Linear.Projection:inverseInfinitePerspective from linear-1.19.1.3

Average Error: 6.9 → 1.4

Time: 2.8s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot y \le -5.3674001281639158 \cdot 10^{214}:\\ \;\;\;\;y \cdot \left(\left(x - z\right) \cdot t\right)\\ \mathbf{elif}\;x \cdot y - z \cdot y \le 6.42449616954347382 \cdot 10^{211}:\\ \;\;\;\;\left(x \cdot y - z \cdot y\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;\left(t \cdot y\right) \cdot \left(x - z\right)\\ \end{array}\]

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 ((((x * y) - (z * y)) <= -5.367400128163916e+214)) {
		VAR = (y * ((x - z) * t));
	} else {
		double VAR_1;
		if ((((x * y) - (z * y)) <= 6.424496169543474e+211)) {
			VAR_1 = (((x * y) - (z * y)) * t);
		} 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	6.9
Target	3.2
Herbie	1.4

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

Derivation

1. Split input into 3 regimes

2. if (- (* x y) (* z y)) < -5.367400128163916e+214

   1. Initial program 30.3

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

   3. Applied distribute-rgt-out-- 30.3

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

   4. Applied associate-*l* 1.2

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

   if -5.367400128163916e+214 < (- (* x y) (* z y)) < 6.424496169543474e+211

   1. Initial program 1.5

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

   if 6.424496169543474e+211 < (- (* x y) (* z y))

   1. Initial program 31.0

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

   3. Applied add-cube-cbrt 31.6

      \[\leadsto \color{blue}{\left(\left(\sqrt[3]{x \cdot y - z \cdot y} \cdot \sqrt[3]{x \cdot y - z \cdot y}\right) \cdot \sqrt[3]{x \cdot y - z \cdot y}\right)} \cdot t\]

   4. Applied associate-*l* 31.6

      \[\leadsto \color{blue}{\left(\sqrt[3]{x \cdot y - z \cdot y} \cdot \sqrt[3]{x \cdot y - z \cdot y}\right) \cdot \left(\sqrt[3]{x \cdot y - z \cdot y} \cdot t\right)}\]

   5. Taylor expanded around inf 31.0

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

   6. Simplified 0.9

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

4. Final simplification 1.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot y \le -5.3674001281639158 \cdot 10^{214}:\\ \;\;\;\;y \cdot \left(\left(x - z\right) \cdot t\right)\\ \mathbf{elif}\;x \cdot y - z \cdot y \le 6.42449616954347382 \cdot 10^{211}:\\ \;\;\;\;\left(x \cdot y - z \cdot y\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;\left(t \cdot y\right) \cdot \left(x - z\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020091 
(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))