Average Error: 15.2 → 1.0
Time: 2.3s
Precision: binary64
\[\frac{\left(x \cdot 2\right) \cdot y}{x - y} \]
↓
\[\begin{array}{l} t_0 := \frac{\left(x \cdot 2\right) \cdot y}{x - y}\\ t_1 := \frac{x}{\mathsf{fma}\left(0.5, \frac{x}{y}, -0.5\right)}\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_0 \leq -4.9413458312946296 \cdot 10^{-293}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_0 \leq 3.172930057284486 \cdot 10^{-84}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
\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}\\
t_1 := \frac{x}{\mathsf{fma}\left(0.5, \frac{x}{y}, -0.5\right)}\\
\mathbf{if}\;t_0 \leq -\infty:\\
\;\;\;\;t_1\\

\mathbf{elif}\;t_0 \leq -4.9413458312946296 \cdot 10^{-293}:\\
\;\;\;\;t_0\\

\mathbf{elif}\;t_0 \leq 0:\\
\;\;\;\;t_1\\

\mathbf{elif}\;t_0 \leq 3.172930057284486 \cdot 10^{-84}:\\
\;\;\;\;t_0\\

\mathbf{else}:\\
\;\;\;\;t_1\\


\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))) (t_1 (/ x (fma 0.5 (/ x y) -0.5))))
   (if (<= t_0 (- INFINITY))
     t_1
     (if (<= t_0 -4.9413458312946296e-293)
       t_0
       (if (<= t_0 0.0) t_1 (if (<= t_0 3.172930057284486e-84) t_0 t_1))))))
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 t_1 = x / fma(0.5, (x / y), -0.5);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_0 <= -4.9413458312946296e-293) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = t_1;
	} else if (t_0 <= 3.172930057284486e-84) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Target

Original15.2
Target0.3
Herbie1.0
\[\begin{array}{l} \mathbf{if}\;x < -1.7210442634149447 \cdot 10^{+81}:\\ \;\;\;\;\frac{2 \cdot x}{x - y} \cdot y\\ \mathbf{elif}\;x < 83645045635564430:\\ \;\;\;\;\frac{x \cdot 2}{\frac{x - y}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot x}{x - y} \cdot y\\ \end{array} \]

Derivation

  1. Split input into 2 regimes

  2. if (/.f64 (*.f64 (*.f64 x 2) y) (-.f64 x y)) < -inf.0 or -4.94134583129462956e-293 < (/.f64 (*.f64 (*.f64 x 2) y) (-.f64 x y)) < 0.0 or 3.1729300572844862e-84 < (/.f64 (*.f64 (*.f64 x 2) y) (-.f64 x y))

    1. Initial program 37.7

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

    2. Simplified1.7

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

    if -inf.0 < (/.f64 (*.f64 (*.f64 x 2) y) (-.f64 x y)) < -4.94134583129462956e-293 or 0.0 < (/.f64 (*.f64 (*.f64 x 2) y) (-.f64 x y)) < 3.1729300572844862e-84

    1. Initial program 0.6

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

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(x \cdot 2\right) \cdot y}{x - y} \leq -\infty:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(0.5, \frac{x}{y}, -0.5\right)}\\ \mathbf{elif}\;\frac{\left(x \cdot 2\right) \cdot y}{x - y} \leq -4.9413458312946296 \cdot 10^{-293}:\\ \;\;\;\;\frac{\left(x \cdot 2\right) \cdot y}{x - y}\\ \mathbf{elif}\;\frac{\left(x \cdot 2\right) \cdot y}{x - y} \leq 0:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(0.5, \frac{x}{y}, -0.5\right)}\\ \mathbf{elif}\;\frac{\left(x \cdot 2\right) \cdot y}{x - y} \leq 3.172930057284486 \cdot 10^{-84}:\\ \;\;\;\;\frac{\left(x \cdot 2\right) \cdot y}{x - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(0.5, \frac{x}{y}, -0.5\right)}\\ \end{array} \]

Reproduce

herbie shell --seed 2022077 
(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)))