Average Error: 37.6 → 30.5
Time: 26.6s
Precision: 64
\[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;im \le -1.8776884943596414 \cdot 10^{-117}:\\ \;\;\;\;\sqrt{2.0 \cdot \left(re + \sqrt{\sqrt{\sqrt[3]{im \cdot im + re \cdot re}}} \cdot \left(\sqrt{\sqrt{im \cdot im + re \cdot re}} \cdot \sqrt{\sqrt{\left(\left(\sqrt[3]{\sqrt[3]{im \cdot im + re \cdot re}} \cdot \sqrt[3]{\sqrt[3]{im \cdot im + re \cdot re}}\right) \cdot \sqrt[3]{\sqrt[3]{im \cdot im + re \cdot re}}\right) \cdot \sqrt[3]{im \cdot im + re \cdot re}}}\right)\right)} \cdot 0.5\\ \mathbf{elif}\;im \le 1.2794987088213298 \cdot 10^{-198}:\\ \;\;\;\;\sqrt{\left(re + re\right) \cdot 2.0} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(re + im\right) \cdot 2.0} \cdot 0.5\\ \end{array}\]
0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}
↓
\begin{array}{l}
\mathbf{if}\;im \le -1.8776884943596414 \cdot 10^{-117}:\\
\;\;\;\;\sqrt{2.0 \cdot \left(re + \sqrt{\sqrt{\sqrt[3]{im \cdot im + re \cdot re}}} \cdot \left(\sqrt{\sqrt{im \cdot im + re \cdot re}} \cdot \sqrt{\sqrt{\left(\left(\sqrt[3]{\sqrt[3]{im \cdot im + re \cdot re}} \cdot \sqrt[3]{\sqrt[3]{im \cdot im + re \cdot re}}\right) \cdot \sqrt[3]{\sqrt[3]{im \cdot im + re \cdot re}}\right) \cdot \sqrt[3]{im \cdot im + re \cdot re}}}\right)\right)} \cdot 0.5\\

\mathbf{elif}\;im \le 1.2794987088213298 \cdot 10^{-198}:\\
\;\;\;\;\sqrt{\left(re + re\right) \cdot 2.0} \cdot 0.5\\

\mathbf{else}:\\
\;\;\;\;\sqrt{\left(re + im\right) \cdot 2.0} \cdot 0.5\\

\end{array}
double f(double re, double im) {
        double r30441053 = 0.5;
        double r30441054 = 2.0;
        double r30441055 = re;
        double r30441056 = r30441055 * r30441055;
        double r30441057 = im;
        double r30441058 = r30441057 * r30441057;
        double r30441059 = r30441056 + r30441058;
        double r30441060 = sqrt(r30441059);
        double r30441061 = r30441060 + r30441055;
        double r30441062 = r30441054 * r30441061;
        double r30441063 = sqrt(r30441062);
        double r30441064 = r30441053 * r30441063;
        return r30441064;
}
↓
double f(double re, double im) {
        double r30441065 = im;
        double r30441066 = -1.8776884943596414e-117;
        bool r30441067 = r30441065 <= r30441066;
        double r30441068 = 2.0;
        double r30441069 = re;
        double r30441070 = r30441065 * r30441065;
        double r30441071 = r30441069 * r30441069;
        double r30441072 = r30441070 + r30441071;
        double r30441073 = cbrt(r30441072);
        double r30441074 = sqrt(r30441073);
        double r30441075 = sqrt(r30441074);
        double r30441076 = sqrt(r30441072);
        double r30441077 = sqrt(r30441076);
        double r30441078 = cbrt(r30441073);
        double r30441079 = r30441078 * r30441078;
        double r30441080 = r30441079 * r30441078;
        double r30441081 = r30441080 * r30441073;
        double r30441082 = sqrt(r30441081);
        double r30441083 = sqrt(r30441082);
        double r30441084 = r30441077 * r30441083;
        double r30441085 = r30441075 * r30441084;
        double r30441086 = r30441069 + r30441085;
        double r30441087 = r30441068 * r30441086;
        double r30441088 = sqrt(r30441087);
        double r30441089 = 0.5;
        double r30441090 = r30441088 * r30441089;
        double r30441091 = 1.2794987088213298e-198;
        bool r30441092 = r30441065 <= r30441091;
        double r30441093 = r30441069 + r30441069;
        double r30441094 = r30441093 * r30441068;
        double r30441095 = sqrt(r30441094);
        double r30441096 = r30441095 * r30441089;
        double r30441097 = r30441069 + r30441065;
        double r30441098 = r30441097 * r30441068;
        double r30441099 = sqrt(r30441098);
        double r30441100 = r30441099 * r30441089;
        double r30441101 = r30441092 ? r30441096 : r30441100;
        double r30441102 = r30441067 ? r30441090 : r30441101;
        return r30441102;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original37.6
Target32.6
Herbie30.5
\[\begin{array}{l} \mathbf{if}\;re \lt 0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if im < -1.8776884943596414e-117

    1. Initial program 36.4

      \[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
    2. Using strategy rm

    3. Applied add-sqr-sqrt36.4

      \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \left(\color{blue}{\sqrt{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt{\sqrt{re \cdot re + im \cdot im}}} + re\right)}\]
    4. Using strategy rm

    5. Applied add-cube-cbrt36.4

      \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt{\sqrt{\color{blue}{\left(\sqrt[3]{re \cdot re + im \cdot im} \cdot \sqrt[3]{re \cdot re + im \cdot im}\right) \cdot \sqrt[3]{re \cdot re + im \cdot im}}}} + re\right)}\]

    6. Applied sqrt-prod36.4

      \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt{\color{blue}{\sqrt{\sqrt[3]{re \cdot re + im \cdot im} \cdot \sqrt[3]{re \cdot re + im \cdot im}} \cdot \sqrt{\sqrt[3]{re \cdot re + im \cdot im}}}} + re\right)}\]

    7. Applied sqrt-prod36.4

      \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{\sqrt{re \cdot re + im \cdot im}} \cdot \color{blue}{\left(\sqrt{\sqrt{\sqrt[3]{re \cdot re + im \cdot im} \cdot \sqrt[3]{re \cdot re + im \cdot im}}} \cdot \sqrt{\sqrt{\sqrt[3]{re \cdot re + im \cdot im}}}\right)} + re\right)}\]

    8. Applied associate-*r*36.4

      \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \left(\color{blue}{\left(\sqrt{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt{\sqrt{\sqrt[3]{re \cdot re + im \cdot im} \cdot \sqrt[3]{re \cdot re + im \cdot im}}}\right) \cdot \sqrt{\sqrt{\sqrt[3]{re \cdot re + im \cdot im}}}} + re\right)}\]
    9. Using strategy rm

    10. Applied add-cube-cbrt36.5

      \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \left(\left(\sqrt{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt{\sqrt{\color{blue}{\left(\left(\sqrt[3]{\sqrt[3]{re \cdot re + im \cdot im}} \cdot \sqrt[3]{\sqrt[3]{re \cdot re + im \cdot im}}\right) \cdot \sqrt[3]{\sqrt[3]{re \cdot re + im \cdot im}}\right)} \cdot \sqrt[3]{re \cdot re + im \cdot im}}}\right) \cdot \sqrt{\sqrt{\sqrt[3]{re \cdot re + im \cdot im}}} + re\right)}\]

    if -1.8776884943596414e-117 < im < 1.2794987088213298e-198

    1. Initial program 40.1

      \[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]

    2. Taylor expanded around inf 33.9

      \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \left(\color{blue}{re} + re\right)}\]

    if 1.2794987088213298e-198 < im

    1. Initial program 37.3

      \[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]

    2. Taylor expanded around 0 23.4

      \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \color{blue}{\left(re + im\right)}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification30.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;im \le -1.8776884943596414 \cdot 10^{-117}:\\ \;\;\;\;\sqrt{2.0 \cdot \left(re + \sqrt{\sqrt{\sqrt[3]{im \cdot im + re \cdot re}}} \cdot \left(\sqrt{\sqrt{im \cdot im + re \cdot re}} \cdot \sqrt{\sqrt{\left(\left(\sqrt[3]{\sqrt[3]{im \cdot im + re \cdot re}} \cdot \sqrt[3]{\sqrt[3]{im \cdot im + re \cdot re}}\right) \cdot \sqrt[3]{\sqrt[3]{im \cdot im + re \cdot re}}\right) \cdot \sqrt[3]{im \cdot im + re \cdot re}}}\right)\right)} \cdot 0.5\\ \mathbf{elif}\;im \le 1.2794987088213298 \cdot 10^{-198}:\\ \;\;\;\;\sqrt{\left(re + re\right) \cdot 2.0} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(re + im\right) \cdot 2.0} \cdot 0.5\\ \end{array}\]

Reproduce

herbie shell --seed 2019107 
(FPCore (re im)
  :name "math.sqrt on complex, real part"

  :herbie-target
  (if (< re 0) (* 0.5 (* (sqrt 2) (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)))))