Average Error: 13.2 → 14.2
Time: 5.6s
Precision: 64
\[1.00000000000000001 \cdot 10^{-150} \lt \left|x\right| \lt 9.99999999999999981 \cdot 10^{149}\]
\[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\]
↓
\[\log \left(e^{\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot \sqrt{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}}\right)}}\right)\]
\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}
↓
\log \left(e^{\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}} \cdot \sqrt{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}}\right)}}\right)
double code(double p, double x) {
	return sqrt((0.5 * (1.0 + (x / sqrt((((4.0 * p) * p) + (x * x)))))));
}
↓
double code(double p, double x) {
	return log(exp(sqrt((0.5 * (1.0 + (x / (sqrt(sqrt((((4.0 * p) * p) + (x * x)))) * sqrt(sqrt((((4.0 * p) * p) + (x * x)))))))))));
}

Error

Bits error versus p

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original13.2
Target13.2
Herbie14.2
\[\sqrt{0.5 + \frac{\mathsf{copysign}\left(0.5, x\right)}{\mathsf{hypot}\left(1, \frac{2 \cdot p}{x}\right)}}\]

Derivation

  1. Initial program 13.2

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

  3. Applied add-log-exp13.2

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

  4. Applied add-log-exp13.2

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

  5. Applied sum-log13.2

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

  6. Simplified13.2

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

  8. Applied add-log-exp13.2

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

  10. Applied add-sqr-sqrt13.2

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

  11. Applied sqrt-prod14.2

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

  13. Applied exp-sum14.2

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

  14. Applied log-prod14.2

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

  15. Simplified14.2

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

  16. Simplified14.2

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

  17. Final simplification14.2

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

Reproduce

herbie shell --seed 2020106 +o rules:numerics
(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 (/ (* 2 p) x)))))

  (sqrt (* 0.5 (+ 1 (/ x (sqrt (+ (* (* 4 p) p) (* x x))))))))