Average Error: 13.3 → 14.3
Time: 8.4s
Precision: binary64
\[10^{-150} < \left|x\right| \land \left|x\right| < 10^{+150}\]
\[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\]
↓
\[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\sqrt{p \cdot \left(p \cdot 4\right) + x \cdot x}} \cdot \sqrt{\sqrt{p \cdot \left(p \cdot 4\right) + x \cdot x}}}\right)}\]
\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}
↓
\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\sqrt{p \cdot \left(p \cdot 4\right) + x \cdot x}} \cdot \sqrt{\sqrt{p \cdot \left(p \cdot 4\right) + x \cdot x}}}\right)}
(FPCore (p x)
 :precision binary64
 (sqrt (* 0.5 (+ 1.0 (/ x (sqrt (+ (* (* 4.0 p) p) (* x x))))))))
↓
(FPCore (p x)
 :precision binary64
 (sqrt
  (*
   0.5
   (+
    1.0
    (/
     x
     (*
      (sqrt (sqrt (+ (* p (* p 4.0)) (* x x))))
      (sqrt (sqrt (+ (* p (* p 4.0)) (* x x))))))))))
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 * (1.0 + (x / (sqrt(sqrt((p * (p * 4.0)) + (x * x))) * sqrt(sqrt((p * (p * 4.0)) + (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

Original13.3
Target13.3
Herbie14.3
\[\sqrt{0.5 + \frac{\mathsf{copysign}\left(0.5, x\right)}{\mathsf{hypot}\left(1, \frac{2 \cdot p}{x}\right)}}\]

Derivation

  1. Initial program 13.3

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

  3. Applied add-sqr-sqrt_binary64_184314.3

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

  4. Simplified14.3

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

  5. Simplified14.3

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

  6. Final simplification14.3

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

Reproduce

herbie shell --seed 2020231 
(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))))))))