Linear.Projection:inverseInfinitePerspective from linear-1.19.1.3

Average Error: 6.9 → 0.5

Time: 7.2s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \cdot y - y \cdot z \leq -8.96330847105403 \cdot 10^{+255}:\\ \;\;\;\;y \cdot \left(t \cdot \left(x - z\right)\right)\\ \mathbf{elif}\;x \cdot y - y \cdot z \leq -2.110507631270343 \cdot 10^{-193}:\\ \;\;\;\;\left(x \cdot y - y \cdot z\right) \cdot t\\ \mathbf{elif}\;x \cdot y - y \cdot z \leq 6.674273054962094 \cdot 10^{-148} \lor \neg \left(x \cdot y - y \cdot z \leq 3.69106269764638 \cdot 10^{+220}\right):\\ \;\;\;\;y \cdot \left(t \cdot \left(x - z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(y \cdot \left(x - z\right)\right)\\ \end{array}\]

FPCore

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

↓

(FPCore (x y z t)
 :precision binary64
 (if (<= (- (* x y) (* y z)) -8.96330847105403e+255)
   (* y (* t (- x z)))
   (if (<= (- (* x y) (* y z)) -2.110507631270343e-193)
     (* (- (* x y) (* y z)) t)
     (if (or (<= (- (* x y) (* y z)) 6.674273054962094e-148)
             (not (<= (- (* x y) (* y z)) 3.69106269764638e+220)))
       (* y (* t (- x z)))
       (* t (* y (- 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 tmp;
	if (((x * y) - (y * z)) <= -8.96330847105403e+255) {
		tmp = y * (t * (x - z));
	} else if (((x * y) - (y * z)) <= -2.110507631270343e-193) {
		tmp = ((x * y) - (y * z)) * t;
	} else if ((((x * y) - (y * z)) <= 6.674273054962094e-148) || !(((x * y) - (y * z)) <= 3.69106269764638e+220)) {
		tmp = y * (t * (x - z));
	} else {
		tmp = t * (y * (x - z));
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	6.9
Target	3.0
Herbie	0.5

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

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 x y) (*.f64 z y)) < -8.96330847105402962e255 or -2.11050763127034311e-193 < (-.f64 (*.f64 x y) (*.f64 z y)) < 6.6742730549620938e-148 or 3.69106269764637988e220 < (-.f64 (*.f64 x y) (*.f64 z y))

   1. Initial program 20.3

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

   2. Simplified 1.1

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

   if -8.96330847105402962e255 < (-.f64 (*.f64 x y) (*.f64 z y)) < -2.11050763127034311e-193

   1. Initial program 0.3

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

   if 6.6742730549620938e-148 < (-.f64 (*.f64 x y) (*.f64 z y)) < 3.69106269764637988e220

   1. Initial program 0.3

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

   3. Applied *-un-lft-identity_binary64_16787 0.3

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

   4. Applied associate-*r*_binary64_16727 0.3

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

   5. Simplified 0.3

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

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - y \cdot z \leq -8.96330847105403 \cdot 10^{+255}:\\ \;\;\;\;y \cdot \left(t \cdot \left(x - z\right)\right)\\ \mathbf{elif}\;x \cdot y - y \cdot z \leq -2.110507631270343 \cdot 10^{-193}:\\ \;\;\;\;\left(x \cdot y - y \cdot z\right) \cdot t\\ \mathbf{elif}\;x \cdot y - y \cdot z \leq 6.674273054962094 \cdot 10^{-148} \lor \neg \left(x \cdot y - y \cdot z \leq 3.69106269764638 \cdot 10^{+220}\right):\\ \;\;\;\;y \cdot \left(t \cdot \left(x - z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(y \cdot \left(x - z\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020342 
(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))