Average Error: 37.5 → 13.0
Time: 18.8s
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 r7591390 = 0.5;
        double r7591391 = 2.0;
        double r7591392 = re;
        double r7591393 = r7591392 * r7591392;
        double r7591394 = im;
        double r7591395 = r7591394 * r7591394;
        double r7591396 = r7591393 + r7591395;
        double r7591397 = sqrt(r7591396);
        double r7591398 = r7591397 + r7591392;
        double r7591399 = r7591391 * r7591398;
        double r7591400 = sqrt(r7591399);
        double r7591401 = r7591390 * r7591400;
        return r7591401;
}
↓
double f(double re, double im) {
        double r7591402 = re;
        double r7591403 = im;
        double r7591404 = hypot(r7591402, r7591403);
        double r7591405 = r7591402 + r7591404;
        double r7591406 = 2.0;
        double r7591407 = r7591405 * r7591406;
        double r7591408 = sqrt(r7591407);
        double r7591409 = 0.5;
        double r7591410 = r7591408 * r7591409;
        return r7591410;
}

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

Derivation

  1. Initial program 37.5

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

  2. Simplified13.0

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

  3. Final simplification13.0

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

Reproduce

herbie shell --seed 2019129 +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)))))