Linear.Projection:infinitePerspective from linear-1.19.1.3, A

Average Error: 7.0 → 2.9

Time: 4.7s

Precision: binary64

Math

\[\frac{x \cdot 2}{y \cdot z - t \cdot z}\]

↓

\[\begin{array}{l} \mathbf{if}\;z \le -4.0387324098205055 \cdot 10^{42}:\\ \;\;\;\;\left(x \cdot \frac{\sqrt{2}}{\sqrt[3]{y - t} \cdot \sqrt[3]{y - t}}\right) \cdot \frac{\frac{\sqrt{2}}{z}}{\sqrt[3]{y - t}}\\ \mathbf{elif}\;z \le 4.57265390019630824 \cdot 10^{110}:\\ \;\;\;\;\frac{x \cdot 2}{z \cdot \left(y - t\right)}\\ \mathbf{elif}\;z \le 3.5651713612658004 \cdot 10^{199}:\\ \;\;\;\;\left(x \cdot \frac{\sqrt{2}}{\sqrt[3]{y - t} \cdot \sqrt[3]{y - t}}\right) \cdot \frac{\frac{\sqrt{2}}{z}}{\sqrt[3]{y - t}}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \frac{2}{z}\right) \cdot \frac{1}{y - t}\\ \end{array}\]

C

double code(double x, double y, double z, double t) {
	return ((double) (((double) (x * 2.0)) / ((double) (((double) (y * z)) - ((double) (t * z))))));
}

↓

double code(double x, double y, double z, double t) {
	double VAR;
	if ((z <= -4.0387324098205055e+42)) {
		VAR = ((double) (((double) (x * ((double) (((double) sqrt(2.0)) / ((double) (((double) cbrt(((double) (y - t)))) * ((double) cbrt(((double) (y - t)))))))))) * ((double) (((double) (((double) sqrt(2.0)) / z)) / ((double) cbrt(((double) (y - t))))))));
	} else {
		double VAR_1;
		if ((z <= 4.572653900196308e+110)) {
			VAR_1 = ((double) (((double) (x * 2.0)) / ((double) (z * ((double) (y - t))))));
		} else {
			double VAR_2;
			if ((z <= 3.5651713612658004e+199)) {
				VAR_2 = ((double) (((double) (x * ((double) (((double) sqrt(2.0)) / ((double) (((double) cbrt(((double) (y - t)))) * ((double) cbrt(((double) (y - t)))))))))) * ((double) (((double) (((double) sqrt(2.0)) / z)) / ((double) cbrt(((double) (y - t))))))));
			} else {
				VAR_2 = ((double) (((double) (x * ((double) (2.0 / z)))) * ((double) (1.0 / ((double) (y - t))))));
			}
			VAR_1 = VAR_2;
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	7.0
Target	2.4
Herbie	2.9

\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot 2}{y \cdot z - t \cdot z} \lt -2.559141628295061 \cdot 10^{-13}:\\ \;\;\;\;\frac{x}{\left(y - t\right) \cdot z} \cdot 2\\ \mathbf{elif}\;\frac{x \cdot 2}{y \cdot z - t \cdot z} \lt 1.04502782733012586 \cdot 10^{-269}:\\ \;\;\;\;\frac{\frac{x}{z} \cdot 2}{y - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(y - t\right) \cdot z} \cdot 2\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if z < -4.0387324098205055e42 or 4.57265390019630824e110 < z < 3.5651713612658004e199

   1. Initial program 12.3

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z}\]

   2. Simplified 10.2

      \[\leadsto \color{blue}{x \cdot \frac{2}{z \cdot \left(y - t\right)}}\]
   3. Using strategy rm

   4. Applied associate-/r* 9.5

      \[\leadsto x \cdot \color{blue}{\frac{\frac{2}{z}}{y - t}}\]
   5. Using strategy rm

   6. Applied add-cube-cbrt 9.9

      \[\leadsto x \cdot \frac{\frac{2}{z}}{\color{blue}{\left(\sqrt[3]{y - t} \cdot \sqrt[3]{y - t}\right) \cdot \sqrt[3]{y - t}}}\]

   7. Applied *-un-lft-identity 9.9

      \[\leadsto x \cdot \frac{\frac{2}{\color{blue}{1 \cdot z}}}{\left(\sqrt[3]{y - t} \cdot \sqrt[3]{y - t}\right) \cdot \sqrt[3]{y - t}}\]

   8. Applied add-sqr-sqrt 10.0

      \[\leadsto x \cdot \frac{\frac{\color{blue}{\sqrt{2} \cdot \sqrt{2}}}{1 \cdot z}}{\left(\sqrt[3]{y - t} \cdot \sqrt[3]{y - t}\right) \cdot \sqrt[3]{y - t}}\]

   9. Applied times-frac 9.9

      \[\leadsto x \cdot \frac{\color{blue}{\frac{\sqrt{2}}{1} \cdot \frac{\sqrt{2}}{z}}}{\left(\sqrt[3]{y - t} \cdot \sqrt[3]{y - t}\right) \cdot \sqrt[3]{y - t}}\]

   10. Applied times-frac 9.9

       \[\leadsto x \cdot \color{blue}{\left(\frac{\frac{\sqrt{2}}{1}}{\sqrt[3]{y - t} \cdot \sqrt[3]{y - t}} \cdot \frac{\frac{\sqrt{2}}{z}}{\sqrt[3]{y - t}}\right)}\]

   11. Applied associate-*r* 3.1

       \[\leadsto \color{blue}{\left(x \cdot \frac{\frac{\sqrt{2}}{1}}{\sqrt[3]{y - t} \cdot \sqrt[3]{y - t}}\right) \cdot \frac{\frac{\sqrt{2}}{z}}{\sqrt[3]{y - t}}}\]

   12. Simplified 3.1

       \[\leadsto \color{blue}{\left(x \cdot \frac{\sqrt{2}}{\sqrt[3]{y - t} \cdot \sqrt[3]{y - t}}\right)} \cdot \frac{\frac{\sqrt{2}}{z}}{\sqrt[3]{y - t}}\]

   if -4.0387324098205055e42 < z < 4.57265390019630824e110

   1. Initial program 2.9

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z}\]

   2. Simplified 3.1

      \[\leadsto \color{blue}{x \cdot \frac{2}{z \cdot \left(y - t\right)}}\]
   3. Using strategy rm

   4. Applied associate-*r/ 2.9

      \[\leadsto \color{blue}{\frac{x \cdot 2}{z \cdot \left(y - t\right)}}\]

   if 3.5651713612658004e199 < z

   1. Initial program 15.1

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z}\]

   2. Simplified 10.2

      \[\leadsto \color{blue}{x \cdot \frac{2}{z \cdot \left(y - t\right)}}\]
   3. Using strategy rm

   4. Applied associate-/r* 9.7

      \[\leadsto x \cdot \color{blue}{\frac{\frac{2}{z}}{y - t}}\]
   5. Using strategy rm

   6. Applied div-inv 9.7

      \[\leadsto x \cdot \color{blue}{\left(\frac{2}{z} \cdot \frac{1}{y - t}\right)}\]

   7. Applied associate-*r* 2.6

      \[\leadsto \color{blue}{\left(x \cdot \frac{2}{z}\right) \cdot \frac{1}{y - t}}\]
3. Recombined 3 regimes into one program.

4. Final simplification 2.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -4.0387324098205055 \cdot 10^{42}:\\ \;\;\;\;\left(x \cdot \frac{\sqrt{2}}{\sqrt[3]{y - t} \cdot \sqrt[3]{y - t}}\right) \cdot \frac{\frac{\sqrt{2}}{z}}{\sqrt[3]{y - t}}\\ \mathbf{elif}\;z \le 4.57265390019630824 \cdot 10^{110}:\\ \;\;\;\;\frac{x \cdot 2}{z \cdot \left(y - t\right)}\\ \mathbf{elif}\;z \le 3.5651713612658004 \cdot 10^{199}:\\ \;\;\;\;\left(x \cdot \frac{\sqrt{2}}{\sqrt[3]{y - t} \cdot \sqrt[3]{y - t}}\right) \cdot \frac{\frac{\sqrt{2}}{z}}{\sqrt[3]{y - t}}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \frac{2}{z}\right) \cdot \frac{1}{y - t}\\ \end{array}\]

Reproduce

herbie shell --seed 2020184 
(FPCore (x y z t)
  :name "Linear.Projection:infinitePerspective from linear-1.19.1.3, A"
  :precision binary64

  :herbie-target
  (if (< (/ (* x 2.0) (- (* y z) (* t z))) -2.559141628295061e-13) (* (/ x (* (- y t) z)) 2.0) (if (< (/ (* x 2.0) (- (* y z) (* t z))) 1.0450278273301259e-269) (/ (* (/ x z) 2.0) (- y t)) (* (/ x (* (- y t) z)) 2.0)))

  (/ (* x 2.0) (- (* y z) (* t z))))