Linear.Projection:inverseInfinitePerspective from linear-1.19.1.3

Average Error: 7.0 → 2.3

Time: 6.6s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot y \le -2.75024393033002904 \cdot 10^{126}:\\ \;\;\;\;\left(t \cdot y\right) \cdot \left(x - z\right)\\ \mathbf{elif}\;x \cdot y - z \cdot y \le 6.55468647688102877 \cdot 10^{122}:\\ \;\;\;\;\left(x \cdot y - z \cdot y\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\left(x - z\right) \cdot t\right)\\ \end{array}\]

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 ((((double) (((double) (x * y)) - ((double) (z * y)))) <= -2.750243930330029e+126)) {
		VAR = ((double) (((double) (t * y)) * ((double) (x - z))));
	} else {
		double VAR_1;
		if ((((double) (((double) (x * y)) - ((double) (z * y)))) <= 6.554686476881029e+122)) {
			VAR_1 = ((double) (((double) (((double) (x * y)) - ((double) (z * y)))) * t));
		} else {
			VAR_1 = ((double) (y * ((double) (((double) (x - z)) * t))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	7.0
Target	2.9
Herbie	2.3

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

Derivation

1. Split input into 3 regimes

2. if (- (* x y) (* z y)) < -2.75024393033002904e126

   1. Initial program 17.0

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

   3. Applied add-cube-cbrt 17.8

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

   4. Applied associate-*l* 17.8

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

   5. Taylor expanded around inf 17.0

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

   6. Simplified 2.8

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

   if -2.75024393033002904e126 < (- (* x y) (* z y)) < 6.55468647688102877e122

   1. Initial program 1.9

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

   if 6.55468647688102877e122 < (- (* x y) (* z y))

   1. Initial program 19.2

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

   3. Applied distribute-rgt-out-- 19.2

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

   4. Applied associate-*l* 3.4

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

4. Final simplification 2.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot y \le -2.75024393033002904 \cdot 10^{126}:\\ \;\;\;\;\left(t \cdot y\right) \cdot \left(x - z\right)\\ \mathbf{elif}\;x \cdot y - z \cdot y \le 6.55468647688102877 \cdot 10^{122}:\\ \;\;\;\;\left(x \cdot y - z \cdot y\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\left(x - z\right) \cdot t\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020168 
(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))