Average Error: 13.5 → 0.1
Time: 4.9s
Precision: binary64
\[10^{-150} < \left|x\right| \land \left|x\right| < 10^{+150}\]
\[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \]
↓
\[\begin{array}{l} t_0 := \frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}}\\ \mathbf{if}\;t_0 \leq -0.9999999999987923:\\ \;\;\;\;\left|\frac{p}{x}\right|\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \log \left(e^{t_0 + 1}\right)}\\ \end{array} \]
\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}
↓
\begin{array}{l}
t_0 := \frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}}\\
\mathbf{if}\;t_0 \leq -0.9999999999987923:\\
\;\;\;\;\left|\frac{p}{x}\right|\\

\mathbf{else}:\\
\;\;\;\;\sqrt{0.5 \cdot \log \left(e^{t_0 + 1}\right)}\\


\end{array}

(FPCore (p x)
 :precision binary64
 (sqrt (* 0.5 (+ 1.0 (/ x (sqrt (+ (* (* 4.0 p) p) (* x x))))))))
↓
(FPCore (p x)
 :precision binary64
 (let* ((t_0 (/ x (sqrt (+ (* p (* 4.0 p)) (* x x))))))
   (if (<= t_0 -0.9999999999987923)
     (fabs (/ p x))
     (sqrt (* 0.5 (log (exp (+ t_0 1.0))))))))
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) {
	double t_0 = x / sqrt((p * (4.0 * p)) + (x * x));
	double tmp;
	if (t_0 <= -0.9999999999987923) {
		tmp = fabs(p / x);
	} else {
		tmp = sqrt(0.5 * log(exp(t_0 + 1.0)));
	}
	return tmp;
}

Error

Bits error versus p

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original13.5
Target13.5
Herbie0.1
\[\sqrt{0.5 + \frac{\mathsf{copysign}\left(0.5, x\right)}{\mathsf{hypot}\left(1, \frac{2 \cdot p}{x}\right)}} \]

Derivation

  1. Split input into 2 regimes

  2. if (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 4 p) p) (*.f64 x x)))) < -0.9999999999987923

    1. Initial program 53.5

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

    2. Taylor expanded around -inf 31.2

      \[\leadsto \sqrt{0.5 \cdot \color{blue}{\left(2 \cdot \frac{{p}^{2}}{{x}^{2}}\right)}} \]

    3. Simplified22.9

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

    5. Applied add-sqr-sqrt_binary6423.1

      \[\leadsto \color{blue}{\sqrt{\sqrt{0.5 \cdot \left(2 \cdot \frac{p}{\frac{x \cdot x}{p}}\right)}} \cdot \sqrt{\sqrt{0.5 \cdot \left(2 \cdot \frac{p}{\frac{x \cdot x}{p}}\right)}}} \]

    6. Simplified23.0

      \[\leadsto \color{blue}{\sqrt{\left|\frac{p}{x}\right|}} \cdot \sqrt{\sqrt{0.5 \cdot \left(2 \cdot \frac{p}{\frac{x \cdot x}{p}}\right)}} \]

    7. Simplified0.5

      \[\leadsto \sqrt{\left|\frac{p}{x}\right|} \cdot \color{blue}{\sqrt{\left|\frac{p}{x}\right|}} \]
    8. Using strategy rm

    9. Applied rem-square-sqrt_binary640.0

      \[\leadsto \color{blue}{\left|\frac{p}{x}\right|} \]

    if -0.9999999999987923 < (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 4 p) p) (*.f64 x x))))

    1. Initial program 0.1

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

    3. Applied add-log-exp_binary640.1

      \[\leadsto \sqrt{0.5 \cdot \color{blue}{\log \left(e^{1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}}\right)}} \]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -0.9999999999987923:\\ \;\;\;\;\left|\frac{p}{x}\right|\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot \log \left(e^{\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} + 1}\right)}\\ \end{array} \]

Reproduce

herbie shell --seed 2021205 
(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))))))))