Linear.Projection:inverseInfinitePerspective from linear-1.19.1.3

Average Error: 7.1 → 3.9

Time: 5.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \le -3.63358414705648937 \cdot 10^{163} \lor \neg \left(y \le 4.75754250327531405 \cdot 10^{93}\right):\\ \;\;\;\;y \cdot \left(t \cdot \left(x - z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(x \cdot y\right) + t \cdot \left(\left(-z\right) \cdot y\right)\\ \end{array}\]

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 temp;
	if (((y <= -3.6335841470564894e+163) || !(y <= 4.757542503275314e+93))) {
		temp = (y * (t * (x - z)));
	} else {
		temp = ((t * (x * y)) + (t * (-z * y)));
	}
	return temp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	7.1
Target	3.0
Herbie	3.9

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

Derivation

1. Split input into 2 regimes

2. if y < -3.6335841470564894e+163 or 4.757542503275314e+93 < y

   1. Initial program 22.2

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

   2. Simplified 22.2

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

   4. Applied sub-neg 22.2

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

   5. Applied distribute-lft-in 22.2

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

   6. Applied distribute-lft-in 22.2

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

   7. Simplified 22.2

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

   8. Simplified 22.2

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

   10. Applied associate-*r* 15.5

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

   11. Applied associate-*r* 4.9

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

   12. Applied distribute-rgt-out 4.9

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

   13. Simplified 4.9

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

   if -3.6335841470564894e+163 < y < 4.757542503275314e+93

   1. Initial program 3.6

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

   2. Simplified 3.6

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

   4. Applied sub-neg 3.6

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

   5. Applied distribute-lft-in 3.6

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

   6. Applied distribute-lft-in 3.6

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

   7. Simplified 3.6

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

   8. Simplified 3.6

      \[\leadsto t \cdot \left(x \cdot y\right) + \color{blue}{t \cdot \left(\left(-z\right) \cdot y\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification 3.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -3.63358414705648937 \cdot 10^{163} \lor \neg \left(y \le 4.75754250327531405 \cdot 10^{93}\right):\\ \;\;\;\;y \cdot \left(t \cdot \left(x - z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(x \cdot y\right) + t \cdot \left(\left(-z\right) \cdot y\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020066 +o rules:numerics
(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))