Average Error: 37.3 → 12.8
Time: 19.8s
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 r7008667 = 0.5;
        double r7008668 = 2.0;
        double r7008669 = re;
        double r7008670 = r7008669 * r7008669;
        double r7008671 = im;
        double r7008672 = r7008671 * r7008671;
        double r7008673 = r7008670 + r7008672;
        double r7008674 = sqrt(r7008673);
        double r7008675 = r7008674 + r7008669;
        double r7008676 = r7008668 * r7008675;
        double r7008677 = sqrt(r7008676);
        double r7008678 = r7008667 * r7008677;
        return r7008678;
}
↓
double f(double re, double im) {
        double r7008679 = re;
        double r7008680 = im;
        double r7008681 = hypot(r7008679, r7008680);
        double r7008682 = r7008679 + r7008681;
        double r7008683 = 2.0;
        double r7008684 = r7008682 * r7008683;
        double r7008685 = sqrt(r7008684);
        double r7008686 = 0.5;
        double r7008687 = r7008685 * r7008686;
        return r7008687;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.3
Target32.5
Herbie12.8
\[\begin{array}{l} \mathbf{if}\;re \lt 0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Initial program 37.3

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

  2. Simplified12.8

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

  3. Final simplification12.8

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

Reproduce

herbie shell --seed 2019163 +o rules:numerics
(FPCore (re im)
  :name "math.sqrt on complex, real part"

  :herbie-target
  (if (< re 0) (* 0.5 (* (sqrt 2) (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)))))