Linear.Projection:inverseInfinitePerspective from linear-1.19.1.3

Average Error: 6.7 → 1.2

Time: 3.6s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \cdot y - y \cdot z \leq -3.397707030939296 \cdot 10^{+265}:\\ \;\;\;\;y \cdot \left(x \cdot t\right) - y \cdot \left(z \cdot t\right)\\ \mathbf{elif}\;x \cdot y - y \cdot z \leq 2.294972121944136 \cdot 10^{+173}:\\ \;\;\;\;\left(x \cdot y - y \cdot z\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot t\right) \cdot \left(x - z\right)\\ \end{array}\]

FPCore

(FPCore (x y z t) :precision binary64 (* (- (* x y) (* z y)) t))

↓

(FPCore (x y z t)
 :precision binary64
 (if (<= (- (* x y) (* y z)) -3.397707030939296e+265)
   (- (* y (* x t)) (* y (* z t)))
   (if (<= (- (* x y) (* y z)) 2.294972121944136e+173)
     (* (- (* x y) (* y z)) t)
     (* (* y t) (- x z)))))

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) (y * z)))) <= -3.397707030939296e+265)) {
		VAR = ((double) (((double) (y * ((double) (x * t)))) - ((double) (y * ((double) (z * t))))));
	} else {
		double VAR_1;
		if ((((double) (((double) (x * y)) - ((double) (y * z)))) <= 2.294972121944136e+173)) {
			VAR_1 = ((double) (((double) (((double) (x * y)) - ((double) (y * z)))) * t));
		} else {
			VAR_1 = ((double) (((double) (y * t)) * ((double) (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	6.7
Target	3.1
Herbie	1.2

\[\begin{array}{l} \mathbf{if}\;t < -9.231879582886777 \cdot 10^{-80}:\\ \;\;\;\;\left(y \cdot t\right) \cdot \left(x - z\right)\\ \mathbf{elif}\;t < 2.543067051564877 \cdot 10^{+83}:\\ \;\;\;\;y \cdot \left(t \cdot \left(x - z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot \left(x - z\right)\right) \cdot t\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if (- (* x y) (* z y)) < -3.39770703093929569e265

   1. Initial program 42.7

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

   3. Applied add-cube-cbrt 43.2

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

   4. Applied associate-*r* 43.2

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

   5. Simplified 10.0

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

   7. Applied associate-*r* 9.9

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

   8. Simplified 9.9

      \[\leadsto \left(\color{blue}{\left(y \cdot {\left(\sqrt[3]{t}\right)}^{2}\right)} \cdot \left(x - z\right)\right) \cdot \sqrt[3]{t}\]
   9. Using strategy rm

   10. Applied associate-*l* 1.5

       \[\leadsto \color{blue}{\left(y \cdot {\left(\sqrt[3]{t}\right)}^{2}\right) \cdot \left(\left(x - z\right) \cdot \sqrt[3]{t}\right)}\]

   11. Simplified 1.5

       \[\leadsto \left(y \cdot {\left(\sqrt[3]{t}\right)}^{2}\right) \cdot \color{blue}{\left(\sqrt[3]{t} \cdot \left(x - z\right)\right)}\]
   12. Using strategy rm

   13. Applied sub-neg 1.5

       \[\leadsto \left(y \cdot {\left(\sqrt[3]{t}\right)}^{2}\right) \cdot \left(\sqrt[3]{t} \cdot \color{blue}{\left(x + \left(-z\right)\right)}\right)\]

   14. Applied distribute-lft-in 1.5

       \[\leadsto \left(y \cdot {\left(\sqrt[3]{t}\right)}^{2}\right) \cdot \color{blue}{\left(\sqrt[3]{t} \cdot x + \sqrt[3]{t} \cdot \left(-z\right)\right)}\]

   15. Applied distribute-lft-in 1.5

       \[\leadsto \color{blue}{\left(y \cdot {\left(\sqrt[3]{t}\right)}^{2}\right) \cdot \left(\sqrt[3]{t} \cdot x\right) + \left(y \cdot {\left(\sqrt[3]{t}\right)}^{2}\right) \cdot \left(\sqrt[3]{t} \cdot \left(-z\right)\right)}\]

   16. Simplified 0.9

       \[\leadsto \color{blue}{y \cdot \left(t \cdot x\right)} + \left(y \cdot {\left(\sqrt[3]{t}\right)}^{2}\right) \cdot \left(\sqrt[3]{t} \cdot \left(-z\right)\right)\]

   17. Simplified 0.3

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

   if -3.39770703093929569e265 < (- (* x y) (* z y)) < 2.29497212194413589e173

   1. Initial program 1.3

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

   if 2.29497212194413589e173 < (- (* x y) (* z y))

   1. Initial program 24.7

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

   3. Applied add-cube-cbrt 25.4

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

   4. Applied associate-*r* 25.4

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

   5. Simplified 8.5

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

   7. Applied associate-*r* 8.0

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

   8. Simplified 8.0

      \[\leadsto \left(\color{blue}{\left(y \cdot {\left(\sqrt[3]{t}\right)}^{2}\right)} \cdot \left(x - z\right)\right) \cdot \sqrt[3]{t}\]
   9. Using strategy rm

   10. Applied associate-*l* 1.9

       \[\leadsto \color{blue}{\left(y \cdot {\left(\sqrt[3]{t}\right)}^{2}\right) \cdot \left(\left(x - z\right) \cdot \sqrt[3]{t}\right)}\]

   11. Simplified 1.9

       \[\leadsto \left(y \cdot {\left(\sqrt[3]{t}\right)}^{2}\right) \cdot \color{blue}{\left(\sqrt[3]{t} \cdot \left(x - z\right)\right)}\]
   12. Using strategy rm

   13. Applied associate-*r* 2.6

       \[\leadsto \color{blue}{\left(\left(y \cdot {\left(\sqrt[3]{t}\right)}^{2}\right) \cdot \sqrt[3]{t}\right) \cdot \left(x - z\right)}\]

   14. Simplified 1.4

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

4. Final simplification 1.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - y \cdot z \leq -3.397707030939296 \cdot 10^{+265}:\\ \;\;\;\;y \cdot \left(x \cdot t\right) - y \cdot \left(z \cdot t\right)\\ \mathbf{elif}\;x \cdot y - y \cdot z \leq 2.294972121944136 \cdot 10^{+173}:\\ \;\;\;\;\left(x \cdot y - y \cdot z\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot t\right) \cdot \left(x - z\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020198 
(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))