Linear.Projection:inverseInfinitePerspective from linear-1.19.1.3

Average Error: 6.9 → 2.9

Time: 3.7s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \le -2.68352422182606035 \cdot 10^{45}:\\ \;\;\;\;\left(y \cdot t\right) \cdot \left(x - z\right)\\ \mathbf{elif}\;y \le 2.165575148717488 \cdot 10^{-97}:\\ \;\;\;\;t \cdot \left(y \cdot \left(x - z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(t \cdot x\right) - y \cdot \left(t \cdot z\right)\\ \end{array}\]

C

double code(double x, double y, double z, double t) {
	return ((double) (((double) (((double) (x * y)) - ((double) (z * y)))) * t));
}

↓

double code(double x, double y, double z, double t) {
	double VAR;
	if ((y <= -2.6835242218260603e+45)) {
		VAR = ((double) (((double) (y * t)) * ((double) (x - z))));
	} else {
		double VAR_1;
		if ((y <= 2.165575148717488e-97)) {
			VAR_1 = ((double) (t * ((double) (y * ((double) (x - z))))));
		} else {
			VAR_1 = ((double) (((double) (y * ((double) (t * x)))) - ((double) (y * ((double) (t * 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	6.9
Target	3.3
Herbie	2.9

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

Derivation

1. Split input into 3 regimes

2. if y < -2.68352422182606035e45

   1. Initial program 18.0

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

   2. Simplified 4.1

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

   4. Applied associate-*r* 4.3

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

   if -2.68352422182606035e45 < y < 2.165575148717488e-97

   1. Initial program 2.4

      \[\left(x \cdot y - z \cdot y\right) \cdot t\]
   2. Using strategy rm

   3. Applied *-un-lft-identity 2.4

      \[\leadsto \left(x \cdot y - z \cdot y\right) \cdot \color{blue}{\left(1 \cdot t\right)}\]

   4. Applied associate-*r* 2.4

      \[\leadsto \color{blue}{\left(\left(x \cdot y - z \cdot y\right) \cdot 1\right) \cdot t}\]

   5. Simplified 2.4

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

   if 2.165575148717488e-97 < y

   1. Initial program 10.3

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

   2. Simplified 3.1

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

   4. Applied sub-neg 3.1

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

   5. Applied distribute-lft-in 3.1

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

   6. Applied distribute-lft-in 3.1

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

4. Final simplification 2.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -2.68352422182606035 \cdot 10^{45}:\\ \;\;\;\;\left(y \cdot t\right) \cdot \left(x - z\right)\\ \mathbf{elif}\;y \le 2.165575148717488 \cdot 10^{-97}:\\ \;\;\;\;t \cdot \left(y \cdot \left(x - z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(t \cdot x\right) - y \cdot \left(t \cdot z\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020185 
(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))