Average Error: 20.5 → 5.2
Time: 2.2s
Precision: 64
\[0.0 \lt x \lt 1 \land y \lt 1\]
\[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -1.10115633480067138 \cdot 10^{153}:\\ \;\;\;\;\frac{x + y}{-\left(x + y\right)}\\ \mathbf{elif}\;y \le -4.6546003190309958 \cdot 10^{-157}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{elif}\;y \le 3.32161448822691038 \cdot 10^{-168}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \end{array}\]
\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}
↓
\begin{array}{l}
\mathbf{if}\;y \le -1.10115633480067138 \cdot 10^{153}:\\
\;\;\;\;\frac{x + y}{-\left(x + y\right)}\\

\mathbf{elif}\;y \le -4.6546003190309958 \cdot 10^{-157}:\\
\;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\

\mathbf{elif}\;y \le 3.32161448822691038 \cdot 10^{-168}:\\
\;\;\;\;1\\

\mathbf{else}:\\
\;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\

\end{array}
double code(double x, double y) {
	return (((x - y) * (x + y)) / ((x * x) + (y * y)));
}
↓
double code(double x, double y) {
	double VAR;
	if ((y <= -1.1011563348006714e+153)) {
		VAR = ((x + y) / -(x + y));
	} else {
		double VAR_1;
		if ((y <= -4.654600319030996e-157)) {
			VAR_1 = (((x - y) * (x + y)) / ((x * x) + (y * y)));
		} else {
			double VAR_2;
			if ((y <= 3.3216144882269104e-168)) {
				VAR_2 = 1.0;
			} else {
				VAR_2 = (((x - y) * (x + y)) / ((x * x) + (y * y)));
			}
			VAR_1 = VAR_2;
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original20.5
Target0.1
Herbie5.2
\[\begin{array}{l} \mathbf{if}\;0.5 \lt \left|\frac{x}{y}\right| \lt 2:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{2}{1 + \frac{x}{y} \cdot \frac{x}{y}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if y < -1.1011563348006714e+153

    1. Initial program 63.7

      \[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\]
    2. Using strategy rm

    3. Applied *-commutative63.7

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

    4. Applied associate-/l*61.7

      \[\leadsto \color{blue}{\frac{x + y}{\frac{x \cdot x + y \cdot y}{x - y}}}\]

    5. Simplified61.7

      \[\leadsto \frac{x + y}{\color{blue}{\frac{{x}^{2} + {y}^{2}}{x - y}}}\]

    6. Taylor expanded around 0 0

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

    if -1.1011563348006714e+153 < y < -4.654600319030996e-157 or 3.3216144882269104e-168 < y

    1. Initial program 0.3

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

    if -4.654600319030996e-157 < y < 3.3216144882269104e-168

    1. Initial program 30.2

      \[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\]
    2. Using strategy rm

    3. Applied *-commutative30.2

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

    4. Applied associate-/l*31.0

      \[\leadsto \color{blue}{\frac{x + y}{\frac{x \cdot x + y \cdot y}{x - y}}}\]

    5. Simplified31.0

      \[\leadsto \frac{x + y}{\color{blue}{\frac{{x}^{2} + {y}^{2}}{x - y}}}\]
    6. Using strategy rm

    7. Applied flip--30.2

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

    8. Applied associate-/r/30.2

      \[\leadsto \frac{x + y}{\color{blue}{\frac{{x}^{2} + {y}^{2}}{x \cdot x - y \cdot y} \cdot \left(x + y\right)}}\]

    9. Simplified31.0

      \[\leadsto \frac{x + y}{\color{blue}{\frac{\frac{{x}^{2} + {y}^{2}}{x - y}}{x + y}} \cdot \left(x + y\right)}\]

    10. Taylor expanded around inf 16.1

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

  4. Final simplification5.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -1.10115633480067138 \cdot 10^{153}:\\ \;\;\;\;\frac{x + y}{-\left(x + y\right)}\\ \mathbf{elif}\;y \le -4.6546003190309958 \cdot 10^{-157}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{elif}\;y \le 3.32161448822691038 \cdot 10^{-168}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \end{array}\]

Reproduce

herbie shell --seed 2020078 
(FPCore (x y)
  :name "Kahan p9 Example"
  :precision binary64
  :pre (and (< 0.0 x 1) (< y 1))

  :herbie-target
  (if (< 0.5 (fabs (/ x y)) 2) (/ (* (- x y) (+ x y)) (+ (* x x) (* y y))) (- 1 (/ 2 (+ 1 (* (/ x y) (/ x y))))))

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