Average Error: 37.7 → 13.6
Time: 13.6s
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 r293745 = 0.5;
        double r293746 = 2.0;
        double r293747 = re;
        double r293748 = r293747 * r293747;
        double r293749 = im;
        double r293750 = r293749 * r293749;
        double r293751 = r293748 + r293750;
        double r293752 = sqrt(r293751);
        double r293753 = r293752 - r293747;
        double r293754 = r293746 * r293753;
        double r293755 = sqrt(r293754);
        double r293756 = r293745 * r293755;
        return r293756;
}
↓
double f(double re, double im) {
        double r293757 = re;
        double r293758 = im;
        double r293759 = hypot(r293757, r293758);
        double r293760 = r293759 - r293757;
        double r293761 = 2.0;
        double r293762 = r293760 * r293761;
        double r293763 = sqrt(r293762);
        double r293764 = 0.5;
        double r293765 = r293763 * r293764;
        return r293765;
}

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 37.7

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

  2. Simplified13.6

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

  3. Final simplification13.6

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

Reproduce

herbie shell --seed 2019128 +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)))))