Linear.Projection:inverseInfinitePerspective from linear-1.19.1.3

Average Error: 7.2 → 2.8

Time: 1.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le -4.60750742140281364 \cdot 10^{-51}:\\ \;\;\;\;t \cdot \left(y \cdot \left(x - z\right)\right)\\ \mathbf{elif}\;t \le 9.97795893985270688:\\ \;\;\;\;\left(t \cdot \left(x - z\right)\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(t \cdot y\right) \cdot \left(x - z\right)\\ \end{array}\]

C

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 VAR;
	if ((t <= -4.6075074214028136e-51)) {
		VAR = (t * (y * (x - z)));
	} else {
		double VAR_1;
		if ((t <= 9.977958939852707)) {
			VAR_1 = ((t * (x - z)) * y);
		} else {
			VAR_1 = ((t * y) * (x - z));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	7.2
Target	3.0
Herbie	2.8

\[\begin{array}{l} \mathbf{if}\;t \lt -9.2318795828867769 \cdot 10^{-80}:\\ \;\;\;\;\left(y \cdot t\right) \cdot \left(x - z\right)\\ \mathbf{elif}\;t \lt 2.5430670515648771 \cdot 10^{83}:\\ \;\;\;\;y \cdot \left(t \cdot \left(x - z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot \left(x - z\right)\right) \cdot t\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if t < -4.6075074214028136e-51

   1. Initial program 3.1

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

   2. Simplified 3.1

      \[\leadsto \color{blue}{t \cdot \left(y \cdot \left(x - z\right)\right)}\]
   3. Using strategy rm

   4. Applied associate-*r* 3.2

      \[\leadsto \color{blue}{\left(t \cdot y\right) \cdot \left(x - z\right)}\]
   5. Using strategy rm

   6. Applied associate-*l* 3.1

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

   if -4.6075074214028136e-51 < t < 9.977958939852707

   1. Initial program 10.3

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

   2. Simplified 10.3

      \[\leadsto \color{blue}{t \cdot \left(y \cdot \left(x - z\right)\right)}\]
   3. Using strategy rm

   4. Applied *-commutative 10.3

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

   5. Applied associate-*r* 2.2

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

   if 9.977958939852707 < t

   1. Initial program 3.0

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

   2. Simplified 3.0

      \[\leadsto \color{blue}{t \cdot \left(y \cdot \left(x - z\right)\right)}\]
   3. Using strategy rm

   4. Applied associate-*r* 4.0

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

4. Final simplification 2.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -4.60750742140281364 \cdot 10^{-51}:\\ \;\;\;\;t \cdot \left(y \cdot \left(x - z\right)\right)\\ \mathbf{elif}\;t \le 9.97795893985270688:\\ \;\;\;\;\left(t \cdot \left(x - z\right)\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(t \cdot y\right) \cdot \left(x - z\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020071 +o rules:numerics
(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))