Average Error: 38.9 → 15.8
Time: 18.2s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
↓
\[0.5 \cdot \sqrt{\left(re + \left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{\frac{3}{4}} \cdot \sqrt[3]{\sqrt{\mathsf{hypot}\left(re, im\right)}}\right) \cdot \sqrt[3]{\sqrt{\sqrt{\mathsf{hypot}\left(re, im\right)}}}\right) \cdot 2}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}
↓
0.5 \cdot \sqrt{\left(re + \left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{\frac{3}{4}} \cdot \sqrt[3]{\sqrt{\mathsf{hypot}\left(re, im\right)}}\right) \cdot \sqrt[3]{\sqrt{\sqrt{\mathsf{hypot}\left(re, im\right)}}}\right) \cdot 2}
double f(double re, double im) {
        double r111150 = 0.5;
        double r111151 = 2.0;
        double r111152 = re;
        double r111153 = r111152 * r111152;
        double r111154 = im;
        double r111155 = r111154 * r111154;
        double r111156 = r111153 + r111155;
        double r111157 = sqrt(r111156);
        double r111158 = r111157 + r111152;
        double r111159 = r111151 * r111158;
        double r111160 = sqrt(r111159);
        double r111161 = r111150 * r111160;
        return r111161;
}
↓
double f(double re, double im) {
        double r111162 = 0.5;
        double r111163 = re;
        double r111164 = im;
        double r111165 = hypot(r111163, r111164);
        double r111166 = 0.75;
        double r111167 = pow(r111165, r111166);
        double r111168 = sqrt(r111165);
        double r111169 = cbrt(r111168);
        double r111170 = r111167 * r111169;
        double r111171 = sqrt(r111168);
        double r111172 = cbrt(r111171);
        double r111173 = r111170 * r111172;
        double r111174 = r111163 + r111173;
        double r111175 = 2.0;
        double r111176 = r111174 * r111175;
        double r111177 = sqrt(r111176);
        double r111178 = r111162 * r111177;
        return r111178;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original38.9
Target33.7
Herbie15.8
\[\begin{array}{l} \mathbf{if}\;re \lt 0.0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Initial program 38.9

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

  2. Simplified14.1

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

  4. Applied add-sqr-sqrt15.3

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

  6. Applied add-sqr-sqrt15.3

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

  7. Applied sqrt-prod15.5

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

  8. Applied associate-*r*15.5

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

  9. Simplified15.8

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

  11. Applied add-cube-cbrt16.3

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

  12. Applied associate-*r*16.4

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

  13. Simplified16.0

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

  15. Applied pow1/216.0

    \[\leadsto 0.5 \cdot \sqrt{\left(re + \left({\left(\sqrt{\color{blue}{{\left(\mathsf{hypot}\left(re, im\right)\right)}^{\frac{1}{2}}}}\right)}^{3} \cdot \sqrt[3]{\sqrt{\mathsf{hypot}\left(re, im\right)}}\right) \cdot \sqrt[3]{\sqrt{\sqrt{\mathsf{hypot}\left(re, im\right)}}}\right) \cdot 2}\]

  16. Applied sqrt-pow116.0

    \[\leadsto 0.5 \cdot \sqrt{\left(re + \left({\color{blue}{\left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}}^{3} \cdot \sqrt[3]{\sqrt{\mathsf{hypot}\left(re, im\right)}}\right) \cdot \sqrt[3]{\sqrt{\sqrt{\mathsf{hypot}\left(re, im\right)}}}\right) \cdot 2}\]

  17. Applied pow-pow15.8

    \[\leadsto 0.5 \cdot \sqrt{\left(re + \left(\color{blue}{{\left(\mathsf{hypot}\left(re, im\right)\right)}^{\left(\frac{\frac{1}{2}}{2} \cdot 3\right)}} \cdot \sqrt[3]{\sqrt{\mathsf{hypot}\left(re, im\right)}}\right) \cdot \sqrt[3]{\sqrt{\sqrt{\mathsf{hypot}\left(re, im\right)}}}\right) \cdot 2}\]

  18. Simplified15.8

    \[\leadsto 0.5 \cdot \sqrt{\left(re + \left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{\color{blue}{\frac{3}{4}}} \cdot \sqrt[3]{\sqrt{\mathsf{hypot}\left(re, im\right)}}\right) \cdot \sqrt[3]{\sqrt{\sqrt{\mathsf{hypot}\left(re, im\right)}}}\right) \cdot 2}\]

  19. Final simplification15.8

    \[\leadsto 0.5 \cdot \sqrt{\left(re + \left({\left(\mathsf{hypot}\left(re, im\right)\right)}^{\frac{3}{4}} \cdot \sqrt[3]{\sqrt{\mathsf{hypot}\left(re, im\right)}}\right) \cdot \sqrt[3]{\sqrt{\sqrt{\mathsf{hypot}\left(re, im\right)}}}\right) \cdot 2}\]

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)))))