Average Error: 13.5 → 6.0
Time: 4.3s
Precision: binary64
\[10^{-150} < \left|x\right| \land \left|x\right| < 10^{+150}\]
\[\sqrt{0.5 \cdot \left(1 + \frac{x}{\sqrt{\left(4 \cdot p\right) \cdot p + x \cdot x}}\right)} \]
↓
\[\begin{array}{l} t_0 := p \cdot \left(4 \cdot p\right)\\ \mathbf{if}\;\frac{x}{\sqrt{t_0 + x \cdot x}} \leq -1:\\ \;\;\;\;\sqrt{\frac{p}{\frac{x \cdot x}{p}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\log \left(e^{\mathsf{fma}\left(0.5, \frac{x}{\sqrt{\mathsf{fma}\left(x, x, t_0\right)}}, 0.5\right)}\right)}\\ \end{array} \]
\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 := p \cdot \left(4 \cdot p\right)\\
\mathbf{if}\;\frac{x}{\sqrt{t_0 + x \cdot x}} \leq -1:\\
\;\;\;\;\sqrt{\frac{p}{\frac{x \cdot x}{p}}}\\

\mathbf{else}:\\
\;\;\;\;\sqrt{\log \left(e^{\mathsf{fma}\left(0.5, \frac{x}{\sqrt{\mathsf{fma}\left(x, x, t_0\right)}}, 0.5\right)}\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 (* p (* 4.0 p))))
   (if (<= (/ x (sqrt (+ t_0 (* x x)))) -1.0)
     (sqrt (/ p (/ (* x x) p)))
     (sqrt (log (exp (fma 0.5 (/ x (sqrt (fma x x t_0))) 0.5)))))))
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 = p * (4.0 * p);
	double tmp;
	if ((x / sqrt(t_0 + (x * x))) <= -1.0) {
		tmp = sqrt(p / ((x * x) / p));
	} else {
		tmp = sqrt(log(exp(fma(0.5, (x / sqrt(fma(x, x, t_0))), 0.5))));
	}
	return tmp;
}

Error

Bits error versus p

Bits error versus x

Target

Original13.5
Target13.5
Herbie6.0
\[\sqrt{0.5 + \frac{\mathsf{copysign}\left(0.5, x\right)}{\mathsf{hypot}\left(1, \frac{2 \cdot p}{x}\right)}} \]

Derivation

  1. Split input into 2 regimes

  2. if (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 4 p) p) (*.f64 x x)))) < -1

    1. Initial program 54.1

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

    2. Simplified54.1

      \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(0.5, \frac{x}{\sqrt{\mathsf{fma}\left(x, x, p \cdot \left(4 \cdot p\right)\right)}}, 0.5\right)}} \]

    3. Taylor expanded in x around -inf 30.7

      \[\leadsto \sqrt{\color{blue}{\frac{{p}^{2}}{{x}^{2}}}} \]

    4. Simplified23.6

      \[\leadsto \sqrt{\color{blue}{\frac{p}{\frac{x \cdot x}{p}}}} \]

    if -1 < (/.f64 x (sqrt.f64 (+.f64 (*.f64 (*.f64 4 p) p) (*.f64 x x))))

    1. Initial program 0.2

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

    2. Simplified0.2

      \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(0.5, \frac{x}{\sqrt{\mathsf{fma}\left(x, x, p \cdot \left(4 \cdot p\right)\right)}}, 0.5\right)}} \]

    3. Applied add-log-exp_binary640.2

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

  4. Final simplification6.0

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

Reproduce

herbie shell --seed 2022067 
(FPCore (p x)
  :name "Given's Rotation SVD example"
  :precision binary64
  :pre (and (< 1e-150 (fabs x)) (< (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))))))))