Average Error: 20.2 → 4.8
Time: 2.9s
Precision: binary64
\[0.0 \lt x \lt 1 \land y \lt 1\]
\[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -8.15983082784838867 \cdot 10^{152}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le -5.8412403973205966 \cdot 10^{-164}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{elif}\;y \le -4.5550794910755097 \cdot 10^{-167}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le 1.87266605808828046 \cdot 10^{-175}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \le 1.23673946718731825 \cdot 10^{-162}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\log \left({\left(e^{1}\right)}^{\left(\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\right)}\right)\\ \end{array}\]
\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}
↓
\begin{array}{l}
\mathbf{if}\;y \le -8.15983082784838867 \cdot 10^{152}:\\
\;\;\;\;-1\\

\mathbf{elif}\;y \le -5.8412403973205966 \cdot 10^{-164}:\\
\;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\

\mathbf{elif}\;y \le -4.5550794910755097 \cdot 10^{-167}:\\
\;\;\;\;-1\\

\mathbf{elif}\;y \le 1.87266605808828046 \cdot 10^{-175}:\\
\;\;\;\;1\\

\mathbf{elif}\;y \le 1.23673946718731825 \cdot 10^{-162}:\\
\;\;\;\;-1\\

\mathbf{else}:\\
\;\;\;\;\log \left({\left(e^{1}\right)}^{\left(\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\right)}\right)\\

\end{array}
double code(double x, double y) {
	return ((double) (((double) (((double) (x - y)) * ((double) (x + y)))) / ((double) (((double) (x * x)) + ((double) (y * y))))));
}
↓
double code(double x, double y) {
	double VAR;
	if ((y <= -8.159830827848389e+152)) {
		VAR = -1.0;
	} else {
		double VAR_1;
		if ((y <= -5.841240397320597e-164)) {
			VAR_1 = ((double) (((double) (((double) (x - y)) * ((double) (x + y)))) / ((double) (((double) (x * x)) + ((double) (y * y))))));
		} else {
			double VAR_2;
			if ((y <= -4.55507949107551e-167)) {
				VAR_2 = -1.0;
			} else {
				double VAR_3;
				if ((y <= 1.8726660580882805e-175)) {
					VAR_3 = 1.0;
				} else {
					double VAR_4;
					if ((y <= 1.2367394671873183e-162)) {
						VAR_4 = -1.0;
					} else {
						VAR_4 = ((double) log(((double) pow(((double) exp(1.0)), ((double) (((double) (((double) (x - y)) * ((double) (x + y)))) / ((double) (((double) (x * x)) + ((double) (y * y))))))))));
					}
					VAR_3 = VAR_4;
				}
				VAR_2 = VAR_3;
			}
			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

Original20.2
Target0.1
Herbie4.8
\[\begin{array}{l} \mathbf{if}\;0.5 \lt \left|\frac{x}{y}\right| \lt 2:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{2}{1 + \frac{x}{y} \cdot \frac{x}{y}}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if y < -8.15983082784838867e152 or -5.8412403973205966e-164 < y < -4.5550794910755097e-167 or 1.87266605808828046e-175 < y < 1.23673946718731825e-162

    1. Initial program 61.1

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

    2. Taylor expanded around 0 2.4

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

    if -8.15983082784838867e152 < y < -5.8412403973205966e-164

    1. Initial program 0.1

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

    if -4.5550794910755097e-167 < y < 1.87266605808828046e-175

    1. Initial program 30.0

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

    2. Taylor expanded around inf 14.2

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

    if 1.23673946718731825e-162 < y

    1. Initial program 0.0

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

    3. Applied add-sqr-sqrt0.1

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

    4. Applied times-frac0.5

      \[\leadsto \color{blue}{\frac{x - y}{\sqrt{x \cdot x + y \cdot y}} \cdot \frac{x + y}{\sqrt{x \cdot x + y \cdot y}}}\]
    5. Using strategy rm

    6. Applied add-log-exp0.5

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

    7. Simplified0.1

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

  4. Final simplification4.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -8.15983082784838867 \cdot 10^{152}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le -5.8412403973205966 \cdot 10^{-164}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{elif}\;y \le -4.5550794910755097 \cdot 10^{-167}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \le 1.87266605808828046 \cdot 10^{-175}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \le 1.23673946718731825 \cdot 10^{-162}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\log \left({\left(e^{1}\right)}^{\left(\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\right)}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020161 
(FPCore (x y)
  :name "Kahan p9 Example"
  :precision binary64
  :pre (and (< 0.0 x 1.0) (< y 1.0))

  :herbie-target
  (if (< 0.5 (fabs (/ x y)) 2.0) (/ (* (- x y) (+ x y)) (+ (* x x) (* y y))) (- 1.0 (/ 2.0 (+ 1.0 (* (/ x y) (/ x y))))))

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