Average Error: 37.9 → 13.1
Time: 18.0s
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 r5362506 = 0.5;
        double r5362507 = 2.0;
        double r5362508 = re;
        double r5362509 = r5362508 * r5362508;
        double r5362510 = im;
        double r5362511 = r5362510 * r5362510;
        double r5362512 = r5362509 + r5362511;
        double r5362513 = sqrt(r5362512);
        double r5362514 = r5362513 + r5362508;
        double r5362515 = r5362507 * r5362514;
        double r5362516 = sqrt(r5362515);
        double r5362517 = r5362506 * r5362516;
        return r5362517;
}
↓
double f(double re, double im) {
        double r5362518 = re;
        double r5362519 = im;
        double r5362520 = hypot(r5362518, r5362519);
        double r5362521 = r5362518 + r5362520;
        double r5362522 = 2.0;
        double r5362523 = r5362521 * r5362522;
        double r5362524 = sqrt(r5362523);
        double r5362525 = 0.5;
        double r5362526 = r5362524 * r5362525;
        return r5362526;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.9
Target32.7
Herbie13.1
\[\begin{array}{l} \mathbf{if}\;re \lt 0.0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Initial program 37.9

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

  2. Simplified13.1

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

  3. Final simplification13.1

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

Reproduce

herbie shell --seed 2019172 +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)))))