Average Error: 15.3 → 0.9
Time: 4.0s
Precision: binary64
\[\frac{\left(x \cdot 2\right) \cdot y}{x - y} \]
↓
\[\begin{array}{l} t_0 := \frac{\left(x \cdot 2\right) \cdot y}{x - y}\\ \mathbf{if}\;t_0 \leq -\infty \lor \neg \left(t_0 \leq -1.4103516884251033 \cdot 10^{-289}\right) \land \left(t_0 \leq 0 \lor \neg \left(t_0 \leq 3.7717846339651095 \cdot 10^{-56}\right)\right):\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(0.5, \frac{x}{y}, -0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
\frac{\left(x \cdot 2\right) \cdot y}{x - y}
↓
\begin{array}{l}
t_0 := \frac{\left(x \cdot 2\right) \cdot y}{x - y}\\
\mathbf{if}\;t_0 \leq -\infty \lor \neg \left(t_0 \leq -1.4103516884251033 \cdot 10^{-289}\right) \land \left(t_0 \leq 0 \lor \neg \left(t_0 \leq 3.7717846339651095 \cdot 10^{-56}\right)\right):\\
\;\;\;\;\frac{x}{\mathsf{fma}\left(0.5, \frac{x}{y}, -0.5\right)}\\

\mathbf{else}:\\
\;\;\;\;t_0\\


\end{array}

(FPCore (x y) :precision binary64 (/ (* (* x 2.0) y) (- x y)))
↓
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (* x 2.0) y) (- x y))))
   (if (or (<= t_0 (- INFINITY))
           (and (not (<= t_0 -1.4103516884251033e-289))
                (or (<= t_0 0.0) (not (<= t_0 3.7717846339651095e-56)))))
     (/ x (fma 0.5 (/ x y) -0.5))
     t_0)))
double code(double x, double y) {
	return ((x * 2.0) * y) / (x - y);
}
↓
double code(double x, double y) {
	double t_0 = ((x * 2.0) * y) / (x - y);
	double tmp;
	if ((t_0 <= -((double) INFINITY)) || (!(t_0 <= -1.4103516884251033e-289) && ((t_0 <= 0.0) || !(t_0 <= 3.7717846339651095e-56)))) {
		tmp = x / fma(0.5, (x / y), -0.5);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Target

Original15.3
Target0.3
Herbie0.9
\[\begin{array}{l} \mathbf{if}\;x < -1.7210442634149447 \cdot 10^{+81}:\\ \;\;\;\;\frac{2 \cdot x}{x - y} \cdot y\\ \mathbf{elif}\;x < 83645045635564430:\\ \;\;\;\;\frac{x \cdot 2}{\frac{x - y}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot x}{x - y} \cdot y\\ \end{array} \]

Derivation

  1. Split input into 2 regimes

  2. if (/.f64 (*.f64 (*.f64 x 2) y) (-.f64 x y)) < -inf.0 or -1.41035168842510327e-289 < (/.f64 (*.f64 (*.f64 x 2) y) (-.f64 x y)) < -0.0 or 3.77178463396510947e-56 < (/.f64 (*.f64 (*.f64 x 2) y) (-.f64 x y))

    1. Initial program 40.6

      \[\frac{\left(x \cdot 2\right) \cdot y}{x - y} \]

    2. Simplified1.4

      \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(0.5, \frac{x}{y}, -0.5\right)}} \]

    if -inf.0 < (/.f64 (*.f64 (*.f64 x 2) y) (-.f64 x y)) < -1.41035168842510327e-289 or -0.0 < (/.f64 (*.f64 (*.f64 x 2) y) (-.f64 x y)) < 3.77178463396510947e-56

    1. Initial program 0.6

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

  4. Final simplification0.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(x \cdot 2\right) \cdot y}{x - y} \leq -\infty \lor \neg \left(\frac{\left(x \cdot 2\right) \cdot y}{x - y} \leq -1.4103516884251033 \cdot 10^{-289}\right) \land \left(\frac{\left(x \cdot 2\right) \cdot y}{x - y} \leq 0 \lor \neg \left(\frac{\left(x \cdot 2\right) \cdot y}{x - y} \leq 3.7717846339651095 \cdot 10^{-56}\right)\right):\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(0.5, \frac{x}{y}, -0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x \cdot 2\right) \cdot y}{x - y}\\ \end{array} \]

Reproduce

herbie shell --seed 2021275 
(FPCore (x y)
  :name "Linear.Projection:perspective from linear-1.19.1.3, B"
  :precision binary64

  :herbie-target
  (if (< x -1.7210442634149447e+81) (* (/ (* 2.0 x) (- x y)) y) (if (< x 83645045635564430.0) (/ (* x 2.0) (/ (- x y) y)) (* (/ (* 2.0 x) (- x y)) y)))

  (/ (* (* x 2.0) y) (- x y)))