Average Error: 38.6 → 13.2
Time: 23.4s
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 r31003 = 0.5;
        double r31004 = 2.0;
        double r31005 = re;
        double r31006 = r31005 * r31005;
        double r31007 = im;
        double r31008 = r31007 * r31007;
        double r31009 = r31006 + r31008;
        double r31010 = sqrt(r31009);
        double r31011 = r31010 - r31005;
        double r31012 = r31004 * r31011;
        double r31013 = sqrt(r31012);
        double r31014 = r31003 * r31013;
        return r31014;
}
↓
double f(double re, double im) {
        double r31015 = 0.5;
        double r31016 = re;
        double r31017 = im;
        double r31018 = hypot(r31016, r31017);
        double r31019 = r31018 - r31016;
        double r31020 = 2.0;
        double r31021 = r31019 * r31020;
        double r31022 = sqrt(r31021);
        double r31023 = r31015 * r31022;
        return r31023;
}

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)))))