Average Error: 13.4 → 5.6
Time: 5.3s
Precision: binary64
\[10^{-150} < \left|x\right| \land \left|x\right| < 10^{+150}\]
\[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -0.9999999996172134:\\ \;\;\;\;\sqrt{0.5 \cdot \left(2 \cdot {\left(\frac{-1}{x} \cdot \left(-p\right)\right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot e^{\log \left(\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} + 1\right)}}\\ \end{array}\]
\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)}
↓
\begin{array}{l}
\mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -0.9999999996172134:\\
\;\;\;\;\sqrt{0.5 \cdot \left(2 \cdot {\left(\frac{-1}{x} \cdot \left(-p\right)\right)}^{2}\right)}\\

\mathbf{else}:\\
\;\;\;\;\sqrt{0.5 \cdot e^{\log \left(\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} + 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
 (if (<= (/ x (sqrt (+ (* p (* 4.0 p)) (* x x)))) -0.9999999996172134)
   (sqrt (* 0.5 (* 2.0 (pow (* (/ -1.0 x) (- p)) 2.0))))
   (sqrt
    (* 0.5 (exp (log (+ (/ x (sqrt (+ (* p (* 4.0 p)) (* x x)))) 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 tmp;
	if ((x / sqrt((p * (4.0 * p)) + (x * x))) <= -0.9999999996172134) {
		tmp = sqrt(0.5 * (2.0 * pow(((-1.0 / x) * -p), 2.0)));
	} else {
		tmp = sqrt(0.5 * exp(log((x / sqrt((p * (4.0 * p)) + (x * x))) + 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

Original13.4
Target13.4
Herbie5.6
\[\sqrt{0.5 + \frac{\mathsf{copysign}\left(0.5, x\right)}{\mathsf{hypot}\left(1, \frac{2 \cdot p}{x}\right)}}\]

Derivation

  1. Split input into 2 regimes

  2. if (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 4 p) p) (*.f64 x x)))) < -0.99999999961721342

    1. Initial program 53.7

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

    3. Applied add-exp-log_binary64_216253.7

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

    4. Simplified53.7

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

    5. Taylor expanded around -inf 44.9

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

    6. Simplified22.3

      \[\leadsto \sqrt{0.5 \cdot \color{blue}{\left(2 \cdot {\left(\frac{-1}{x} \cdot \left(-p\right)\right)}^{2}\right)}}\]

    if -0.99999999961721342 < (/.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-exp-log_binary64_21620.1

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

    4. Simplified0.1

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

  4. Final simplification5.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} \leq -0.9999999996172134:\\ \;\;\;\;\sqrt{0.5 \cdot \left(2 \cdot {\left(\frac{-1}{x} \cdot \left(-p\right)\right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot e^{\log \left(\frac{x}{\sqrt{p \cdot \left(4 \cdot p\right) + x \cdot x}} + 1\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020346 
(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))))))))