Average Error: 37.5 → 13.6
Time: 22.5s
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 r4527525 = 0.5;
        double r4527526 = 2.0;
        double r4527527 = re;
        double r4527528 = r4527527 * r4527527;
        double r4527529 = im;
        double r4527530 = r4527529 * r4527529;
        double r4527531 = r4527528 + r4527530;
        double r4527532 = sqrt(r4527531);
        double r4527533 = r4527532 + r4527527;
        double r4527534 = r4527526 * r4527533;
        double r4527535 = sqrt(r4527534);
        double r4527536 = r4527525 * r4527535;
        return r4527536;
}
↓
double f(double re, double im) {
        double r4527537 = re;
        double r4527538 = im;
        double r4527539 = hypot(r4527537, r4527538);
        double r4527540 = r4527537 + r4527539;
        double r4527541 = 2.0;
        double r4527542 = r4527540 * r4527541;
        double r4527543 = sqrt(r4527542);
        double r4527544 = 0.5;
        double r4527545 = r4527543 * r4527544;
        return r4527545;
}

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
Target32.7
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 2019162 +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)))))