Average Error: 6.3 → 1.1
Time: 2.9s
Precision: binary64
\[[y, t] = \mathsf{sort}([y, t]) \\]
\[\left(x \cdot y - z \cdot y\right) \cdot t \]
↓
\[\begin{array}{l} \mathbf{if}\;\begin{array}{l} t_1 := x \cdot y - y \cdot z\\ t_1 \leq -2.1823359454349023 \cdot 10^{+265} \lor \neg \left(t_1 \leq 2.1350391858894197 \cdot 10^{+306}\right) \end{array}:\\ \;\;\;\;\left(x - z\right) \cdot \left(y \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(y \cdot \left(x - z\right)\right)\\ \end{array} \]
\left(x \cdot y - z \cdot y\right) \cdot t
↓
\begin{array}{l}
\mathbf{if}\;\begin{array}{l}
t_1 := x \cdot y - y \cdot z\\
t_1 \leq -2.1823359454349023 \cdot 10^{+265} \lor \neg \left(t_1 \leq 2.1350391858894197 \cdot 10^{+306}\right)
\end{array}:\\
\;\;\;\;\left(x - z\right) \cdot \left(y \cdot t\right)\\

\mathbf{else}:\\
\;\;\;\;t \cdot \left(y \cdot \left(x - z\right)\right)\\


\end{array}

(FPCore (x y z t) :precision binary64 (* (- (* x y) (* z y)) t))
↓
(FPCore (x y z t)
 :precision binary64
 (if (let* ((t_1 (- (* x y) (* y z))))
       (or (<= t_1 -2.1823359454349023e+265)
           (not (<= t_1 2.1350391858894197e+306))))
   (* (- x z) (* y t))
   (* t (* y (- x z)))))
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 tmp;
	if ((t_1 <= -2.1823359454349023e+265) || !(t_1 <= 2.1350391858894197e+306)) {
		tmp = (x - z) * (y * t);
	} else {
		tmp = t * (y * (x - z));
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original6.3
Target2.5
Herbie1.1
\[\begin{array}{l} \mathbf{if}\;t < -9.231879582886777 \cdot 10^{-80}:\\ \;\;\;\;\left(y \cdot t\right) \cdot \left(x - z\right)\\ \mathbf{elif}\;t < 2.543067051564877 \cdot 10^{+83}:\\ \;\;\;\;y \cdot \left(t \cdot \left(x - z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot \left(x - z\right)\right) \cdot t\\ \end{array} \]

Derivation

  1. Split input into 2 regimes

  2. if (-.f64 (*.f64 x y) (*.f64 z y)) < -2.1823359454349023e265 or 2.13503918588941969e306 < (-.f64 (*.f64 x y) (*.f64 z y))

    1. Initial program 21.9

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

    2. Simplified0.1

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

    3. Taylor expanded in y around inf 0.1

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

    if -2.1823359454349023e265 < (-.f64 (*.f64 x y) (*.f64 z y)) < 2.13503918588941969e306

    1. Initial program 1.4

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

    2. Simplified6.4

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

    3. Taylor expanded in y around 0 6.9

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

    4. Simplified1.4

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

  4. Final simplification1.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - y \cdot z \leq -2.1823359454349023 \cdot 10^{+265} \lor \neg \left(x \cdot y - y \cdot z \leq 2.1350391858894197 \cdot 10^{+306}\right):\\ \;\;\;\;\left(x - z\right) \cdot \left(y \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(y \cdot \left(x - z\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022088 
(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))