Average Error: 39.1 → 13.2
Time: 10.8s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
↓
\[0.5 \cdot \sqrt{\left(re + \mathsf{hypot}\left(re, im\right)\right) \cdot 2}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}
↓
0.5 \cdot \sqrt{\left(re + \mathsf{hypot}\left(re, im\right)\right) \cdot 2}
double f(double re, double im) {
        double r233487 = 0.5;
        double r233488 = 2.0;
        double r233489 = re;
        double r233490 = r233489 * r233489;
        double r233491 = im;
        double r233492 = r233491 * r233491;
        double r233493 = r233490 + r233492;
        double r233494 = sqrt(r233493);
        double r233495 = r233494 + r233489;
        double r233496 = r233488 * r233495;
        double r233497 = sqrt(r233496);
        double r233498 = r233487 * r233497;
        return r233498;
}
↓
double f(double re, double im) {
        double r233499 = 0.5;
        double r233500 = re;
        double r233501 = im;
        double r233502 = hypot(r233500, r233501);
        double r233503 = r233500 + r233502;
        double r233504 = 2.0;
        double r233505 = r233503 * r233504;
        double r233506 = sqrt(r233505);
        double r233507 = r233499 * r233506;
        return r233507;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original39.1
Target34.1
Herbie13.2
\[\begin{array}{l} \mathbf{if}\;re \lt 0.0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Initial program 39.1

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

  2. Simplified13.2

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

  3. Final simplification13.2

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

Reproduce

herbie shell --seed 2020046 +o rules:numerics
(FPCore (re im)
  :name "math.sqrt on complex, real part"
  :precision binary64

  :herbie-target
  (if (< re 0.0) (* 0.5 (* (sqrt 2) (sqrt (/ (* im im) (- (sqrt (+ (* re re) (* im im))) re))))) (* 0.5 (sqrt (* 2 (+ (sqrt (+ (* re re) (* im im))) re)))))

  (* 0.5 (sqrt (* 2 (+ (sqrt (+ (* re re) (* im im))) re)))))