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

Average Error: 3.5 → 2.3

Time: 3.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \cdot 3 \le -1.19441357929082014 \cdot 10^{-136}:\\ \;\;\;\;\left(x - \frac{\frac{y}{3}}{z}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y}\\ \mathbf{elif}\;z \cdot 3 \le 4.1001370634933834 \cdot 10^{-235}:\\ \;\;\;\;\left(x - \frac{y}{z \cdot 3}\right) + \frac{\frac{t}{z \cdot 3}}{y}\\ \mathbf{elif}\;z \cdot 3 \le 1.2163856744143447 \cdot 10^{-91}:\\ \;\;\;\;\left(x - \frac{\frac{y}{3}}{z}\right) + \frac{t}{3} \cdot \frac{\frac{1}{z}}{y}\\ \mathbf{else}:\\ \;\;\;\;\left(x - \frac{y}{3} \cdot \frac{1}{z}\right) + \frac{\frac{t}{z}}{3 \cdot y}\\ \end{array}\]

C

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

↓

double code(double x, double y, double z, double t) {
	double VAR;
	if (((z * 3.0) <= -1.1944135792908201e-136)) {
		VAR = ((x - ((y / 3.0) / z)) + (t / ((z * 3.0) * y)));
	} else {
		double VAR_1;
		if (((z * 3.0) <= 4.1001370634933834e-235)) {
			VAR_1 = ((x - (y / (z * 3.0))) + ((t / (z * 3.0)) / y));
		} else {
			double VAR_2;
			if (((z * 3.0) <= 1.2163856744143447e-91)) {
				VAR_2 = ((x - ((y / 3.0) / z)) + ((t / 3.0) * ((1.0 / z) / y)));
			} else {
				VAR_2 = ((x - ((y / 3.0) * (1.0 / z))) + ((t / z) / (3.0 * y)));
			}
			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	3.5
Target	1.9
Herbie	2.3

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

Derivation

1. Split input into 4 regimes

2. if (* z 3.0) < -1.1944135792908201e-136

   1. Initial program 1.1

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

   3. Applied *-commutative 1.1

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

   4. Applied associate-/r* 1.0

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

   if -1.1944135792908201e-136 < (* z 3.0) < 4.1001370634933834e-235

   1. Initial program 22.0

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

   3. Applied associate-/r* 2.6

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

   if 4.1001370634933834e-235 < (* z 3.0) < 1.2163856744143447e-91

   1. Initial program 13.1

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

   3. Applied associate-*l* 13.1

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

   4. Applied associate-/r* 5.0

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

   6. Applied *-commutative 5.0

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

   7. Applied associate-/r* 5.0

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

   9. Applied div-inv 5.0

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

   10. Applied times-frac 14.2

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

   if 1.2163856744143447e-91 < (* z 3.0)

   1. Initial program 1.0

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

   3. Applied associate-*l* 0.9

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

   4. Applied associate-/r* 1.4

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

   6. Applied *-commutative 1.4

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

   7. Applied associate-/r* 1.4

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

   9. Applied div-inv 1.5

      \[\leadsto \left(x - \color{blue}{\frac{y}{3} \cdot \frac{1}{z}}\right) + \frac{\frac{t}{z}}{3 \cdot y}\]
3. Recombined 4 regimes into one program.

4. Final simplification 2.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot 3 \le -1.19441357929082014 \cdot 10^{-136}:\\ \;\;\;\;\left(x - \frac{\frac{y}{3}}{z}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y}\\ \mathbf{elif}\;z \cdot 3 \le 4.1001370634933834 \cdot 10^{-235}:\\ \;\;\;\;\left(x - \frac{y}{z \cdot 3}\right) + \frac{\frac{t}{z \cdot 3}}{y}\\ \mathbf{elif}\;z \cdot 3 \le 1.2163856744143447 \cdot 10^{-91}:\\ \;\;\;\;\left(x - \frac{\frac{y}{3}}{z}\right) + \frac{t}{3} \cdot \frac{\frac{1}{z}}{y}\\ \mathbf{else}:\\ \;\;\;\;\left(x - \frac{y}{3} \cdot \frac{1}{z}\right) + \frac{\frac{t}{z}}{3 \cdot y}\\ \end{array}\]

Reproduce

herbie shell --seed 2020071 +o rules:numerics
(FPCore (x y z t)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, H"
  :precision binary64

  :herbie-target
  (+ (- x (/ y (* z 3))) (/ (/ t (* z 3)) y))

  (+ (- x (/ y (* z 3))) (/ t (* (* z 3) y))))