Average Error: 37.0 → 13.7
Time: 20.7s
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 r7825677 = 0.5;
        double r7825678 = 2.0;
        double r7825679 = re;
        double r7825680 = r7825679 * r7825679;
        double r7825681 = im;
        double r7825682 = r7825681 * r7825681;
        double r7825683 = r7825680 + r7825682;
        double r7825684 = sqrt(r7825683);
        double r7825685 = r7825684 + r7825679;
        double r7825686 = r7825678 * r7825685;
        double r7825687 = sqrt(r7825686);
        double r7825688 = r7825677 * r7825687;
        return r7825688;
}
↓
double f(double re, double im) {
        double r7825689 = re;
        double r7825690 = im;
        double r7825691 = hypot(r7825689, r7825690);
        double r7825692 = r7825689 + r7825691;
        double r7825693 = 2.0;
        double r7825694 = r7825692 * r7825693;
        double r7825695 = sqrt(r7825694);
        double r7825696 = 0.5;
        double r7825697 = r7825695 * r7825696;
        return r7825697;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.0
Target32.4
Herbie13.7
\[\begin{array}{l} \mathbf{if}\;re \lt 0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Initial program 37.0

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

  2. Simplified13.7

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

  3. Final simplification13.7

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

Reproduce

herbie shell --seed 2019138 +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)))))