Average Error: 6.8 → 1.3
Time: 8.2s
Precision: binary64
\[[y, t] = \mathsf{sort}([y, t]) \\]
\[\left(x \cdot y - z \cdot y\right) \cdot t \]
↓
\[\begin{array}{l} t_1 := x \cdot y - y \cdot z\\ t_2 := y \cdot \left(t \cdot \left(x - z\right)\right)\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq 1.5878990421270284 \cdot 10^{+302}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot \left(x - z\right), t, t \cdot \mathsf{fma}\left(z, -y, y \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]
\left(x \cdot y - z \cdot y\right) \cdot t
↓
\begin{array}{l}
t_1 := x \cdot y - y \cdot z\\
t_2 := y \cdot \left(t \cdot \left(x - z\right)\right)\\
\mathbf{if}\;t_1 \leq -\infty:\\
\;\;\;\;t_2\\

\mathbf{elif}\;t_1 \leq 1.5878990421270284 \cdot 10^{+302}:\\
\;\;\;\;\mathsf{fma}\left(y \cdot \left(x - z\right), t, t \cdot \mathsf{fma}\left(z, -y, y \cdot z\right)\right)\\

\mathbf{else}:\\
\;\;\;\;t_2\\


\end{array}

(FPCore (x y z t) :precision binary64 (* (- (* x y) (* z y)) t))
↓
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x y) (* y z))) (t_2 (* y (* t (- x z)))))
   (if (<= t_1 (- INFINITY))
     t_2
     (if (<= t_1 1.5878990421270284e+302)
       (fma (* y (- x z)) t (* t (fma z (- y) (* y z))))
       t_2))))
double code(double x, double y, double z, double t) {
	return ((x * y) - (z * y)) * t;
}
↓
double code(double x, double y, double z, double t) {
	double t_1 = (x * y) - (y * z);
	double t_2 = y * (t * (x - z));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_2;
	} else if (t_1 <= 1.5878990421270284e+302) {
		tmp = fma((y * (x - z)), t, (t * fma(z, -y, (y * z))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original6.8
Target3.7
Herbie1.3
\[\begin{array}{l} \mathbf{if}\;t < -9.231879582886777 \cdot 10^{-80}:\\ \;\;\;\;\left(y \cdot t\right) \cdot \left(x - z\right)\\ \mathbf{elif}\;t < 2.543067051564877 \cdot 10^{+83}:\\ \;\;\;\;y \cdot \left(t \cdot \left(x - z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot \left(x - z\right)\right) \cdot t\\ \end{array} \]

Derivation

  1. Split input into 2 regimes

  2. if (-.f64 (*.f64 x y) (*.f64 z y)) < -inf.0 or 1.58789904212702844e302 < (-.f64 (*.f64 x y) (*.f64 z y))

    1. Initial program 62.1

      \[\left(x \cdot y - z \cdot y\right) \cdot t \]

    2. Simplified0.2

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

    if -inf.0 < (-.f64 (*.f64 x y) (*.f64 z y)) < 1.58789904212702844e302

    1. Initial program 1.5

      \[\left(x \cdot y - z \cdot y\right) \cdot t \]

    2. Applied egg-rr1.4

      \[\leadsto \color{blue}{\mathsf{fma}\left(y \cdot \left(x - z\right), t, \mathsf{fma}\left(z, -y, y \cdot z\right) \cdot t\right)} \]
  3. Recombined 2 regimes into one program.

  4. Final simplification1.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - y \cdot z \leq -\infty:\\ \;\;\;\;y \cdot \left(t \cdot \left(x - z\right)\right)\\ \mathbf{elif}\;x \cdot y - y \cdot z \leq 1.5878990421270284 \cdot 10^{+302}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot \left(x - z\right), t, t \cdot \mathsf{fma}\left(z, -y, y \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(t \cdot \left(x - z\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022130 
(FPCore (x y z t)
  :name "Linear.Projection:inverseInfinitePerspective from linear-1.19.1.3"
  :precision binary64

  :herbie-target
  (if (< t -9.231879582886777e-80) (* (* y t) (- x z)) (if (< t 2.543067051564877e+83) (* y (* t (- x z))) (* (* y (- x z)) t)))

  (* (- (* x y) (* z y)) t))