Average Error: 37.5 → 13.2
Time: 25.2s
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 r611727 = 0.5;
        double r611728 = 2.0;
        double r611729 = re;
        double r611730 = r611729 * r611729;
        double r611731 = im;
        double r611732 = r611731 * r611731;
        double r611733 = r611730 + r611732;
        double r611734 = sqrt(r611733);
        double r611735 = r611734 - r611729;
        double r611736 = r611728 * r611735;
        double r611737 = sqrt(r611736);
        double r611738 = r611727 * r611737;
        return r611738;
}
↓
double f(double re, double im) {
        double r611739 = re;
        double r611740 = im;
        double r611741 = hypot(r611739, r611740);
        double r611742 = r611741 - r611739;
        double r611743 = 2.0;
        double r611744 = r611742 * r611743;
        double r611745 = sqrt(r611744);
        double r611746 = 0.5;
        double r611747 = r611745 * r611746;
        return r611747;
}

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 2019121 +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)))))