Average Error: 13.0 → 13.0
Time: 5.9s
Precision: 64
\[1.00000000000000001 \cdot 10^{-150} \lt \left|x\right| \lt 9.99999999999999981 \cdot 10^{149}\]
\[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\]
↓
\[\sqrt{0.5 \cdot \log \left(e^{\frac{{1}^{3} + {\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}^{3}}{\mathsf{fma}\left(1, 1 - \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}, \frac{{x}^{2}}{\mathsf{fma}\left(4 \cdot p, p, x \cdot x\right)}\right)}}\right)}\]
\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}
↓
\sqrt{0.5 \cdot \log \left(e^{\frac{{1}^{3} + {\left(\frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}^{3}}{\mathsf{fma}\left(1, 1 - \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}, \frac{{x}^{2}}{\mathsf{fma}\left(4 \cdot p, p, x \cdot x\right)}\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 sqrt((0.5 * log(exp(((pow(1.0, 3.0) + pow((x / sqrt((((4.0 * p) * p) + (x * x)))), 3.0)) / fma(1.0, (1.0 - (x / sqrt((((4.0 * p) * p) + (x * x))))), (pow(x, 2.0) / fma((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

Original13.0
Target13.0
Herbie13.0
\[\sqrt{0.5 + \frac{\mathsf{copysign}\left(0.5, x\right)}{\mathsf{hypot}\left(1, \frac{2 \cdot p}{x}\right)}}\]

Derivation

  1. Initial program 13.0

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

  3. Applied add-log-exp13.0

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

  4. Applied add-log-exp13.0

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

  5. Applied sum-log13.0

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

  6. Simplified13.0

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

  8. Applied flip3-+13.0

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

  9. Simplified13.0

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

  10. Final simplification13.0

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

Reproduce

herbie shell --seed 2020053 +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))))))))