Average Error: 38.6 → 13.5
Time: 57.9s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
↓
\[0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\mathsf{hypot}\left(re, im\right) + re}\right)\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}
↓
0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\mathsf{hypot}\left(re, im\right) + re}\right)
double f(double re, double im) {
        double r6576684 = 0.5;
        double r6576685 = 2.0;
        double r6576686 = re;
        double r6576687 = r6576686 * r6576686;
        double r6576688 = im;
        double r6576689 = r6576688 * r6576688;
        double r6576690 = r6576687 + r6576689;
        double r6576691 = sqrt(r6576690);
        double r6576692 = r6576691 + r6576686;
        double r6576693 = r6576685 * r6576692;
        double r6576694 = sqrt(r6576693);
        double r6576695 = r6576684 * r6576694;
        return r6576695;
}
↓
double f(double re, double im) {
        double r6576696 = 0.5;
        double r6576697 = 2.0;
        double r6576698 = sqrt(r6576697);
        double r6576699 = re;
        double r6576700 = im;
        double r6576701 = hypot(r6576699, r6576700);
        double r6576702 = r6576701 + r6576699;
        double r6576703 = sqrt(r6576702);
        double r6576704 = r6576698 * r6576703;
        double r6576705 = r6576696 * r6576704;
        return r6576705;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original38.6
Target33.5
Herbie13.5
\[\begin{array}{l} \mathbf{if}\;re \lt 0.0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

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

    \[\leadsto \color{blue}{0.5 \cdot \sqrt{\left(re + \mathsf{hypot}\left(re, im\right)\right) \cdot 2}}\]
  3. Using strategy rm

  4. Applied sqrt-prod13.5

    \[\leadsto 0.5 \cdot \color{blue}{\left(\sqrt{re + \mathsf{hypot}\left(re, im\right)} \cdot \sqrt{2}\right)}\]

  5. Final simplification13.5

    \[\leadsto 0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\mathsf{hypot}\left(re, im\right) + re}\right)\]

Reproduce

herbie shell --seed 2019200 +o rules:numerics
(FPCore (re im)
  :name "math.sqrt on complex, real part"

  :herbie-target
  (if (< re 0.0) (* 0.5 (* (sqrt 2.0) (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)))))