Linear.Projection:inverseInfinitePerspective from linear-1.19.1.3

Average Error: 7.0 → 0.3

Time: 10.5s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} t_1 := x \cdot y - y \cdot z\\ t_2 := \left(x - z\right) \cdot \left(y \cdot t\right)\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\begin{array}{l} t_3 := t \cdot \left(y \cdot \left(x - z\right)\right)\\ \mathbf{if}\;t_1 \leq -4.6578095093216616 \cdot 10^{-250}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t_1 \leq 8.22540275056901 \cdot 10^{-271}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq 1.0849371537542345 \cdot 10^{+225}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\left(x - z\right) \cdot t\right)\\ \end{array}\\ \end{array} \]

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 (* (- x z) (* y t))))
   (if (<= t_1 (- INFINITY))
     t_2
     (let* ((t_3 (* t (* y (- x z)))))
       (if (<= t_1 -4.6578095093216616e-250)
         t_3
         (if (<= t_1 8.22540275056901e-271)
           t_2
           (if (<= t_1 1.0849371537542345e+225) t_3 (* y (* (- x z) t)))))))))

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 = (x - z) * (y * t);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_2;
	} else {
		double t_3 = t * (y * (x - z));
		double tmp_1;
		if (t_1 <= -4.6578095093216616e-250) {
			tmp_1 = t_3;
		} else if (t_1 <= 8.22540275056901e-271) {
			tmp_1 = t_2;
		} else if (t_1 <= 1.0849371537542345e+225) {
			tmp_1 = t_3;
		} else {
			tmp_1 = y * ((x - z) * t);
		}
		tmp = tmp_1;
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	7.0
Target	3.2
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 3 regimes

2. if (-.f64 (*.f64 x y) (*.f64 z y)) < -inf.0 or -4.6578095093216616e-250 < (-.f64 (*.f64 x y) (*.f64 z y)) < 8.22540275056901014e-271

   1. Initial program 27.8

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

   2. Taylor expanded in y around inf 0.2

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

   if -inf.0 < (-.f64 (*.f64 x y) (*.f64 z y)) < -4.6578095093216616e-250 or 8.22540275056901014e-271 < (-.f64 (*.f64 x y) (*.f64 z y)) < 1.0849371537542345e225

   1. Initial program 0.2

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

   2. Taylor expanded in y around 0 0.2

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

   if 1.0849371537542345e225 < (-.f64 (*.f64 x y) (*.f64 z y))

   1. Initial program 34.8

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

   2. Simplified 0.8

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

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - y \cdot z \leq -\infty:\\ \;\;\;\;\left(x - z\right) \cdot \left(y \cdot t\right)\\ \mathbf{elif}\;x \cdot y - y \cdot z \leq -4.6578095093216616 \cdot 10^{-250}:\\ \;\;\;\;t \cdot \left(y \cdot \left(x - z\right)\right)\\ \mathbf{elif}\;x \cdot y - y \cdot z \leq 8.22540275056901 \cdot 10^{-271}:\\ \;\;\;\;\left(x - z\right) \cdot \left(y \cdot t\right)\\ \mathbf{elif}\;x \cdot y - y \cdot z \leq 1.0849371537542345 \cdot 10^{+225}:\\ \;\;\;\;t \cdot \left(y \cdot \left(x - z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\left(x - z\right) \cdot t\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2021275 
(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))