Average Error: 39.4 → 13.8
Time: 3.7s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;im \le -1.2115064360128906 \cdot 10^{154} \lor \neg \left(im \le -1.0469681604918512 \cdot 10^{27} \lor \neg \left(im \le 1.1016319640223668 \cdot 10^{-7} \lor \neg \left(im \le 2.963711736123439 \cdot 10^{58}\right)\right)\right):\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(1 \cdot \mathsf{hypot}\left(re, im\right) + re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im \cdot im}{\mathsf{hypot}\left(re, im\right) - re}}\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}
↓
\begin{array}{l}
\mathbf{if}\;im \le -1.2115064360128906 \cdot 10^{154} \lor \neg \left(im \le -1.0469681604918512 \cdot 10^{27} \lor \neg \left(im \le 1.1016319640223668 \cdot 10^{-7} \lor \neg \left(im \le 2.963711736123439 \cdot 10^{58}\right)\right)\right):\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(1 \cdot \mathsf{hypot}\left(re, im\right) + re\right)}\\

\mathbf{else}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im \cdot im}{\mathsf{hypot}\left(re, im\right) - re}}\\

\end{array}
double f(double re, double im) {
        double r222086 = 0.5;
        double r222087 = 2.0;
        double r222088 = re;
        double r222089 = r222088 * r222088;
        double r222090 = im;
        double r222091 = r222090 * r222090;
        double r222092 = r222089 + r222091;
        double r222093 = sqrt(r222092);
        double r222094 = r222093 + r222088;
        double r222095 = r222087 * r222094;
        double r222096 = sqrt(r222095);
        double r222097 = r222086 * r222096;
        return r222097;
}
↓
double f(double re, double im) {
        double r222098 = im;
        double r222099 = -1.2115064360128906e+154;
        bool r222100 = r222098 <= r222099;
        double r222101 = -1.0469681604918512e+27;
        bool r222102 = r222098 <= r222101;
        double r222103 = 1.1016319640223668e-07;
        bool r222104 = r222098 <= r222103;
        double r222105 = 2.963711736123439e+58;
        bool r222106 = r222098 <= r222105;
        double r222107 = !r222106;
        bool r222108 = r222104 || r222107;
        double r222109 = !r222108;
        bool r222110 = r222102 || r222109;
        double r222111 = !r222110;
        bool r222112 = r222100 || r222111;
        double r222113 = 0.5;
        double r222114 = 2.0;
        double r222115 = 1.0;
        double r222116 = re;
        double r222117 = hypot(r222116, r222098);
        double r222118 = r222115 * r222117;
        double r222119 = r222118 + r222116;
        double r222120 = r222114 * r222119;
        double r222121 = sqrt(r222120);
        double r222122 = r222113 * r222121;
        double r222123 = r222098 * r222098;
        double r222124 = r222117 - r222116;
        double r222125 = r222123 / r222124;
        double r222126 = r222114 * r222125;
        double r222127 = sqrt(r222126);
        double r222128 = r222113 * r222127;
        double r222129 = r222112 ? r222122 : r222128;
        return r222129;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original39.4
Target33.7
Herbie13.8
\[\begin{array}{l} \mathbf{if}\;re \lt 0.0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if im < -1.2115064360128906e+154 or -1.0469681604918512e+27 < im < 1.1016319640223668e-07 or 2.963711736123439e+58 < im

    1. Initial program 42.5

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

    3. Applied *-un-lft-identity42.5

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\color{blue}{1 \cdot \left(re \cdot re + im \cdot im\right)}} + re\right)}\]

    4. Applied sqrt-prod42.5

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{\sqrt{1} \cdot \sqrt{re \cdot re + im \cdot im}} + re\right)}\]

    5. Simplified42.5

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{1} \cdot \sqrt{re \cdot re + im \cdot im} + re\right)}\]

    6. Simplified14.0

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(1 \cdot \color{blue}{\mathsf{hypot}\left(re, im\right)} + re\right)}\]

    if -1.2115064360128906e+154 < im < -1.0469681604918512e+27 or 1.1016319640223668e-07 < im < 2.963711736123439e+58

    1. Initial program 22.7

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

    3. Applied flip-+26.5

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\frac{\sqrt{re \cdot re + im \cdot im} \cdot \sqrt{re \cdot re + im \cdot im} - re \cdot re}{\sqrt{re \cdot re + im \cdot im} - re}}}\]

    4. Simplified21.3

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{\color{blue}{im \cdot im}}{\sqrt{re \cdot re + im \cdot im} - re}}\]

    5. Simplified12.6

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

  4. Final simplification13.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;im \le -1.2115064360128906 \cdot 10^{154} \lor \neg \left(im \le -1.0469681604918512 \cdot 10^{27} \lor \neg \left(im \le 1.1016319640223668 \cdot 10^{-7} \lor \neg \left(im \le 2.963711736123439 \cdot 10^{58}\right)\right)\right):\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(1 \cdot \mathsf{hypot}\left(re, im\right) + re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im \cdot im}{\mathsf{hypot}\left(re, im\right) - re}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020020 +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)))))