Average Error: 31.6 → 12.0
Time: 2.5s
Precision: binary64
\[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
↓
\[\begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ \mathbf{if}\;x \cdot x \leq 1.8455816858840264 \cdot 10^{-266}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{x}{y}\right)}^{2}\right)\right), -1\right)\\ \mathbf{elif}\;x \cdot x \leq 2.087941773223886 \cdot 10^{+218}:\\ \;\;\;\;\frac{x \cdot x - t_0}{x \cdot x + t_0}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}
↓
\begin{array}{l}
t_0 := y \cdot \left(y \cdot 4\right)\\
\mathbf{if}\;x \cdot x \leq 1.8455816858840264 \cdot 10^{-266}:\\
\;\;\;\;\mathsf{fma}\left(0.5, \mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{x}{y}\right)}^{2}\right)\right), -1\right)\\

\mathbf{elif}\;x \cdot x \leq 2.087941773223886 \cdot 10^{+218}:\\
\;\;\;\;\frac{x \cdot x - t_0}{x \cdot x + t_0}\\

\mathbf{else}:\\
\;\;\;\;1\\


\end{array}

(FPCore (x y)
 :precision binary64
 (/ (- (* x x) (* (* y 4.0) y)) (+ (* x x) (* (* y 4.0) y))))
↓
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* y 4.0))))
   (if (<= (* x x) 1.8455816858840264e-266)
     (fma 0.5 (expm1 (log1p (pow (/ x y) 2.0))) -1.0)
     (if (<= (* x x) 2.087941773223886e+218)
       (/ (- (* x x) t_0) (+ (* x x) t_0))
       1.0))))
double code(double x, double y) {
	return ((x * x) - ((y * 4.0) * y)) / ((x * x) + ((y * 4.0) * y));
}
↓
double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double tmp;
	if ((x * x) <= 1.8455816858840264e-266) {
		tmp = fma(0.5, expm1(log1p(pow((x / y), 2.0))), -1.0);
	} else if ((x * x) <= 2.087941773223886e+218) {
		tmp = ((x * x) - t_0) / ((x * x) + t_0);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Target

Original31.6
Target31.2
Herbie12.0
\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} < 0.9743233849626781:\\ \;\;\;\;\frac{x \cdot x}{x \cdot x + \left(y \cdot y\right) \cdot 4} - \frac{\left(y \cdot y\right) \cdot 4}{x \cdot x + \left(y \cdot y\right) \cdot 4}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{x}{\sqrt{x \cdot x + \left(y \cdot y\right) \cdot 4}}\right)}^{2} - \frac{\left(y \cdot y\right) \cdot 4}{x \cdot x + \left(y \cdot y\right) \cdot 4}\\ \end{array} \]

Derivation

  1. Split input into 3 regimes

  2. if (*.f64 x x) < 1.8455816858840264e-266

    1. Initial program 28.9

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

    2. Simplified29.1

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-4, y \cdot y, x \cdot x\right)}{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot 4\right)\right)}} \]

    3. Taylor expanded in y around inf 14.5

      \[\leadsto \color{blue}{0.5 \cdot \frac{{x}^{2}}{{y}^{2}} - 1} \]

    4. Simplified14.5

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \frac{x \cdot x}{y \cdot y}, -1\right)} \]

    5. Applied egg-rr7.5

      \[\leadsto \mathsf{fma}\left(0.5, \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{x}{y}\right)}^{2}\right)\right)}, -1\right) \]

    if 1.8455816858840264e-266 < (*.f64 x x) < 2.0879417732238861e218

    1. Initial program 16.3

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

    if 2.0879417732238861e218 < (*.f64 x x)

    1. Initial program 52.9

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

    2. Simplified52.9

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-4, y \cdot y, x \cdot x\right)}{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot 4\right)\right)}} \]

    3. Taylor expanded in y around 0 10.6

      \[\leadsto \color{blue}{1} \]
  3. Recombined 3 regimes into one program.

  4. Final simplification12.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 1.8455816858840264 \cdot 10^{-266}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{x}{y}\right)}^{2}\right)\right), -1\right)\\ \mathbf{elif}\;x \cdot x \leq 2.087941773223886 \cdot 10^{+218}:\\ \;\;\;\;\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{x \cdot x + y \cdot \left(y \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Reproduce

herbie shell --seed 2022130 
(FPCore (x y)
  :name "Diagrams.TwoD.Arc:arcBetween from diagrams-lib-1.3.0.3"
  :precision binary64

  :herbie-target
  (if (< (/ (- (* x x) (* (* y 4.0) y)) (+ (* x x) (* (* y 4.0) y))) 0.9743233849626781) (- (/ (* x x) (+ (* x x) (* (* y y) 4.0))) (/ (* (* y y) 4.0) (+ (* x x) (* (* y y) 4.0)))) (- (pow (/ x (sqrt (+ (* x x) (* (* y y) 4.0)))) 2.0) (/ (* (* y y) 4.0) (+ (* x x) (* (* y y) 4.0)))))

  (/ (- (* x x) (* (* y 4.0) y)) (+ (* x x) (* (* y 4.0) y))))