.6 → 13.2
Time: 22.3s
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 r31030 = 0.5;
        double r31031 = 2.0;
        double r31032 = re;
        double r31033 = r31032 * r31032;
        double r31034 = im;
        double r31035 = r31034 * r31034;
        double r31036 = r31033 + r31035;
        double r31037 = sqrt(r31036);
        double r31038 = r31037 - r31032;
        double r31039 = r31031 * r31038;
        double r31040 = sqrt(r31039);
        double r31041 = r31030 * r31040;
        return r31041;
}
↓
double f(double re, double im) {
        double r31042 = 0.5;
        double r31043 = re;
        double r31044 = im;
        double r31045 = hypot(r31043, r31044);
        double r31046 = r31045 - r31043;
        double r31047 = 2.0;
        double r31048 = r31046 * r31047;
        double r31049 = sqrt(r31048);
        double r31050 = r31042 * r31049;
        return r31050;
}

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 38.6

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

  2. Simplified13.2

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

  3. Final simplification13.2

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

Reproduce

herbie shell --seed 2019323 +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)))))