Average Error: 38.3 → 12.9
Time: 33.7s
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 r4433405 = 0.5;
        double r4433406 = 2.0;
        double r4433407 = re;
        double r4433408 = r4433407 * r4433407;
        double r4433409 = im;
        double r4433410 = r4433409 * r4433409;
        double r4433411 = r4433408 + r4433410;
        double r4433412 = sqrt(r4433411);
        double r4433413 = r4433412 + r4433407;
        double r4433414 = r4433406 * r4433413;
        double r4433415 = sqrt(r4433414);
        double r4433416 = r4433405 * r4433415;
        return r4433416;
}
↓
double f(double re, double im) {
        double r4433417 = re;
        double r4433418 = im;
        double r4433419 = hypot(r4433417, r4433418);
        double r4433420 = r4433417 + r4433419;
        double r4433421 = 2.0;
        double r4433422 = r4433420 * r4433421;
        double r4433423 = sqrt(r4433422);
        double r4433424 = 0.5;
        double r4433425 = r4433423 * r4433424;
        return r4433425;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original38.3
Target33.0
Herbie12.9
\[\begin{array}{l} \mathbf{if}\;re \lt 0.0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2.0} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Initial program 38.3

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

  2. Simplified12.9

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

  3. Final simplification12.9

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

Reproduce

herbie shell --seed 2019165 +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)))))