Average Error: 38.8 → 13.7
Time: 19.3s
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 r101618 = 0.5;
        double r101619 = 2.0;
        double r101620 = re;
        double r101621 = r101620 * r101620;
        double r101622 = im;
        double r101623 = r101622 * r101622;
        double r101624 = r101621 + r101623;
        double r101625 = sqrt(r101624);
        double r101626 = r101625 + r101620;
        double r101627 = r101619 * r101626;
        double r101628 = sqrt(r101627);
        double r101629 = r101618 * r101628;
        return r101629;
}
↓
double f(double re, double im) {
        double r101630 = 0.5;
        double r101631 = re;
        double r101632 = im;
        double r101633 = hypot(r101631, r101632);
        double r101634 = r101631 + r101633;
        double r101635 = sqrt(r101634);
        double r101636 = 2.0;
        double r101637 = sqrt(r101636);
        double r101638 = r101635 * r101637;
        double r101639 = r101630 * r101638;
        return r101639;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original38.8
Target33.8
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 38.8

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

  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 2019347 +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)))))