Average Error: 38.7 → 13.7
Time: 15.5s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
↓
\[0.5 \cdot \left(\sqrt{re + \mathsf{hypot}\left(re, im\right)} \cdot \sqrt{2}\right)\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}
↓
0.5 \cdot \left(\sqrt{re + \mathsf{hypot}\left(re, im\right)} \cdot \sqrt{2}\right)
double f(double re, double im) {
        double r182415 = 0.5;
        double r182416 = 2.0;
        double r182417 = re;
        double r182418 = r182417 * r182417;
        double r182419 = im;
        double r182420 = r182419 * r182419;
        double r182421 = r182418 + r182420;
        double r182422 = sqrt(r182421);
        double r182423 = r182422 + r182417;
        double r182424 = r182416 * r182423;
        double r182425 = sqrt(r182424);
        double r182426 = r182415 * r182425;
        return r182426;
}
↓
double f(double re, double im) {
        double r182427 = 0.5;
        double r182428 = re;
        double r182429 = im;
        double r182430 = hypot(r182428, r182429);
        double r182431 = r182428 + r182430;
        double r182432 = sqrt(r182431);
        double r182433 = 2.0;
        double r182434 = sqrt(r182433);
        double r182435 = r182432 * r182434;
        double r182436 = r182427 * r182435;
        return r182436;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original38.7
Target33.7
Herbie13.7
\[\begin{array}{l} \mathbf{if}\;re \lt 0.0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Initial program 38.7

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

  2. Simplified13.4

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

  4. Applied sqrt-prod13.7

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

  5. Final simplification13.7

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

Reproduce

herbie shell --seed 2019208 +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)))))