Average Error: 2.1 → 0.8
Time: 13.9s
Precision: 64
\[\left(0.5\right) \cdot \left(\sqrt{\left(\left(2.0\right) \cdot \left(\left(\sqrt{\left(\frac{\left(re \cdot re\right)}{\left(im \cdot im\right)}\right)}\right) - re\right)\right)}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -0.024383544921875:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} + re}}\\ \end{array}\]
\left(0.5\right) \cdot \left(\sqrt{\left(\left(2.0\right) \cdot \left(\left(\sqrt{\left(\frac{\left(re \cdot re\right)}{\left(im \cdot im\right)}\right)}\right) - re\right)\right)}\right)
↓
\begin{array}{l}
\mathbf{if}\;re \le -0.024383544921875:\\
\;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\\

\mathbf{else}:\\
\;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} + re}}\\

\end{array}
double f(double re, double im) {
        double r1049643 = 0.5;
        double r1049644 = /* ERROR: no posit support in C */;
        double r1049645 = 2.0;
        double r1049646 = /* ERROR: no posit support in C */;
        double r1049647 = re;
        double r1049648 = r1049647 * r1049647;
        double r1049649 = im;
        double r1049650 = r1049649 * r1049649;
        double r1049651 = r1049648 + r1049650;
        double r1049652 = sqrt(r1049651);
        double r1049653 = r1049652 - r1049647;
        double r1049654 = r1049646 * r1049653;
        double r1049655 = sqrt(r1049654);
        double r1049656 = r1049644 * r1049655;
        return r1049656;
}
↓
double f(double re, double im) {
        double r1049657 = re;
        double r1049658 = -0.024383544921875;
        bool r1049659 = r1049657 <= r1049658;
        double r1049660 = 0.5;
        double r1049661 = 2.0;
        double r1049662 = r1049657 * r1049657;
        double r1049663 = im;
        double r1049664 = r1049663 * r1049663;
        double r1049665 = r1049662 + r1049664;
        double r1049666 = sqrt(r1049665);
        double r1049667 = r1049666 - r1049657;
        double r1049668 = r1049661 * r1049667;
        double r1049669 = sqrt(r1049668);
        double r1049670 = r1049660 * r1049669;
        double r1049671 = r1049666 + r1049657;
        double r1049672 = r1049664 / r1049671;
        double r1049673 = r1049661 * r1049672;
        double r1049674 = sqrt(r1049673);
        double r1049675 = r1049660 * r1049674;
        double r1049676 = r1049659 ? r1049670 : r1049675;
        return r1049676;
}

Error

Bits error versus re

Bits error versus im

Derivation

  1. Split input into 2 regimes

  2. if re < -0.024383544921875

    1. Initial program 0.7

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

    if -0.024383544921875 < re

    1. Initial program 3.2

      \[\left(0.5\right) \cdot \left(\sqrt{\left(\left(2.0\right) \cdot \left(\left(\sqrt{\left(\frac{\left(re \cdot re\right)}{\left(im \cdot im\right)}\right)}\right) - re\right)\right)}\right)\]
    2. Using strategy rm

    3. Applied p16-flip--3.0

      \[\leadsto \left(0.5\right) \cdot \left(\sqrt{\left(\left(2.0\right) \cdot \color{blue}{\left(\frac{\left(\left(\left(\sqrt{\left(\frac{\left(re \cdot re\right)}{\left(im \cdot im\right)}\right)}\right) \cdot \left(\sqrt{\left(\frac{\left(re \cdot re\right)}{\left(im \cdot im\right)}\right)}\right)\right) - \left(re \cdot re\right)\right)}{\left(\frac{\left(\sqrt{\left(\frac{\left(re \cdot re\right)}{\left(im \cdot im\right)}\right)}\right)}{re}\right)}\right)}\right)}\right)\]

    4. Simplified0.9

      \[\leadsto \left(0.5\right) \cdot \left(\sqrt{\left(\left(2.0\right) \cdot \left(\frac{\color{blue}{\left(im \cdot im\right)}}{\left(\frac{\left(\sqrt{\left(\frac{\left(re \cdot re\right)}{\left(im \cdot im\right)}\right)}\right)}{re}\right)}\right)\right)}\right)\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -0.024383544921875:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} + re}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019125 +o rules:numerics
(FPCore (re im)
  :name "math.sqrt on complex, imaginary part, im greater than 0 branch"
  (*.p16 (real->posit16 0.5) (sqrt.p16 (*.p16 (real->posit16 2.0) (-.p16 (sqrt.p16 (+.p16 (*.p16 re re) (*.p16 im im))) re)))))