Average Error: 38.4 → 13.5
Time: 24.6s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
↓
\[0.5 \cdot \sqrt{\left(\sqrt{\mathsf{hypot}\left(re, im\right) + re} \cdot \sqrt{\mathsf{hypot}\left(re, im\right) + re}\right) \cdot 2}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}
↓
0.5 \cdot \sqrt{\left(\sqrt{\mathsf{hypot}\left(re, im\right) + re} \cdot \sqrt{\mathsf{hypot}\left(re, im\right) + re}\right) \cdot 2}
double f(double re, double im) {
        double r145116 = 0.5;
        double r145117 = 2.0;
        double r145118 = re;
        double r145119 = r145118 * r145118;
        double r145120 = im;
        double r145121 = r145120 * r145120;
        double r145122 = r145119 + r145121;
        double r145123 = sqrt(r145122);
        double r145124 = r145123 + r145118;
        double r145125 = r145117 * r145124;
        double r145126 = sqrt(r145125);
        double r145127 = r145116 * r145126;
        return r145127;
}
↓
double f(double re, double im) {
        double r145128 = 0.5;
        double r145129 = re;
        double r145130 = im;
        double r145131 = hypot(r145129, r145130);
        double r145132 = r145131 + r145129;
        double r145133 = sqrt(r145132);
        double r145134 = r145133 * r145133;
        double r145135 = 2.0;
        double r145136 = r145134 * r145135;
        double r145137 = sqrt(r145136);
        double r145138 = r145128 * r145137;
        return r145138;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original38.4
Target33.5
Herbie13.5
\[\begin{array}{l} \mathbf{if}\;re \lt 0.0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Initial program 38.4

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

  2. Simplified13.3

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

  4. Applied add-sqr-sqrt14.4

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

  6. Applied add-sqr-sqrt14.4

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

  7. Simplified14.1

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

  8. Simplified13.5

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

  9. Final simplification13.5

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

Reproduce

herbie shell --seed 2019326 +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)))))