Linear.Projection:inverseInfinitePerspective from linear-1.19.1.3

Average Error: 6.6 → 1.3

Time: 6.0s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot y \le -1.60349984449973093 \cdot 10^{176}:\\ \;\;\;\;y \cdot \left(\left(x - z\right) \cdot t\right)\\ \mathbf{elif}\;x \cdot y - z \cdot y \le -3.4409840354552192 \cdot 10^{-163}:\\ \;\;\;\;t \cdot \left(\left(\sqrt[3]{y \cdot \left(x - z\right)} \cdot \sqrt[3]{y \cdot \left(x - z\right)}\right) \cdot \sqrt[3]{y \cdot \left(x - z\right)}\right)\\ \mathbf{elif}\;x \cdot y - z \cdot y \le 1.161429321917945 \cdot 10^{-170}:\\ \;\;\;\;\left(t \cdot y\right) \cdot \left(x - z\right)\\ \mathbf{elif}\;x \cdot y - z \cdot y \le 1.36586232090869967 \cdot 10^{176}:\\ \;\;\;\;\sqrt{x \cdot y - z \cdot y} \cdot \left(\sqrt{x \cdot y - z \cdot y} \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t \cdot y\right) \cdot \left(x - z\right)\\ \end{array}\]

C

double code(double x, double y, double z, double t) {
	return ((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)))) <= -1.603499844499731e+176)) {
		VAR = ((double) (y * ((double) (((double) (x - z)) * t))));
	} else {
		double VAR_1;
		if ((((double) (((double) (x * y)) - ((double) (z * y)))) <= -3.440984035455219e-163)) {
			VAR_1 = ((double) (t * ((double) (((double) (((double) cbrt(((double) (y * ((double) (x - z)))))) * ((double) cbrt(((double) (y * ((double) (x - z)))))))) * ((double) cbrt(((double) (y * ((double) (x - z))))))))));
		} else {
			double VAR_2;
			if ((((double) (((double) (x * y)) - ((double) (z * y)))) <= 1.161429321917945e-170)) {
				VAR_2 = ((double) (((double) (t * y)) * ((double) (x - z))));
			} else {
				double VAR_3;
				if ((((double) (((double) (x * y)) - ((double) (z * y)))) <= 1.3658623209086997e+176)) {
					VAR_3 = ((double) (((double) sqrt(((double) (((double) (x * y)) - ((double) (z * y)))))) * ((double) (((double) sqrt(((double) (((double) (x * y)) - ((double) (z * y)))))) * t))));
				} else {
					VAR_3 = ((double) (((double) (t * y)) * ((double) (x - z))));
				}
				VAR_2 = VAR_3;
			}
			VAR_1 = VAR_2;
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	6.6
Target	3.1
Herbie	1.3

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

Derivation

1. Split input into 4 regimes

2. if (- (* x y) (* z y)) < -1.60349984449973093e176

   1. Initial program 20.9

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

   3. Applied distribute-rgt-out-- 20.9

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

   4. Applied associate-*l* 2.2

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

   if -1.60349984449973093e176 < (- (* x y) (* z y)) < -3.4409840354552192e-163

   1. Initial program 0.3

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

   2. Simplified 0.2

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

   4. Applied add-cube-cbrt 1.3

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

   if -3.4409840354552192e-163 < (- (* x y) (* z y)) < 1.161429321917945e-170 or 1.36586232090869967e176 < (- (* x y) (* z y))

   1. Initial program 14.4

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

   2. Simplified 14.4

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

   4. Applied associate-*r* 1.6

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

   if 1.161429321917945e-170 < (- (* x y) (* z y)) < 1.36586232090869967e176

   1. Initial program 0.2

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

   3. Applied add-sqr-sqrt 0.6

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

   4. Applied associate-*l* 0.6

      \[\leadsto \color{blue}{\sqrt{x \cdot y - z \cdot y} \cdot \left(\sqrt{x \cdot y - z \cdot y} \cdot t\right)}\]
3. Recombined 4 regimes into one program.

4. Final simplification 1.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot y \le -1.60349984449973093 \cdot 10^{176}:\\ \;\;\;\;y \cdot \left(\left(x - z\right) \cdot t\right)\\ \mathbf{elif}\;x \cdot y - z \cdot y \le -3.4409840354552192 \cdot 10^{-163}:\\ \;\;\;\;t \cdot \left(\left(\sqrt[3]{y \cdot \left(x - z\right)} \cdot \sqrt[3]{y \cdot \left(x - z\right)}\right) \cdot \sqrt[3]{y \cdot \left(x - z\right)}\right)\\ \mathbf{elif}\;x \cdot y - z \cdot y \le 1.161429321917945 \cdot 10^{-170}:\\ \;\;\;\;\left(t \cdot y\right) \cdot \left(x - z\right)\\ \mathbf{elif}\;x \cdot y - z \cdot y \le 1.36586232090869967 \cdot 10^{176}:\\ \;\;\;\;\sqrt{x \cdot y - z \cdot y} \cdot \left(\sqrt{x \cdot y - z \cdot y} \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t \cdot y\right) \cdot \left(x - z\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020169 
(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))