Average Error: 38.7 → 27.3
Time: 3.3s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -6.5586381711852656 \cdot 10^{44}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-1 \cdot re - re\right)}\\ \mathbf{elif}\;re \le 1.8012497291896643 \cdot 10^{-291}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt{\sqrt{re \cdot re + im \cdot im}} - re\right)}\\ \mathbf{elif}\;re \le 3.31122257455487095 \cdot 10^{-196}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \le 1.3228005539027653 \cdot 10^{163}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2}}{\sqrt{re \cdot re + im \cdot im} + re}}\\ \mathbf{elif}\;re \le 2.1596231037151348 \cdot 10^{232}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot 0\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le -6.5586381711852656 \cdot 10^{44}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-1 \cdot re - re\right)}\\

\mathbf{elif}\;re \le 1.8012497291896643 \cdot 10^{-291}:\\
\;\;\;\;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{elif}\;re \le 3.31122257455487095 \cdot 10^{-196}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\

\mathbf{elif}\;re \le 1.3228005539027653 \cdot 10^{163}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2}}{\sqrt{re \cdot re + im \cdot im} + re}}\\

\mathbf{elif}\;re \le 2.1596231037151348 \cdot 10^{232}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\

\mathbf{else}:\\
\;\;\;\;0.5 \cdot 0\\

\end{array}
double f(double re, double im) {
        double r13078 = 0.5;
        double r13079 = 2.0;
        double r13080 = re;
        double r13081 = r13080 * r13080;
        double r13082 = im;
        double r13083 = r13082 * r13082;
        double r13084 = r13081 + r13083;
        double r13085 = sqrt(r13084);
        double r13086 = r13085 - r13080;
        double r13087 = r13079 * r13086;
        double r13088 = sqrt(r13087);
        double r13089 = r13078 * r13088;
        return r13089;
}
↓
double f(double re, double im) {
        double r13090 = re;
        double r13091 = -6.5586381711852656e+44;
        bool r13092 = r13090 <= r13091;
        double r13093 = 0.5;
        double r13094 = 2.0;
        double r13095 = -1.0;
        double r13096 = r13095 * r13090;
        double r13097 = r13096 - r13090;
        double r13098 = r13094 * r13097;
        double r13099 = sqrt(r13098);
        double r13100 = r13093 * r13099;
        double r13101 = 1.8012497291896643e-291;
        bool r13102 = r13090 <= r13101;
        double r13103 = r13090 * r13090;
        double r13104 = im;
        double r13105 = r13104 * r13104;
        double r13106 = r13103 + r13105;
        double r13107 = sqrt(r13106);
        double r13108 = sqrt(r13107);
        double r13109 = r13108 * r13108;
        double r13110 = r13109 - r13090;
        double r13111 = r13094 * r13110;
        double r13112 = sqrt(r13111);
        double r13113 = r13093 * r13112;
        double r13114 = 3.311222574554871e-196;
        bool r13115 = r13090 <= r13114;
        double r13116 = r13104 - r13090;
        double r13117 = r13094 * r13116;
        double r13118 = sqrt(r13117);
        double r13119 = r13093 * r13118;
        double r13120 = 1.3228005539027653e+163;
        bool r13121 = r13090 <= r13120;
        double r13122 = 2.0;
        double r13123 = pow(r13104, r13122);
        double r13124 = r13107 + r13090;
        double r13125 = r13123 / r13124;
        double r13126 = r13094 * r13125;
        double r13127 = sqrt(r13126);
        double r13128 = r13093 * r13127;
        double r13129 = 2.1596231037151348e+232;
        bool r13130 = r13090 <= r13129;
        double r13131 = 0.0;
        double r13132 = r13093 * r13131;
        double r13133 = r13130 ? r13119 : r13132;
        double r13134 = r13121 ? r13128 : r13133;
        double r13135 = r13115 ? r13119 : r13134;
        double r13136 = r13102 ? r13113 : r13135;
        double r13137 = r13092 ? r13100 : r13136;
        return r13137;
}

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 5 regimes

  2. if re < -6.5586381711852656e+44

    1. Initial program 45.2

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

    2. Taylor expanded around -inf 12.6

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{-1 \cdot re} - re\right)}\]

    if -6.5586381711852656e+44 < re < 1.8012497291896643e-291

    1. Initial program 22.0

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

    3. Applied add-sqr-sqrt22.0

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

    4. Applied sqrt-prod22.1

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

    if 1.8012497291896643e-291 < re < 3.311222574554871e-196 or 1.3228005539027653e+163 < re < 2.1596231037151348e+232

    1. Initial program 45.3

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

    2. Taylor expanded around 0 44.2

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

    if 3.311222574554871e-196 < re < 1.3228005539027653e+163

    1. Initial program 42.8

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

    3. Applied flip--42.8

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

    4. Simplified30.5

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

    if 2.1596231037151348e+232 < re

    1. Initial program 64.0

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

    2. Taylor expanded around inf 48.9

      \[\leadsto 0.5 \cdot \color{blue}{0}\]
  3. Recombined 5 regimes into one program.

  4. Final simplification27.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -6.5586381711852656 \cdot 10^{44}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-1 \cdot re - re\right)}\\ \mathbf{elif}\;re \le 1.8012497291896643 \cdot 10^{-291}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt{\sqrt{re \cdot re + im \cdot im}} - re\right)}\\ \mathbf{elif}\;re \le 3.31122257455487095 \cdot 10^{-196}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{elif}\;re \le 1.3228005539027653 \cdot 10^{163}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2}}{\sqrt{re \cdot re + im \cdot im} + re}}\\ \mathbf{elif}\;re \le 2.1596231037151348 \cdot 10^{232}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot 0\\ \end{array}\]

Reproduce

herbie shell --seed 2020059 
(FPCore (re im)
  :name "math.sqrt on complex, imaginary part, im greater than 0 branch"
  :precision binary64
  (* 0.5 (sqrt (* 2 (- (sqrt (+ (* re re) (* im im))) re)))))