Average Error: 38.4 → 26.3
Time: 5.1s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -5.30337460350832072 \cdot 10^{-234}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\\ \mathbf{elif}\;re \le 1.164285257168749 \cdot 10^{137}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(2 \cdot re\right)}\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le -5.30337460350832072 \cdot 10^{-234}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\\

\mathbf{elif}\;re \le 1.164285257168749 \cdot 10^{137}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\

\mathbf{else}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(2 \cdot re\right)}\\

\end{array}
double f(double re, double im) {
        double r168488 = 0.5;
        double r168489 = 2.0;
        double r168490 = re;
        double r168491 = r168490 * r168490;
        double r168492 = im;
        double r168493 = r168492 * r168492;
        double r168494 = r168491 + r168493;
        double r168495 = sqrt(r168494);
        double r168496 = r168495 + r168490;
        double r168497 = r168489 * r168496;
        double r168498 = sqrt(r168497);
        double r168499 = r168488 * r168498;
        return r168499;
}
↓
double f(double re, double im) {
        double r168500 = re;
        double r168501 = -5.303374603508321e-234;
        bool r168502 = r168500 <= r168501;
        double r168503 = 0.5;
        double r168504 = 2.0;
        double r168505 = im;
        double r168506 = r168505 * r168505;
        double r168507 = r168500 * r168500;
        double r168508 = r168507 + r168506;
        double r168509 = sqrt(r168508);
        double r168510 = r168509 - r168500;
        double r168511 = r168506 / r168510;
        double r168512 = r168504 * r168511;
        double r168513 = sqrt(r168512);
        double r168514 = r168503 * r168513;
        double r168515 = 1.164285257168749e+137;
        bool r168516 = r168500 <= r168515;
        double r168517 = r168509 + r168500;
        double r168518 = r168504 * r168517;
        double r168519 = sqrt(r168518);
        double r168520 = r168503 * r168519;
        double r168521 = 2.0;
        double r168522 = r168521 * r168500;
        double r168523 = r168504 * r168522;
        double r168524 = sqrt(r168523);
        double r168525 = r168503 * r168524;
        double r168526 = r168516 ? r168520 : r168525;
        double r168527 = r168502 ? r168514 : r168526;
        return r168527;
}

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
Target33.1
Herbie26.3
\[\begin{array}{l} \mathbf{if}\;re \lt 0.0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if re < -5.303374603508321e-234

    1. Initial program 47.0

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

    3. Applied flip-+47.0

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

    4. Simplified35.3

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

    if -5.303374603508321e-234 < re < 1.164285257168749e+137

    1. Initial program 22.1

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

    if 1.164285257168749e+137 < re

    1. Initial program 59.1

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

    2. Taylor expanded around inf 9.1

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

  4. Final simplification26.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -5.30337460350832072 \cdot 10^{-234}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\\ \mathbf{elif}\;re \le 1.164285257168749 \cdot 10^{137}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(2 \cdot re\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020024 
(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)))))