Linear.Projection:inverseInfinitePerspective from linear-1.19.1.3

Average Error: 6.9 → 1.3

Time: 2.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot y = -\infty:\\ \;\;\;\;y \cdot \left(\left(x - z\right) \cdot t\right)\\ \mathbf{elif}\;x \cdot y - z \cdot y \le 1.48975433212958462 \cdot 10^{218}:\\ \;\;\;\;\mathsf{fma}\left(x, y, -z \cdot y\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\left(t \cdot y\right) \cdot x + \left(t \cdot y\right) \cdot \left(-z\right)\right)\\ \end{array}\]

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 ((((x * y) - (z * y)) <= -inf.0)) {
		temp = (y * ((x - z) * t));
	} else {
		double temp_1;
		if ((((x * y) - (z * y)) <= 1.4897543321295846e+218)) {
			temp_1 = (fma(x, y, -(z * y)) * t);
		} else {
			temp_1 = (1.0 * (((t * y) * x) + ((t * y) * -z)));
		}
		temp = temp_1;
	}
	return temp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	6.9
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 3 regimes

2. if (- (* x y) (* z y)) < -inf.0

   1. Initial program 64.0

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

   3. Applied distribute-rgt-out-- 64.0

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

   4. Applied associate-*l* 0.2

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

   if -inf.0 < (- (* x y) (* z y)) < 1.4897543321295846e+218

   1. Initial program 1.4

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

   3. Applied fma-neg 1.4

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

   if 1.4897543321295846e+218 < (- (* x y) (* z y))

   1. Initial program 32.8

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

   3. Applied *-un-lft-identity 32.8

      \[\leadsto \color{blue}{\left(1 \cdot \left(x \cdot y - z \cdot y\right)\right)} \cdot t\]

   4. Applied associate-*l* 32.8

      \[\leadsto \color{blue}{1 \cdot \left(\left(x \cdot y - z \cdot y\right) \cdot t\right)}\]

   5. Simplified 0.8

      \[\leadsto 1 \cdot \color{blue}{\left(\left(t \cdot y\right) \cdot \left(x - z\right)\right)}\]
   6. Using strategy rm

   7. Applied sub-neg 0.8

      \[\leadsto 1 \cdot \left(\left(t \cdot y\right) \cdot \color{blue}{\left(x + \left(-z\right)\right)}\right)\]

   8. Applied distribute-lft-in 0.8

      \[\leadsto 1 \cdot \color{blue}{\left(\left(t \cdot y\right) \cdot x + \left(t \cdot y\right) \cdot \left(-z\right)\right)}\]
3. Recombined 3 regimes into one program.

4. Final simplification 1.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - z \cdot y = -\infty:\\ \;\;\;\;y \cdot \left(\left(x - z\right) \cdot t\right)\\ \mathbf{elif}\;x \cdot y - z \cdot y \le 1.48975433212958462 \cdot 10^{218}:\\ \;\;\;\;\mathsf{fma}\left(x, y, -z \cdot y\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\left(t \cdot y\right) \cdot x + \left(t \cdot y\right) \cdot \left(-z\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020058 +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))