Average Error: 37.3 → 25.2
Time: 18.6s
Precision: 64
\[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -1.1092877325715177 \cdot 10^{+151}:\\ \;\;\;\;\sqrt{\left(-2 \cdot re\right) \cdot 2.0} \cdot 0.5\\ \mathbf{elif}\;re \le 8.845630688714825 \cdot 10^{-307}:\\ \;\;\;\;\sqrt{2.0 \cdot \left(\sqrt{\sqrt{im \cdot im + re \cdot re}} \cdot \sqrt{\sqrt{im \cdot im + re \cdot re}} - re\right)} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{\left(im \cdot im\right) \cdot 2.0}}{\sqrt{\sqrt{im \cdot im + re \cdot re} + re}}\\ \end{array}\]
0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le -1.1092877325715177 \cdot 10^{+151}:\\
\;\;\;\;\sqrt{\left(-2 \cdot re\right) \cdot 2.0} \cdot 0.5\\

\mathbf{elif}\;re \le 8.845630688714825 \cdot 10^{-307}:\\
\;\;\;\;\sqrt{2.0 \cdot \left(\sqrt{\sqrt{im \cdot im + re \cdot re}} \cdot \sqrt{\sqrt{im \cdot im + re \cdot re}} - re\right)} \cdot 0.5\\

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

\end{array}
double f(double re, double im) {
        double r472082 = 0.5;
        double r472083 = 2.0;
        double r472084 = re;
        double r472085 = r472084 * r472084;
        double r472086 = im;
        double r472087 = r472086 * r472086;
        double r472088 = r472085 + r472087;
        double r472089 = sqrt(r472088);
        double r472090 = r472089 - r472084;
        double r472091 = r472083 * r472090;
        double r472092 = sqrt(r472091);
        double r472093 = r472082 * r472092;
        return r472093;
}
↓
double f(double re, double im) {
        double r472094 = re;
        double r472095 = -1.1092877325715177e+151;
        bool r472096 = r472094 <= r472095;
        double r472097 = -2.0;
        double r472098 = r472097 * r472094;
        double r472099 = 2.0;
        double r472100 = r472098 * r472099;
        double r472101 = sqrt(r472100);
        double r472102 = 0.5;
        double r472103 = r472101 * r472102;
        double r472104 = 8.845630688714825e-307;
        bool r472105 = r472094 <= r472104;
        double r472106 = im;
        double r472107 = r472106 * r472106;
        double r472108 = r472094 * r472094;
        double r472109 = r472107 + r472108;
        double r472110 = sqrt(r472109);
        double r472111 = sqrt(r472110);
        double r472112 = r472111 * r472111;
        double r472113 = r472112 - r472094;
        double r472114 = r472099 * r472113;
        double r472115 = sqrt(r472114);
        double r472116 = r472115 * r472102;
        double r472117 = r472107 * r472099;
        double r472118 = sqrt(r472117);
        double r472119 = r472110 + r472094;
        double r472120 = sqrt(r472119);
        double r472121 = r472118 / r472120;
        double r472122 = r472102 * r472121;
        double r472123 = r472105 ? r472116 : r472122;
        double r472124 = r472096 ? r472103 : r472123;
        return r472124;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if re < -1.1092877325715177e+151

    1. Initial program 60.0

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

    2. Taylor expanded around inf 60.0

      \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{\color{blue}{{im}^{2} + {re}^{2}}} - re\right)}\]

    3. Simplified60.0

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

    5. Applied add-sqr-sqrt60.0

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

    6. Taylor expanded around -inf 7.1

      \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \color{blue}{\left(-2 \cdot re\right)}}\]

    if -1.1092877325715177e+151 < re < 8.845630688714825e-307

    1. Initial program 19.9

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

    2. Taylor expanded around inf 19.9

      \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{\color{blue}{{im}^{2} + {re}^{2}}} - re\right)}\]

    3. Simplified19.9

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

    5. Applied add-sqr-sqrt20.0

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

    if 8.845630688714825e-307 < re

    1. Initial program 44.9

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

    3. Applied flip--44.8

      \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \color{blue}{\frac{\sqrt{re \cdot re + im \cdot im} \cdot \sqrt{re \cdot re + im \cdot im} - re \cdot re}{\sqrt{re \cdot re + im \cdot im} + re}}}\]

    4. Applied associate-*r/44.8

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

    5. Applied sqrt-div44.9

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

    6. Simplified33.9

      \[\leadsto 0.5 \cdot \frac{\color{blue}{\sqrt{\left(im \cdot im\right) \cdot 2.0 + 0}}}{\sqrt{\sqrt{re \cdot re + im \cdot im} + re}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification25.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.1092877325715177 \cdot 10^{+151}:\\ \;\;\;\;\sqrt{\left(-2 \cdot re\right) \cdot 2.0} \cdot 0.5\\ \mathbf{elif}\;re \le 8.845630688714825 \cdot 10^{-307}:\\ \;\;\;\;\sqrt{2.0 \cdot \left(\sqrt{\sqrt{im \cdot im + re \cdot re}} \cdot \sqrt{\sqrt{im \cdot im + re \cdot re}} - re\right)} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{\left(im \cdot im\right) \cdot 2.0}}{\sqrt{\sqrt{im \cdot im + re \cdot re} + re}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019141 
(FPCore (re im)
  :name "math.sqrt on complex, imaginary part, im greater than 0 branch"
  (* 0.5 (sqrt (* 2.0 (- (sqrt (+ (* re re) (* im im))) re)))))