Average Error: 7.2 → 2.2
Time: 5.2s
Precision: binary64
\[[y, t]=\mathsf{sort}([y, t])\]
\[\left(x \cdot y - z \cdot y\right) \cdot t \]
↓
\[\begin{array}{l} t_1 := \left(x \cdot y - y \cdot z\right) \cdot t\\ \mathbf{if}\;t_1 \leq -\infty \lor \neg \left(t_1 \leq 1.29529836337512 \cdot 10^{-62}\right):\\ \;\;\;\;\left(x - z\right) \cdot \left(y \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
\left(x \cdot y - z \cdot y\right) \cdot t
↓
\begin{array}{l}
t_1 := \left(x \cdot y - y \cdot z\right) \cdot t\\
\mathbf{if}\;t_1 \leq -\infty \lor \neg \left(t_1 \leq 1.29529836337512 \cdot 10^{-62}\right):\\
\;\;\;\;\left(x - z\right) \cdot \left(y \cdot t\right)\\

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


\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)))
   (if (or (<= t_1 (- INFINITY)) (not (<= t_1 1.29529836337512e-62)))
     (* (- x z) (* y t))
     t_1)))
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)) * t;
	double tmp;
	if ((t_1 <= -((double) INFINITY)) || !(t_1 <= 1.29529836337512e-62)) {
		tmp = (x - z) * (y * t);
	} else {
		tmp = t_1;
	}
	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

Original7.2
Target3.4
Herbie2.2
\[\begin{array}{l} \mathbf{if}\;t < -9.231879582886777 \cdot 10^{-80}:\\ \;\;\;\;\left(y \cdot t\right) \cdot \left(x - z\right)\\ \mathbf{elif}\;t < 2.543067051564877 \cdot 10^{+83}:\\ \;\;\;\;y \cdot \left(t \cdot \left(x - z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot \left(x - z\right)\right) \cdot t\\ \end{array} \]

Derivation

  1. Split input into 2 regimes

  2. if (*.f64 (-.f64 (*.f64 x y) (*.f64 z y)) t) < -inf.0 or 1.29529836337512009e-62 < (*.f64 (-.f64 (*.f64 x y) (*.f64 z y)) t)

    1. Initial program 16.5

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

    2. Simplified8.8

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

    3. Taylor expanded in y around inf 2.7

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

    if -inf.0 < (*.f64 (-.f64 (*.f64 x y) (*.f64 z y)) t) < 1.29529836337512009e-62

    1. Initial program 2.0

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

  4. Final simplification2.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot y - y \cdot z\right) \cdot t \leq -\infty \lor \neg \left(\left(x \cdot y - y \cdot z\right) \cdot t \leq 1.29529836337512 \cdot 10^{-62}\right):\\ \;\;\;\;\left(x - z\right) \cdot \left(y \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y - y \cdot z\right) \cdot t\\ \end{array} \]

Reproduce

herbie shell --seed 2021313 
(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))