Average Error: 31.0 → 12.1
Time: 3.9s
Precision: binary64
\[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -1.511705281598505 \cdot 10^{+159}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq -2.5067818285457853 \cdot 10^{-122}:\\ \;\;\;\;\begin{array}{l} t_0 := \mathsf{fma}\left(4, y \cdot y, x \cdot x\right)\\ \mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(x, \frac{x}{t_0}, \left(y \cdot \frac{y}{t_0}\right) \cdot -4\right)\right)\right) \end{array}\\ \mathbf{elif}\;x \leq 1.9388453121044746 \cdot 10^{-153}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq 1.53922962851806 \cdot 10^{+147}:\\ \;\;\;\;\begin{array}{l} t_1 := y \cdot \left(4 \cdot y\right)\\ \frac{x \cdot x}{x \cdot x + t_1} - \frac{4 \cdot y}{\frac{\mathsf{fma}\left(x, x, t_1\right)}{y}} \end{array}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}
↓
\begin{array}{l}
\mathbf{if}\;x \leq -1.511705281598505 \cdot 10^{+159}:\\
\;\;\;\;1\\

\mathbf{elif}\;x \leq -2.5067818285457853 \cdot 10^{-122}:\\
\;\;\;\;\begin{array}{l}
t_0 := \mathsf{fma}\left(4, y \cdot y, x \cdot x\right)\\
\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(x, \frac{x}{t_0}, \left(y \cdot \frac{y}{t_0}\right) \cdot -4\right)\right)\right)
\end{array}\\

\mathbf{elif}\;x \leq 1.9388453121044746 \cdot 10^{-153}:\\
\;\;\;\;-1\\

\mathbf{elif}\;x \leq 1.53922962851806 \cdot 10^{+147}:\\
\;\;\;\;\begin{array}{l}
t_1 := y \cdot \left(4 \cdot y\right)\\
\frac{x \cdot x}{x \cdot x + t_1} - \frac{4 \cdot y}{\frac{\mathsf{fma}\left(x, x, t_1\right)}{y}}
\end{array}\\

\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
 (if (<= x -1.511705281598505e+159)
   1.0
   (if (<= x -2.5067818285457853e-122)
     (let* ((t_0 (fma 4.0 (* y y) (* x x))))
       (expm1 (log1p (fma x (/ x t_0) (* (* y (/ y t_0)) -4.0)))))
     (if (<= x 1.9388453121044746e-153)
       -1.0
       (if (<= x 1.53922962851806e+147)
         (let* ((t_1 (* y (* 4.0 y))))
           (- (/ (* x x) (+ (* x x) t_1)) (/ (* 4.0 y) (/ (fma x x t_1) y))))
         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 tmp;
	if (x <= -1.511705281598505e+159) {
		tmp = 1.0;
	} else if (x <= -2.5067818285457853e-122) {
		double t_0 = fma(4.0, (y * y), (x * x));
		tmp = expm1(log1p(fma(x, (x / t_0), ((y * (y / t_0)) * -4.0))));
	} else if (x <= 1.9388453121044746e-153) {
		tmp = -1.0;
	} else if (x <= 1.53922962851806e+147) {
		double t_1 = y * (4.0 * y);
		tmp = ((x * x) / ((x * x) + t_1)) - ((4.0 * y) / (fma(x, x, t_1) / y));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Target

Original31.0
Target30.7
Herbie12.1
\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} < 0.9743233849626781:\\ \;\;\;\;\frac{x \cdot x}{x \cdot x + \left(y \cdot y\right) \cdot 4} - \frac{\left(y \cdot y\right) \cdot 4}{x \cdot x + \left(y \cdot y\right) \cdot 4}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{x}{\sqrt{x \cdot x + \left(y \cdot y\right) \cdot 4}}\right)}^{2} - \frac{\left(y \cdot y\right) \cdot 4}{x \cdot x + \left(y \cdot y\right) \cdot 4}\\ \end{array} \]

Derivation

  1. Split input into 4 regimes

  2. if x < -1.51170528159850505e159 or 1.53922962851806007e147 < x

    1. Initial program 62.7

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

    2. Taylor expanded in x around inf 8.4

      \[\leadsto \color{blue}{1} \]

    if -1.51170528159850505e159 < x < -2.5067818285457853e-122

    1. Initial program 16.0

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

    2. Applied div-sub_binary6416.0

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

    3. Applied expm1-log1p-u_binary6416.0

      \[\leadsto \frac{x \cdot x}{x \cdot x + \left(y \cdot 4\right) \cdot y} - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y}\right)\right)} \]

    4. Simplified16.0

      \[\leadsto \frac{x \cdot x}{x \cdot x + \left(y \cdot 4\right) \cdot y} - \mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(\frac{y \cdot \left(y \cdot 4\right)}{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot 4\right)\right)}\right)}\right) \]

    5. Applied expm1-log1p-u_binary6416.0

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x \cdot x}{x \cdot x + \left(y \cdot 4\right) \cdot y} - \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{y \cdot \left(y \cdot 4\right)}{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot 4\right)\right)}\right)\right)\right)\right)} \]

    6. Simplified16.3

      \[\leadsto \mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(\mathsf{fma}\left(x, \frac{x}{\mathsf{fma}\left(4, y \cdot y, x \cdot x\right)}, \left(\frac{y}{\mathsf{fma}\left(4, y \cdot y, x \cdot x\right)} \cdot y\right) \cdot -4\right)\right)}\right) \]

    if -2.5067818285457853e-122 < x < 1.93884531210447457e-153

    1. Initial program 28.8

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

    2. Taylor expanded in x around 0 9.3

      \[\leadsto \color{blue}{-1} \]

    if 1.93884531210447457e-153 < x < 1.53922962851806007e147

    1. Initial program 15.7

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

    2. Applied div-sub_binary6415.7

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

    3. Applied associate-/l*_binary6415.2

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

    4. Simplified15.2

      \[\leadsto \frac{x \cdot x}{x \cdot x + \left(y \cdot 4\right) \cdot y} - \frac{y \cdot 4}{\color{blue}{\frac{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot 4\right)\right)}{y}}} \]
  3. Recombined 4 regimes into one program.

  4. Final simplification12.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.511705281598505 \cdot 10^{+159}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq -2.5067818285457853 \cdot 10^{-122}:\\ \;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(x, \frac{x}{\mathsf{fma}\left(4, y \cdot y, x \cdot x\right)}, \left(y \cdot \frac{y}{\mathsf{fma}\left(4, y \cdot y, x \cdot x\right)}\right) \cdot -4\right)\right)\right)\\ \mathbf{elif}\;x \leq 1.9388453121044746 \cdot 10^{-153}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq 1.53922962851806 \cdot 10^{+147}:\\ \;\;\;\;\frac{x \cdot x}{x \cdot x + y \cdot \left(4 \cdot y\right)} - \frac{4 \cdot y}{\frac{\mathsf{fma}\left(x, x, y \cdot \left(4 \cdot y\right)\right)}{y}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Reproduce

herbie shell --seed 2022088 
(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))))