Average Error: 2.0 → 0.8
Time: 21.5s
Precision: 64
\[\left(0.5\right) \cdot \left(\sqrt{\left(\left(2.0\right) \cdot \left(\left(\sqrt{\left(\frac{\left(re \cdot re\right)}{\left(im \cdot im\right)}\right)}\right) - re\right)\right)}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le 0.02642822265625:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \frac{im}{\frac{\sqrt{re \cdot re + im \cdot im} + re}{im}}}\\ \end{array}\]
\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 0.02642822265625:\\
\;\;\;\;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}{\frac{\sqrt{re \cdot re + im \cdot im} + re}{im}}}\\

\end{array}
double f(double re, double im) {
        double r1020121 = 0.5;
        double r1020122 = /* ERROR: no posit support in C */;
        double r1020123 = 2.0;
        double r1020124 = /* ERROR: no posit support in C */;
        double r1020125 = re;
        double r1020126 = r1020125 * r1020125;
        double r1020127 = im;
        double r1020128 = r1020127 * r1020127;
        double r1020129 = r1020126 + r1020128;
        double r1020130 = sqrt(r1020129);
        double r1020131 = r1020130 - r1020125;
        double r1020132 = r1020124 * r1020131;
        double r1020133 = sqrt(r1020132);
        double r1020134 = r1020122 * r1020133;
        return r1020134;
}
↓
double f(double re, double im) {
        double r1020135 = re;
        double r1020136 = 0.02642822265625;
        bool r1020137 = r1020135 <= r1020136;
        double r1020138 = 0.5;
        double r1020139 = 2.0;
        double r1020140 = r1020135 * r1020135;
        double r1020141 = im;
        double r1020142 = r1020141 * r1020141;
        double r1020143 = r1020140 + r1020142;
        double r1020144 = sqrt(r1020143);
        double r1020145 = r1020144 - r1020135;
        double r1020146 = r1020139 * r1020145;
        double r1020147 = sqrt(r1020146);
        double r1020148 = r1020138 * r1020147;
        double r1020149 = r1020144 + r1020135;
        double r1020150 = r1020149 / r1020141;
        double r1020151 = r1020141 / r1020150;
        double r1020152 = r1020139 * r1020151;
        double r1020153 = sqrt(r1020152);
        double r1020154 = r1020138 * r1020153;
        double r1020155 = r1020137 ? r1020148 : r1020154;
        return r1020155;
}

Error

Bits error versus re

Bits error versus im

Derivation

  1. Split input into 2 regimes

  2. if re < 0.02642822265625

    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 0.02642822265625 < re

    1. Initial program 3.6

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

    3. Applied p16-flip--3.3

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

    4. Simplified0.9

      \[\leadsto \left(0.5\right) \cdot \left(\sqrt{\left(\left(2.0\right) \cdot \left(\frac{\color{blue}{\left(im \cdot im\right)}}{\left(\frac{\left(\sqrt{\left(\frac{\left(re \cdot re\right)}{\left(im \cdot im\right)}\right)}\right)}{re}\right)}\right)\right)}\right)\]
    5. Using strategy rm

    6. Applied associate-/l*0.9

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

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le 0.02642822265625:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \frac{im}{\frac{\sqrt{re \cdot re + im \cdot im} + re}{im}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019135 +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)))))