Average Error: 20.1 → 4.7
Time: 4.0s
Precision: binary64
Cost: 2244
\[0 < x \land x < 1 \land y < 1\]
\[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \leq -1.3513757347195584 \cdot 10^{+154}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq -9.049599147467105 \cdot 10^{-163}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(y + x\right)}{x \cdot x + y \cdot y}\\ \mathbf{elif}\;y \leq -1.0294507382078244 \cdot 10^{-172}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 2.2931306999859318 \cdot 10^{-169}:\\ \;\;\;\;1 - 2 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(y + x\right)}{x \cdot x + y \cdot y}\\ \end{array}\]
\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}
↓
\begin{array}{l}
\mathbf{if}\;y \leq -1.3513757347195584 \cdot 10^{+154}:\\
\;\;\;\;-1\\

\mathbf{elif}\;y \leq -9.049599147467105 \cdot 10^{-163}:\\
\;\;\;\;\frac{\left(x - y\right) \cdot \left(y + x\right)}{x \cdot x + y \cdot y}\\

\mathbf{elif}\;y \leq -1.0294507382078244 \cdot 10^{-172}:\\
\;\;\;\;-1\\

\mathbf{elif}\;y \leq 2.2931306999859318 \cdot 10^{-169}:\\
\;\;\;\;1 - 2 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\

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

\end{array}
(FPCore (x y) :precision binary64 (/ (* (- x y) (+ x y)) (+ (* x x) (* y y))))
↓
(FPCore (x y)
 :precision binary64
 (if (<= y -1.3513757347195584e+154)
   -1.0
   (if (<= y -9.049599147467105e-163)
     (/ (* (- x y) (+ y x)) (+ (* x x) (* y y)))
     (if (<= y -1.0294507382078244e-172)
       -1.0
       (if (<= y 2.2931306999859318e-169)
         (- 1.0 (* 2.0 (* (/ y x) (/ y x))))
         (/ (* (- x y) (+ y x)) (+ (* x x) (* y y))))))))
double code(double x, double y) {
	return ((x - y) * (x + y)) / ((x * x) + (y * y));
}
↓
double code(double x, double y) {
	double tmp;
	if (y <= -1.3513757347195584e+154) {
		tmp = -1.0;
	} else if (y <= -9.049599147467105e-163) {
		tmp = ((x - y) * (y + x)) / ((x * x) + (y * y));
	} else if (y <= -1.0294507382078244e-172) {
		tmp = -1.0;
	} else if (y <= 2.2931306999859318e-169) {
		tmp = 1.0 - (2.0 * ((y / x) * (y / x)));
	} else {
		tmp = ((x - y) * (y + x)) / ((x * x) + (y * y));
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original20.1
Target0.1
Herbie4.7
\[\begin{array}{l} \mathbf{if}\;0.5 < \left|\frac{x}{y}\right| \land \left|\frac{x}{y}\right| < 2:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{2}{1 + \frac{x}{y} \cdot \frac{x}{y}}\\ \end{array}\]

Alternatives

Alternative 1
Error5.0
Cost2244
\[\begin{array}{l} \mathbf{if}\;y \leq -1.14108427891841 \cdot 10^{-30}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq -1.0488401738504467 \cdot 10^{-164}:\\ \;\;\;\;\frac{x - y}{\frac{x \cdot x + y \cdot y}{y + x}}\\ \mathbf{elif}\;y \leq -2.8180354867586467 \cdot 10^{-173}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 2.2931306999859318 \cdot 10^{-169}:\\ \;\;\;\;1 - 2 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y}{\frac{x \cdot x + y \cdot y}{y + x}}\\ \end{array}\]
Alternative 2
Error5.0
Cost2244
\[\begin{array}{l} \mathbf{if}\;y \leq -1.14108427891841 \cdot 10^{-30}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq -3.3476235632622506 \cdot 10^{-164}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{y + x}{x \cdot x + y \cdot y}\\ \mathbf{elif}\;y \leq -1.2247246025880804 \cdot 10^{-172}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 1.5712870211415494 \cdot 10^{-162}:\\ \;\;\;\;1 - 2 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{y + x}{x \cdot x + y \cdot y}\\ \end{array}\]
Alternative 3
Error10.0
Cost1032
\[\begin{array}{l} \mathbf{if}\;y \leq -3.9751550003611877 \cdot 10^{-146} \lor \neg \left(y \leq 2.4978925429798098 \cdot 10^{-124}\right):\\ \;\;\;\;-1 + 2 \cdot \frac{x \cdot x}{y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;1 - 2 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \end{array}\]
Alternative 4
Error10.5
Cost1032
\[\begin{array}{l} \mathbf{if}\;y \leq -1.3932628819163734 \cdot 10^{-141} \lor \neg \left(y \leq 7.73441976440763 \cdot 10^{-124}\right):\\ \;\;\;\;-1 + 2 \cdot \frac{x \cdot x}{y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]
Alternative 5
Error10.7
Cost706
\[\begin{array}{l} \mathbf{if}\;y \leq -2.573943156283327 \cdot 10^{-173}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 4.6831456259906935 \cdot 10^{-125}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array}\]
Alternative 6
Error42.9
Cost64
\[1\]

Error

Derivation

  1. Split input into 3 regimes

  2. if y < -1.35137573471955844e154 or -9.049599147467105e-163 < y < -1.02945073820782443e-172

    1. Initial program 1.9

      \[-1\]

    if -1.35137573471955844e154 < y < -9.049599147467105e-163 or 2.2931306999859318e-169 < y

    1. Initial program 0.4

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

    2. Simplified0.4

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

    if -1.02945073820782443e-172 < y < 2.2931306999859318e-169

    1. Initial program 30.4

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

    3. Applied add-sqr-sqrt_binary64_78230.4

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

    4. Applied times-frac_binary64_76631.1

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

    5. Taylor expanded around inf 30.4

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

    6. Simplified13.9

      \[\leadsto \color{blue}{1 - 2 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)}\]

    7. Simplified13.9

      \[\leadsto \color{blue}{1 - 2 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification4.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.3513757347195584 \cdot 10^{+154}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq -9.049599147467105 \cdot 10^{-163}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(y + x\right)}{x \cdot x + y \cdot y}\\ \mathbf{elif}\;y \leq -1.0294507382078244 \cdot 10^{-172}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 2.2931306999859318 \cdot 10^{-169}:\\ \;\;\;\;1 - 2 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(y + x\right)}{x \cdot x + y \cdot y}\\ \end{array}\]

Reproduce

herbie shell --seed 2021044 
(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))))