Average Error: 37.5 → 13.6
Time: 18.3s
Precision: 64
\[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
↓
\[\sqrt{\left(re + \mathsf{hypot}\left(re, im\right)\right) \cdot 2.0} \cdot 0.5\]
0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}
↓
\sqrt{\left(re + \mathsf{hypot}\left(re, im\right)\right) \cdot 2.0} \cdot 0.5
double f(double re, double im) {
        double r6504801 = 0.5;
        double r6504802 = 2.0;
        double r6504803 = re;
        double r6504804 = r6504803 * r6504803;
        double r6504805 = im;
        double r6504806 = r6504805 * r6504805;
        double r6504807 = r6504804 + r6504806;
        double r6504808 = sqrt(r6504807);
        double r6504809 = r6504808 + r6504803;
        double r6504810 = r6504802 * r6504809;
        double r6504811 = sqrt(r6504810);
        double r6504812 = r6504801 * r6504811;
        return r6504812;
}
↓
double f(double re, double im) {
        double r6504813 = re;
        double r6504814 = im;
        double r6504815 = hypot(r6504813, r6504814);
        double r6504816 = r6504813 + r6504815;
        double r6504817 = 2.0;
        double r6504818 = r6504816 * r6504817;
        double r6504819 = sqrt(r6504818);
        double r6504820 = 0.5;
        double r6504821 = r6504819 * r6504820;
        return r6504821;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.5
Target32.4
Herbie13.6
\[\begin{array}{l} \mathbf{if}\;re \lt 0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Initial program 37.5

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

  2. Simplified13.6

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

  3. Final simplification13.6

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

Reproduce

herbie shell --seed 2019152 +o rules:numerics
(FPCore (re im)
  :name "math.sqrt on complex, real part"

  :herbie-target
  (if (< re 0) (* 0.5 (* (sqrt 2) (sqrt (/ (* im im) (- (sqrt (+ (* re re) (* im im))) re))))) (* 0.5 (sqrt (* 2.0 (+ (sqrt (+ (* re re) (* im im))) re)))))

  (* 0.5 (sqrt (* 2.0 (+ (sqrt (+ (* re re) (* im im))) re)))))