Average Error: 20.5 → 4.5
Time: 4.0s
Precision: binary64
\[0 < x \land x < 1 \land y < 1\]
\[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y} \leq 1:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;-1 + 2 \cdot \left({\left(e^{\log \left({\left(\frac{x}{y}\right)}^{2}\right)}\right)}^{0.6666666666666666} \cdot \sqrt[3]{{\left(\frac{x}{y}\right)}^{2}}\right)\\ \end{array}\]
\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}
↓
\begin{array}{l}
\mathbf{if}\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y} \leq 1:\\
\;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\

\mathbf{else}:\\
\;\;\;\;-1 + 2 \cdot \left({\left(e^{\log \left({\left(\frac{x}{y}\right)}^{2}\right)}\right)}^{0.6666666666666666} \cdot \sqrt[3]{{\left(\frac{x}{y}\right)}^{2}}\right)\\

\end{array}
(FPCore (x y) :precision binary64 (/ (* (- x y) (+ x y)) (+ (* x x) (* y y))))
↓
(FPCore (x y)
 :precision binary64
 (if (<= (/ (* (- x y) (+ x y)) (+ (* x x) (* y y))) 1.0)
   (/ (* (- x y) (+ x y)) (+ (* x x) (* y y)))
   (+
    -1.0
    (*
     2.0
     (*
      (pow (exp (log (pow (/ x y) 2.0))) 0.6666666666666666)
      (cbrt (pow (/ x y) 2.0)))))))
double code(double x, double y) {
	return ((x - y) * (x + y)) / ((x * x) + (y * y));
}
↓
double code(double x, double y) {
	double tmp;
	if ((((x - y) * (x + y)) / ((x * x) + (y * y))) <= 1.0) {
		tmp = ((x - y) * (x + y)) / ((x * x) + (y * y));
	} else {
		tmp = -1.0 + (2.0 * (pow(exp(log(pow((x / y), 2.0))), 0.6666666666666666) * cbrt(pow((x / y), 2.0))));
	}
	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

Original20.5
Target0.1
Herbie4.5
\[\begin{array}{l} \mathbf{if}\;0.5 < \left|\frac{x}{y}\right| \land \left|\frac{x}{y}\right| < 2:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{2}{1 + \frac{x}{y} \cdot \frac{x}{y}}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (/.f64 (*.f64 (-.f64 x y) (+.f64 x y)) (+.f64 (*.f64 x x) (*.f64 y y))) < 1

    1. Initial program 0.0

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

    if 1 < (/.f64 (*.f64 (-.f64 x y) (+.f64 x y)) (+.f64 (*.f64 x x) (*.f64 y y)))

    1. Initial program 64.0

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

    2. Taylor expanded around 0 30.8

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

    3. Simplified30.8

      \[\leadsto \color{blue}{-1 + 2 \cdot \frac{x \cdot x}{y \cdot y}}\]
    4. Using strategy rm

    5. Applied add-cube-cbrt_binary6430.8

      \[\leadsto -1 + 2 \cdot \color{blue}{\left(\left(\sqrt[3]{\frac{x \cdot x}{y \cdot y}} \cdot \sqrt[3]{\frac{x \cdot x}{y \cdot y}}\right) \cdot \sqrt[3]{\frac{x \cdot x}{y \cdot y}}\right)}\]

    6. Simplified30.8

      \[\leadsto -1 + 2 \cdot \left(\color{blue}{\left(\sqrt[3]{{\left(\frac{x}{y}\right)}^{2}} \cdot \sqrt[3]{{\left(\frac{x}{y}\right)}^{2}}\right)} \cdot \sqrt[3]{\frac{x \cdot x}{y \cdot y}}\right)\]

    7. Simplified13.9

      \[\leadsto -1 + 2 \cdot \left(\left(\sqrt[3]{{\left(\frac{x}{y}\right)}^{2}} \cdot \sqrt[3]{{\left(\frac{x}{y}\right)}^{2}}\right) \cdot \color{blue}{\sqrt[3]{{\left(\frac{x}{y}\right)}^{2}}}\right)\]
    8. Using strategy rm

    9. Applied pow1/3_binary6413.9

      \[\leadsto -1 + 2 \cdot \left(\left(\sqrt[3]{{\left(\frac{x}{y}\right)}^{2}} \cdot \color{blue}{{\left({\left(\frac{x}{y}\right)}^{2}\right)}^{0.3333333333333333}}\right) \cdot \sqrt[3]{{\left(\frac{x}{y}\right)}^{2}}\right)\]

    10. Applied pow1/3_binary6413.9

      \[\leadsto -1 + 2 \cdot \left(\left(\color{blue}{{\left({\left(\frac{x}{y}\right)}^{2}\right)}^{0.3333333333333333}} \cdot {\left({\left(\frac{x}{y}\right)}^{2}\right)}^{0.3333333333333333}\right) \cdot \sqrt[3]{{\left(\frac{x}{y}\right)}^{2}}\right)\]

    11. Applied pow-prod-up_binary6413.9

      \[\leadsto -1 + 2 \cdot \left(\color{blue}{{\left({\left(\frac{x}{y}\right)}^{2}\right)}^{\left(0.3333333333333333 + 0.3333333333333333\right)}} \cdot \sqrt[3]{{\left(\frac{x}{y}\right)}^{2}}\right)\]

    12. Simplified13.9

      \[\leadsto -1 + 2 \cdot \left({\left({\left(\frac{x}{y}\right)}^{2}\right)}^{\color{blue}{0.6666666666666666}} \cdot \sqrt[3]{{\left(\frac{x}{y}\right)}^{2}}\right)\]
    13. Using strategy rm

    14. Applied add-exp-log_binary6413.9

      \[\leadsto -1 + 2 \cdot \left({\color{blue}{\left(e^{\log \left({\left(\frac{x}{y}\right)}^{2}\right)}\right)}}^{0.6666666666666666} \cdot \sqrt[3]{{\left(\frac{x}{y}\right)}^{2}}\right)\]
  3. Recombined 2 regimes into one program.

  4. Final simplification4.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y} \leq 1:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;-1 + 2 \cdot \left({\left(e^{\log \left({\left(\frac{x}{y}\right)}^{2}\right)}\right)}^{0.6666666666666666} \cdot \sqrt[3]{{\left(\frac{x}{y}\right)}^{2}}\right)\\ \end{array}\]

Alternatives

Reproduce

herbie shell --seed 2021118 
(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))))