Average Error: 37.2 → 13.3
Time: 19.9s
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 r5098568 = 0.5;
        double r5098569 = 2.0;
        double r5098570 = re;
        double r5098571 = r5098570 * r5098570;
        double r5098572 = im;
        double r5098573 = r5098572 * r5098572;
        double r5098574 = r5098571 + r5098573;
        double r5098575 = sqrt(r5098574);
        double r5098576 = r5098575 + r5098570;
        double r5098577 = r5098569 * r5098576;
        double r5098578 = sqrt(r5098577);
        double r5098579 = r5098568 * r5098578;
        return r5098579;
}
↓
double f(double re, double im) {
        double r5098580 = re;
        double r5098581 = im;
        double r5098582 = hypot(r5098580, r5098581);
        double r5098583 = r5098580 + r5098582;
        double r5098584 = 2.0;
        double r5098585 = r5098583 * r5098584;
        double r5098586 = sqrt(r5098585);
        double r5098587 = 0.5;
        double r5098588 = r5098586 * r5098587;
        return r5098588;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.2
Target32.2
Herbie13.3
\[\begin{array}{l} \mathbf{if}\;re \lt 0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Initial program 37.2

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

  2. Simplified13.3

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

  3. Final simplification13.3

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

Reproduce

herbie shell --seed 2019158 +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)))))