Average Error: 38.1 → 11.4
Time: 4.1s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -1.545247932061966 \cdot 10^{105}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im \cdot im}{\mathsf{hypot}\left(re, im\right) - re}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) + re\right)}\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le -1.545247932061966 \cdot 10^{105}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im \cdot im}{\mathsf{hypot}\left(re, im\right) - re}}\\

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

\end{array}
double f(double re, double im) {
        double r318063 = 0.5;
        double r318064 = 2.0;
        double r318065 = re;
        double r318066 = r318065 * r318065;
        double r318067 = im;
        double r318068 = r318067 * r318067;
        double r318069 = r318066 + r318068;
        double r318070 = sqrt(r318069);
        double r318071 = r318070 + r318065;
        double r318072 = r318064 * r318071;
        double r318073 = sqrt(r318072);
        double r318074 = r318063 * r318073;
        return r318074;
}
↓
double f(double re, double im) {
        double r318075 = re;
        double r318076 = -1.545247932061966e+105;
        bool r318077 = r318075 <= r318076;
        double r318078 = 0.5;
        double r318079 = 2.0;
        double r318080 = im;
        double r318081 = r318080 * r318080;
        double r318082 = hypot(r318075, r318080);
        double r318083 = r318082 - r318075;
        double r318084 = r318081 / r318083;
        double r318085 = r318079 * r318084;
        double r318086 = sqrt(r318085);
        double r318087 = r318078 * r318086;
        double r318088 = r318082 + r318075;
        double r318089 = r318079 * r318088;
        double r318090 = sqrt(r318089);
        double r318091 = r318078 * r318090;
        double r318092 = r318077 ? r318087 : r318091;
        return r318092;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original38.1
Target33.2
Herbie11.4
\[\begin{array}{l} \mathbf{if}\;re \lt 0.0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if re < -1.545247932061966e+105

    1. Initial program 61.4

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

    3. Applied flip-+61.4

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

    4. Simplified45.0

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

    5. Simplified30.8

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

    if -1.545247932061966e+105 < re

    1. Initial program 33.6

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

    3. Applied hypot-def7.7

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

  4. Final simplification11.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.545247932061966 \cdot 10^{105}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im \cdot im}{\mathsf{hypot}\left(re, im\right) - re}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\mathsf{hypot}\left(re, im\right) + re\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020059 +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)))))