Average Error: 38.9 → 15.2
Time: 19.2s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -7.139122052802705008680386727937942984221 \cdot 10^{196}:\\ \;\;\;\;0.5 \cdot \sqrt{e^{\log \left(\mathsf{hypot}\left(re, im\right) + re\right)} \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\mathsf{fma}\left(\sqrt[3]{re} \cdot \sqrt[3]{re}, \sqrt[3]{re}, \mathsf{hypot}\left(re, im\right)\right) \cdot 2}\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le -7.139122052802705008680386727937942984221 \cdot 10^{196}:\\
\;\;\;\;0.5 \cdot \sqrt{e^{\log \left(\mathsf{hypot}\left(re, im\right) + re\right)} \cdot 2}\\

\mathbf{else}:\\
\;\;\;\;0.5 \cdot \sqrt{\mathsf{fma}\left(\sqrt[3]{re} \cdot \sqrt[3]{re}, \sqrt[3]{re}, \mathsf{hypot}\left(re, im\right)\right) \cdot 2}\\

\end{array}
double f(double re, double im) {
        double r88851 = 0.5;
        double r88852 = 2.0;
        double r88853 = re;
        double r88854 = r88853 * r88853;
        double r88855 = im;
        double r88856 = r88855 * r88855;
        double r88857 = r88854 + r88856;
        double r88858 = sqrt(r88857);
        double r88859 = r88858 + r88853;
        double r88860 = r88852 * r88859;
        double r88861 = sqrt(r88860);
        double r88862 = r88851 * r88861;
        return r88862;
}
↓
double f(double re, double im) {
        double r88863 = re;
        double r88864 = -7.139122052802705e+196;
        bool r88865 = r88863 <= r88864;
        double r88866 = 0.5;
        double r88867 = im;
        double r88868 = hypot(r88863, r88867);
        double r88869 = r88868 + r88863;
        double r88870 = log(r88869);
        double r88871 = exp(r88870);
        double r88872 = 2.0;
        double r88873 = r88871 * r88872;
        double r88874 = sqrt(r88873);
        double r88875 = r88866 * r88874;
        double r88876 = cbrt(r88863);
        double r88877 = r88876 * r88876;
        double r88878 = fma(r88877, r88876, r88868);
        double r88879 = r88878 * r88872;
        double r88880 = sqrt(r88879);
        double r88881 = r88866 * r88880;
        double r88882 = r88865 ? r88875 : r88881;
        return r88882;
}

Error

Bits error versus re

Bits error versus im

Target

Original38.9
Target33.7
Herbie15.2
\[\begin{array}{l} \mathbf{if}\;re \lt 0.0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if re < -7.139122052802705e+196

    1. Initial program 64.0

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

    2. Simplified44.1

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{\left(re + \mathsf{hypot}\left(re, im\right)\right) \cdot 2}}\]
    3. Using strategy rm

    4. Applied add-sqr-sqrt49.6

      \[\leadsto 0.5 \cdot \sqrt{\left(re + \color{blue}{\sqrt{\mathsf{hypot}\left(re, im\right)} \cdot \sqrt{\mathsf{hypot}\left(re, im\right)}}\right) \cdot 2}\]
    5. Using strategy rm

    6. Applied add-exp-log50.3

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{e^{\log \left(re + \sqrt{\mathsf{hypot}\left(re, im\right)} \cdot \sqrt{\mathsf{hypot}\left(re, im\right)}\right)}} \cdot 2}\]

    7. Simplified44.8

      \[\leadsto 0.5 \cdot \sqrt{e^{\color{blue}{\log \left(\mathsf{hypot}\left(re, im\right) + re\right)}} \cdot 2}\]

    if -7.139122052802705e+196 < re

    1. Initial program 36.4

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

    2. Simplified11.1

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{\left(re + \mathsf{hypot}\left(re, im\right)\right) \cdot 2}}\]
    3. Using strategy rm

    4. Applied add-cube-cbrt12.0

      \[\leadsto 0.5 \cdot \sqrt{\left(\color{blue}{\left(\sqrt[3]{re} \cdot \sqrt[3]{re}\right) \cdot \sqrt[3]{re}} + \mathsf{hypot}\left(re, im\right)\right) \cdot 2}\]

    5. Applied fma-def12.2

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

  4. Final simplification15.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -7.139122052802705008680386727937942984221 \cdot 10^{196}:\\ \;\;\;\;0.5 \cdot \sqrt{e^{\log \left(\mathsf{hypot}\left(re, im\right) + re\right)} \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\mathsf{fma}\left(\sqrt[3]{re} \cdot \sqrt[3]{re}, \sqrt[3]{re}, \mathsf{hypot}\left(re, im\right)\right) \cdot 2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019303 +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)))))