Linear.Projection:inverseInfinitePerspective from linear-1.19.1.3

Average Error: 7.0 → 0.3

Time: 4.6s

Precision: binary64

\[[y, t]=\mathsf{sort}([y, t])\]

Math

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

↓

\[\begin{array}{l} t_1 := x \cdot y - y \cdot z\\ t_2 := \left(y \cdot t\right) \cdot \left(x - z\right)\\ \mathbf{if}\;t_1 \leq -3.984263960517686 \cdot 10^{+286}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq -1.2389227059510665 \cdot 10^{-248}:\\ \;\;\;\;t_1 \cdot t\\ \mathbf{elif}\;t_1 \leq 4.105332198285192 \cdot 10^{-296}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq 3.5209680185619337 \cdot 10^{+205}:\\ \;\;\;\;t \cdot \left(y \cdot \left(x - z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(t \cdot \left(x - z\right)\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x y) (* y z))) (t_2 (* (* y t) (- x z))))
   (if (<= t_1 -3.984263960517686e+286)
     t_2
     (if (<= t_1 -1.2389227059510665e-248)
       (* t_1 t)
       (if (<= t_1 4.105332198285192e-296)
         t_2
         (if (<= t_1 3.5209680185619337e+205)
           (* t (* y (- x z)))
           (* y (* t (- x z)))))))))

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 t_1 = (x * y) - (y * z);
	double t_2 = (y * t) * (x - z);
	double tmp;
	if (t_1 <= -3.984263960517686e+286) {
		tmp = t_2;
	} else if (t_1 <= -1.2389227059510665e-248) {
		tmp = t_1 * t;
	} else if (t_1 <= 4.105332198285192e-296) {
		tmp = t_2;
	} else if (t_1 <= 3.5209680185619337e+205) {
		tmp = t * (y * (x - z));
	} else {
		tmp = y * (t * (x - z));
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	7.0
Target	3.1
Herbie	0.3

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

Derivation

1. Split input into 4 regimes

2. if (-.f64 (*.f64 x y) (*.f64 z y)) < -3.98426396051768635e286 or -1.2389227059510665e-248 < (-.f64 (*.f64 x y) (*.f64 z y)) < 4.1053321982851922e-296

   1. Initial program 28.5

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

   2. Simplified 0.2

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

   4. Applied associate-*r*_binary64 0.2

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

   5. Simplified 0.2

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

   if -3.98426396051768635e286 < (-.f64 (*.f64 x y) (*.f64 z y)) < -1.2389227059510665e-248

   1. Initial program 0.3

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

   if 4.1053321982851922e-296 < (-.f64 (*.f64 x y) (*.f64 z y)) < 3.5209680185619337e205

   1. Initial program 0.2

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

   2. Simplified 9.4

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

   3. Taylor expanded around 0 9.4

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

   4. Simplified 0.2

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

   if 3.5209680185619337e205 < (-.f64 (*.f64 x y) (*.f64 z y))

   1. Initial program 29.9

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

   2. Simplified 1.0

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

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - y \cdot z \leq -3.984263960517686 \cdot 10^{+286}:\\ \;\;\;\;\left(y \cdot t\right) \cdot \left(x - z\right)\\ \mathbf{elif}\;x \cdot y - y \cdot z \leq -1.2389227059510665 \cdot 10^{-248}:\\ \;\;\;\;\left(x \cdot y - y \cdot z\right) \cdot t\\ \mathbf{elif}\;x \cdot y - y \cdot z \leq 4.105332198285192 \cdot 10^{-296}:\\ \;\;\;\;\left(y \cdot t\right) \cdot \left(x - z\right)\\ \mathbf{elif}\;x \cdot y - y \cdot z \leq 3.5209680185619337 \cdot 10^{+205}:\\ \;\;\;\;t \cdot \left(y \cdot \left(x - z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(t \cdot \left(x - z\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2021204 
(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))