Average Error: 38.9 → 26.7
Time: 14.2s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -3.10856269843066260214885829064035485714 \cdot 10^{-310}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2}}{\sqrt{re \cdot re + im \cdot im} - re}}\\ \mathbf{elif}\;re \le 5.909327001812387299818053178697919247085 \cdot 10^{128}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt{\sqrt{re \cdot re + im \cdot im}} + re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(2 \cdot re\right)}\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le -3.10856269843066260214885829064035485714 \cdot 10^{-310}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2}}{\sqrt{re \cdot re + im \cdot im} - re}}\\

\mathbf{elif}\;re \le 5.909327001812387299818053178697919247085 \cdot 10^{128}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt{\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 r182482 = 0.5;
        double r182483 = 2.0;
        double r182484 = re;
        double r182485 = r182484 * r182484;
        double r182486 = im;
        double r182487 = r182486 * r182486;
        double r182488 = r182485 + r182487;
        double r182489 = sqrt(r182488);
        double r182490 = r182489 + r182484;
        double r182491 = r182483 * r182490;
        double r182492 = sqrt(r182491);
        double r182493 = r182482 * r182492;
        return r182493;
}
↓
double f(double re, double im) {
        double r182494 = re;
        double r182495 = -3.10856269843066e-310;
        bool r182496 = r182494 <= r182495;
        double r182497 = 0.5;
        double r182498 = 2.0;
        double r182499 = im;
        double r182500 = 2.0;
        double r182501 = pow(r182499, r182500);
        double r182502 = r182494 * r182494;
        double r182503 = r182499 * r182499;
        double r182504 = r182502 + r182503;
        double r182505 = sqrt(r182504);
        double r182506 = r182505 - r182494;
        double r182507 = r182501 / r182506;
        double r182508 = r182498 * r182507;
        double r182509 = sqrt(r182508);
        double r182510 = r182497 * r182509;
        double r182511 = 5.909327001812387e+128;
        bool r182512 = r182494 <= r182511;
        double r182513 = sqrt(r182505);
        double r182514 = r182513 * r182513;
        double r182515 = r182514 + r182494;
        double r182516 = r182498 * r182515;
        double r182517 = sqrt(r182516);
        double r182518 = r182497 * r182517;
        double r182519 = r182500 * r182494;
        double r182520 = r182498 * r182519;
        double r182521 = sqrt(r182520);
        double r182522 = r182497 * r182521;
        double r182523 = r182512 ? r182518 : r182522;
        double r182524 = r182496 ? r182510 : r182523;
        return r182524;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original38.9
Target34.2
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 < -3.10856269843066e-310

    1. Initial program 45.8

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

    3. Applied flip-+45.7

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

    4. Simplified36.2

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

    if -3.10856269843066e-310 < re < 5.909327001812387e+128

    1. Initial program 21.1

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

    3. Applied add-sqr-sqrt21.1

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

    4. Applied sqrt-prod21.1

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

    if 5.909327001812387e+128 < re

    1. Initial program 56.7

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

    2. Taylor expanded around inf 8.2

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

  4. Final simplification26.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -3.10856269843066260214885829064035485714 \cdot 10^{-310}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2}}{\sqrt{re \cdot re + im \cdot im} - re}}\\ \mathbf{elif}\;re \le 5.909327001812387299818053178697919247085 \cdot 10^{128}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt{\sqrt{re \cdot re + im \cdot im}} + re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(2 \cdot re\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019294 
(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)))))