Average Error: 13.1 → 13.4
Time: 9.9s
Precision: binary64
\[1.00000000000000001 \cdot 10^{-150} \lt \left|x\right| \lt 9.99999999999999981 \cdot 10^{149}\]
\[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\]
↓
\[\sqrt[3]{{\left(\sqrt{0.5 \cdot \left(2 \cdot \log \left(\sqrt[3]{e^{1 + \frac{x}{\left|\sqrt[3]{\left(4 \cdot p\right) \cdot p + x \cdot x}\right| \cdot \sqrt{\sqrt[3]{\left(4 \cdot p\right) \cdot p + x \cdot x}}}}}\right) + \log \left(\sqrt[3]{e^{1 + \frac{x}{\left|\sqrt[3]{\left(4 \cdot p\right) \cdot p + x \cdot x}\right| \cdot \sqrt{\sqrt[3]{\left(4 \cdot p\right) \cdot p + x \cdot x}}}}}\right)\right)}\right)}^{3}}\]
\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}
↓
\sqrt[3]{{\left(\sqrt{0.5 \cdot \left(2 \cdot \log \left(\sqrt[3]{e^{1 + \frac{x}{\left|\sqrt[3]{\left(4 \cdot p\right) \cdot p + x \cdot x}\right| \cdot \sqrt{\sqrt[3]{\left(4 \cdot p\right) \cdot p + x \cdot x}}}}}\right) + \log \left(\sqrt[3]{e^{1 + \frac{x}{\left|\sqrt[3]{\left(4 \cdot p\right) \cdot p + x \cdot x}\right| \cdot \sqrt{\sqrt[3]{\left(4 \cdot p\right) \cdot p + x \cdot x}}}}}\right)\right)}\right)}^{3}}
double code(double p, double x) {
	return ((double) sqrt(((double) (0.5 * ((double) (1.0 + ((double) (x / ((double) sqrt(((double) (((double) (((double) (4.0 * p)) * p)) + ((double) (x * x))))))))))))));
}
↓
double code(double p, double x) {
	return ((double) cbrt(((double) pow(((double) sqrt(((double) (0.5 * ((double) (((double) (2.0 * ((double) log(((double) cbrt(((double) exp(((double) (1.0 + ((double) (x / ((double) (((double) fabs(((double) cbrt(((double) (((double) (((double) (4.0 * p)) * p)) + ((double) (x * x)))))))) * ((double) sqrt(((double) cbrt(((double) (((double) (((double) (4.0 * p)) * p)) + ((double) (x * x)))))))))))))))))))))) + ((double) log(((double) cbrt(((double) exp(((double) (1.0 + ((double) (x / ((double) (((double) fabs(((double) cbrt(((double) (((double) (((double) (4.0 * p)) * p)) + ((double) (x * x)))))))) * ((double) sqrt(((double) cbrt(((double) (((double) (((double) (4.0 * p)) * p)) + ((double) (x * x)))))))))))))))))))))))))), 3.0))));
}

Error

Bits error versus p

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original13.1
Target13.1
Herbie13.4
\[\sqrt{0.5 + \frac{\mathsf{copysign}\left(0.5, x\right)}{\mathsf{hypot}\left(1, \frac{2 \cdot p}{x}\right)}}\]

Derivation

  1. Initial program 13.1

    \[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\]
  2. Using strategy rm

  3. Applied add-cbrt-cube13.1

    \[\leadsto \color{blue}{\sqrt[3]{\left(\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \cdot \sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\right) \cdot \sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}}}\]

  4. Simplified13.1

    \[\leadsto \sqrt[3]{\color{blue}{{\left(\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\right)}^{3}}}\]
  5. Using strategy rm

  6. Applied add-cube-cbrt14.4

    \[\leadsto \sqrt[3]{{\left(\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\color{blue}{\left(\sqrt[3]{\left(4 \cdot p\right) \cdot p + x \cdot x} \cdot \sqrt[3]{\left(4 \cdot p\right) \cdot p + x \cdot x}\right) \cdot \sqrt[3]{\left(4 \cdot p\right) \cdot p + x \cdot x}}}}\right)}\right)}^{3}}\]

  7. Applied sqrt-prod14.4

    \[\leadsto \sqrt[3]{{\left(\sqrt{0.5 \cdot \left(1 + \frac{x}{\color{blue}{\sqrt{\sqrt[3]{\left(4 \cdot p\right) \cdot p + x \cdot x} \cdot \sqrt[3]{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot \sqrt{\sqrt[3]{\left(4 \cdot p\right) \cdot p + x \cdot x}}}}\right)}\right)}^{3}}\]

  8. Simplified14.4

    \[\leadsto \sqrt[3]{{\left(\sqrt{0.5 \cdot \left(1 + \frac{x}{\color{blue}{\left|\sqrt[3]{\left(4 \cdot p\right) \cdot p + x \cdot x}\right|} \cdot \sqrt{\sqrt[3]{\left(4 \cdot p\right) \cdot p + x \cdot x}}}\right)}\right)}^{3}}\]
  9. Using strategy rm

  10. Applied add-log-exp14.4

    \[\leadsto \sqrt[3]{{\left(\sqrt{0.5 \cdot \left(1 + \color{blue}{\log \left(e^{\frac{x}{\left|\sqrt[3]{\left(4 \cdot p\right) \cdot p + x \cdot x}\right| \cdot \sqrt{\sqrt[3]{\left(4 \cdot p\right) \cdot p + x \cdot x}}}}\right)}\right)}\right)}^{3}}\]

  11. Applied add-log-exp14.4

    \[\leadsto \sqrt[3]{{\left(\sqrt{0.5 \cdot \left(\color{blue}{\log \left(e^{1}\right)} + \log \left(e^{\frac{x}{\left|\sqrt[3]{\left(4 \cdot p\right) \cdot p + x \cdot x}\right| \cdot \sqrt{\sqrt[3]{\left(4 \cdot p\right) \cdot p + x \cdot x}}}}\right)\right)}\right)}^{3}}\]

  12. Applied sum-log14.5

    \[\leadsto \sqrt[3]{{\left(\sqrt{0.5 \cdot \color{blue}{\log \left(e^{1} \cdot e^{\frac{x}{\left|\sqrt[3]{\left(4 \cdot p\right) \cdot p + x \cdot x}\right| \cdot \sqrt{\sqrt[3]{\left(4 \cdot p\right) \cdot p + x \cdot x}}}}\right)}}\right)}^{3}}\]

  13. Simplified14.4

    \[\leadsto \sqrt[3]{{\left(\sqrt{0.5 \cdot \log \color{blue}{\left(e^{1 + \frac{x}{\left|\sqrt[3]{\left(4 \cdot p\right) \cdot p + x \cdot x}\right| \cdot \sqrt{\sqrt[3]{\left(4 \cdot p\right) \cdot p + x \cdot x}}}}\right)}}\right)}^{3}}\]
  14. Using strategy rm

  15. Applied add-cube-cbrt13.4

    \[\leadsto \sqrt[3]{{\left(\sqrt{0.5 \cdot \log \color{blue}{\left(\left(\sqrt[3]{e^{1 + \frac{x}{\left|\sqrt[3]{\left(4 \cdot p\right) \cdot p + x \cdot x}\right| \cdot \sqrt{\sqrt[3]{\left(4 \cdot p\right) \cdot p + x \cdot x}}}}} \cdot \sqrt[3]{e^{1 + \frac{x}{\left|\sqrt[3]{\left(4 \cdot p\right) \cdot p + x \cdot x}\right| \cdot \sqrt{\sqrt[3]{\left(4 \cdot p\right) \cdot p + x \cdot x}}}}}\right) \cdot \sqrt[3]{e^{1 + \frac{x}{\left|\sqrt[3]{\left(4 \cdot p\right) \cdot p + x \cdot x}\right| \cdot \sqrt{\sqrt[3]{\left(4 \cdot p\right) \cdot p + x \cdot x}}}}}\right)}}\right)}^{3}}\]

  16. Applied log-prod13.4

    \[\leadsto \sqrt[3]{{\left(\sqrt{0.5 \cdot \color{blue}{\left(\log \left(\sqrt[3]{e^{1 + \frac{x}{\left|\sqrt[3]{\left(4 \cdot p\right) \cdot p + x \cdot x}\right| \cdot \sqrt{\sqrt[3]{\left(4 \cdot p\right) \cdot p + x \cdot x}}}}} \cdot \sqrt[3]{e^{1 + \frac{x}{\left|\sqrt[3]{\left(4 \cdot p\right) \cdot p + x \cdot x}\right| \cdot \sqrt{\sqrt[3]{\left(4 \cdot p\right) \cdot p + x \cdot x}}}}}\right) + \log \left(\sqrt[3]{e^{1 + \frac{x}{\left|\sqrt[3]{\left(4 \cdot p\right) \cdot p + x \cdot x}\right| \cdot \sqrt{\sqrt[3]{\left(4 \cdot p\right) \cdot p + x \cdot x}}}}}\right)\right)}}\right)}^{3}}\]

  17. Simplified13.4

    \[\leadsto \sqrt[3]{{\left(\sqrt{0.5 \cdot \left(\color{blue}{2 \cdot \log \left(\sqrt[3]{e^{1 + \frac{x}{\left|\sqrt[3]{\left(4 \cdot p\right) \cdot p + x \cdot x}\right| \cdot \sqrt{\sqrt[3]{\left(4 \cdot p\right) \cdot p + x \cdot x}}}}}\right)} + \log \left(\sqrt[3]{e^{1 + \frac{x}{\left|\sqrt[3]{\left(4 \cdot p\right) \cdot p + x \cdot x}\right| \cdot \sqrt{\sqrt[3]{\left(4 \cdot p\right) \cdot p + x \cdot x}}}}}\right)\right)}\right)}^{3}}\]

  18. Final simplification13.4

    \[\leadsto \sqrt[3]{{\left(\sqrt{0.5 \cdot \left(2 \cdot \log \left(\sqrt[3]{e^{1 + \frac{x}{\left|\sqrt[3]{\left(4 \cdot p\right) \cdot p + x \cdot x}\right| \cdot \sqrt{\sqrt[3]{\left(4 \cdot p\right) \cdot p + x \cdot x}}}}}\right) + \log \left(\sqrt[3]{e^{1 + \frac{x}{\left|\sqrt[3]{\left(4 \cdot p\right) \cdot p + x \cdot x}\right| \cdot \sqrt{\sqrt[3]{\left(4 \cdot p\right) \cdot p + x \cdot x}}}}}\right)\right)}\right)}^{3}}\]

Reproduce

herbie shell --seed 2020148 
(FPCore (p x)
  :name "Given's Rotation SVD example"
  :precision binary64
  :pre (< 1e-150 (fabs x) 1e+150)

  :herbie-target
  (sqrt (+ 0.5 (/ (copysign 0.5 x) (hypot 1.0 (/ (* 2.0 p) x)))))

  (sqrt (* 0.5 (+ 1.0 (/ x (sqrt (+ (* (* 4.0 p) p) (* x x))))))))