Average Error: 38.9 → 13.4
Time: 19.4s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
↓
\[\sqrt{\left(re + \mathsf{hypot}\left(re, im\right)\right) \cdot 2} \cdot 0.5\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}
↓
\sqrt{\left(re + \mathsf{hypot}\left(re, im\right)\right) \cdot 2} \cdot 0.5
double f(double re, double im) {
        double r8357602 = 0.5;
        double r8357603 = 2.0;
        double r8357604 = re;
        double r8357605 = r8357604 * r8357604;
        double r8357606 = im;
        double r8357607 = r8357606 * r8357606;
        double r8357608 = r8357605 + r8357607;
        double r8357609 = sqrt(r8357608);
        double r8357610 = r8357609 + r8357604;
        double r8357611 = r8357603 * r8357610;
        double r8357612 = sqrt(r8357611);
        double r8357613 = r8357602 * r8357612;
        return r8357613;
}
↓
double f(double re, double im) {
        double r8357614 = re;
        double r8357615 = im;
        double r8357616 = hypot(r8357614, r8357615);
        double r8357617 = r8357614 + r8357616;
        double r8357618 = 2.0;
        double r8357619 = r8357617 * r8357618;
        double r8357620 = sqrt(r8357619);
        double r8357621 = 0.5;
        double r8357622 = r8357620 * r8357621;
        return r8357622;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original38.9
Target33.8
Herbie13.4
\[\begin{array}{l} \mathbf{if}\;re \lt 0.0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Initial program 38.9

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

  2. Simplified13.4

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

  3. Final simplification13.4

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

Reproduce

herbie shell --seed 2019169 +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)))))