Average Error: 38.6 → 26.7
Time: 37.3s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le 3.22771281436210170982311694014463290839 \cdot 10^{-299}:\\ \;\;\;\;\left(\frac{\sqrt{im \cdot im}}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}} \cdot \sqrt{2}\right) \cdot 0.5\\ \mathbf{elif}\;re \le 3.920451788983100763629106005950912663971 \cdot 10^{83}:\\ \;\;\;\;\left(\sqrt{2} \cdot \sqrt{re + \sqrt{re \cdot re + im \cdot im}}\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\left(re + re\right) \cdot 2}\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le 3.22771281436210170982311694014463290839 \cdot 10^{-299}:\\
\;\;\;\;\left(\frac{\sqrt{im \cdot im}}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}} \cdot \sqrt{2}\right) \cdot 0.5\\

\mathbf{elif}\;re \le 3.920451788983100763629106005950912663971 \cdot 10^{83}:\\
\;\;\;\;\left(\sqrt{2} \cdot \sqrt{re + \sqrt{re \cdot re + im \cdot im}}\right) \cdot 0.5\\

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

\end{array}
double f(double re, double im) {
        double r7857527 = 0.5;
        double r7857528 = 2.0;
        double r7857529 = re;
        double r7857530 = r7857529 * r7857529;
        double r7857531 = im;
        double r7857532 = r7857531 * r7857531;
        double r7857533 = r7857530 + r7857532;
        double r7857534 = sqrt(r7857533);
        double r7857535 = r7857534 + r7857529;
        double r7857536 = r7857528 * r7857535;
        double r7857537 = sqrt(r7857536);
        double r7857538 = r7857527 * r7857537;
        return r7857538;
}
↓
double f(double re, double im) {
        double r7857539 = re;
        double r7857540 = 3.2277128143621017e-299;
        bool r7857541 = r7857539 <= r7857540;
        double r7857542 = im;
        double r7857543 = r7857542 * r7857542;
        double r7857544 = sqrt(r7857543);
        double r7857545 = r7857539 * r7857539;
        double r7857546 = r7857545 + r7857543;
        double r7857547 = sqrt(r7857546);
        double r7857548 = r7857547 - r7857539;
        double r7857549 = sqrt(r7857548);
        double r7857550 = r7857544 / r7857549;
        double r7857551 = 2.0;
        double r7857552 = sqrt(r7857551);
        double r7857553 = r7857550 * r7857552;
        double r7857554 = 0.5;
        double r7857555 = r7857553 * r7857554;
        double r7857556 = 3.920451788983101e+83;
        bool r7857557 = r7857539 <= r7857556;
        double r7857558 = r7857539 + r7857547;
        double r7857559 = sqrt(r7857558);
        double r7857560 = r7857552 * r7857559;
        double r7857561 = r7857560 * r7857554;
        double r7857562 = r7857539 + r7857539;
        double r7857563 = r7857562 * r7857551;
        double r7857564 = sqrt(r7857563);
        double r7857565 = r7857554 * r7857564;
        double r7857566 = r7857557 ? r7857561 : r7857565;
        double r7857567 = r7857541 ? r7857555 : r7857566;
        return r7857567;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original38.6
Target33.8
Herbie26.7
\[\begin{array}{l} \mathbf{if}\;re \lt 0.0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if re < 3.2277128143621017e-299

    1. Initial program 46.2

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

    3. Applied sqrt-prod46.3

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

    5. Applied flip-+46.1

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

    6. Applied sqrt-div46.1

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

    7. Simplified35.8

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

    if 3.2277128143621017e-299 < re < 3.920451788983101e+83

    1. Initial program 20.3

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

    3. Applied sqrt-prod20.6

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

    if 3.920451788983101e+83 < re

    1. Initial program 47.9

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

    2. Taylor expanded around inf 11.5

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

  4. Final simplification26.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le 3.22771281436210170982311694014463290839 \cdot 10^{-299}:\\ \;\;\;\;\left(\frac{\sqrt{im \cdot im}}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}} \cdot \sqrt{2}\right) \cdot 0.5\\ \mathbf{elif}\;re \le 3.920451788983100763629106005950912663971 \cdot 10^{83}:\\ \;\;\;\;\left(\sqrt{2} \cdot \sqrt{re + \sqrt{re \cdot re + im \cdot im}}\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\left(re + re\right) \cdot 2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 
(FPCore (re im)
  :name "math.sqrt on complex, real part"

  :herbie-target
  (if (< re 0.0) (* 0.5 (* (sqrt 2.0) (sqrt (/ (* im im) (- (sqrt (+ (* re re) (* im im))) re))))) (* 0.5 (sqrt (* 2.0 (+ (sqrt (+ (* re re) (* im im))) re)))))

  (* 0.5 (sqrt (* 2.0 (+ (sqrt (+ (* re re) (* im im))) re)))))