Diagrams.Solve.Polynomial:cubForm from diagrams-solve-0.1, H

Average Error: 3.7 → 1.1

Time: 6.9s

Precision: binary64

Math

\[\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y}\]

↓

\[\begin{array}{l} \mathbf{if}\;y \le -4.4045042461635908 \cdot 10^{-280}:\\ \;\;\;\;\left(x - \frac{y}{z \cdot 3}\right) + \frac{\sqrt[3]{t} \cdot \sqrt[3]{t}}{z \cdot 3} \cdot \frac{\sqrt[3]{t}}{y}\\ \mathbf{elif}\;y \le 2.4643778162432899 \cdot 10^{-63}:\\ \;\;\;\;\left(x - \frac{y}{z \cdot 3}\right) + \frac{0.333333333333333315 \cdot \frac{t}{z}}{y}\\ \mathbf{else}:\\ \;\;\;\;\left(x - \frac{\frac{y}{z}}{3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y}\\ \end{array}\]

C

double code(double x, double y, double z, double t) {
	return ((double) (((double) (x - ((double) (y / ((double) (z * 3.0)))))) + ((double) (t / ((double) (((double) (z * 3.0)) * y))))));
}

↓

double code(double x, double y, double z, double t) {
	double VAR;
	if ((y <= -4.404504246163591e-280)) {
		VAR = ((double) (((double) (x - ((double) (y / ((double) (z * 3.0)))))) + ((double) (((double) (((double) (((double) cbrt(t)) * ((double) cbrt(t)))) / ((double) (z * 3.0)))) * ((double) (((double) cbrt(t)) / y))))));
	} else {
		double VAR_1;
		if ((y <= 2.46437781624329e-63)) {
			VAR_1 = ((double) (((double) (x - ((double) (y / ((double) (z * 3.0)))))) + ((double) (((double) (0.3333333333333333 * ((double) (t / z)))) / y))));
		} else {
			VAR_1 = ((double) (((double) (x - ((double) (((double) (y / z)) / 3.0)))) + ((double) (t / ((double) (((double) (z * 3.0)) * y))))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	3.7
Target	1.8
Herbie	1.1

\[\left(x - \frac{y}{z \cdot 3}\right) + \frac{\frac{t}{z \cdot 3}}{y}\]

Derivation

1. Split input into 3 regimes

2. if y < -4.4045042461635908e-280

   1. Initial program 3.6

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

   3. Applied add-cube-cbrt 3.8

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

   4. Applied times-frac 1.1

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

   if -4.4045042461635908e-280 < y < 2.4643778162432899e-63

   1. Initial program 9.1

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

   3. Applied associate-/r* 2.2

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

   4. Taylor expanded around 0 2.2

      \[\leadsto \left(x - \frac{y}{z \cdot 3}\right) + \frac{\color{blue}{0.333333333333333315 \cdot \frac{t}{z}}}{y}\]

   if 2.4643778162432899e-63 < y

   1. Initial program 0.5

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

   3. Applied associate-/r* 0.6

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

4. Final simplification 1.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -4.4045042461635908 \cdot 10^{-280}:\\ \;\;\;\;\left(x - \frac{y}{z \cdot 3}\right) + \frac{\sqrt[3]{t} \cdot \sqrt[3]{t}}{z \cdot 3} \cdot \frac{\sqrt[3]{t}}{y}\\ \mathbf{elif}\;y \le 2.4643778162432899 \cdot 10^{-63}:\\ \;\;\;\;\left(x - \frac{y}{z \cdot 3}\right) + \frac{0.333333333333333315 \cdot \frac{t}{z}}{y}\\ \mathbf{else}:\\ \;\;\;\;\left(x - \frac{\frac{y}{z}}{3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y}\\ \end{array}\]

Reproduce

herbie shell --seed 2020174 
(FPCore (x y z t)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, H"
  :precision binary64

  :herbie-target
  (+ (- x (/ y (* z 3.0))) (/ (/ t (* z 3.0)) y))

  (+ (- x (/ y (* z 3.0))) (/ t (* (* z 3.0) y))))