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

Average Error: 3.6 → 0.6

Time: 4.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \cdot 3 \le -1.6156012038310011 \cdot 10^{-15}:\\ \;\;\;\;\left(x - \frac{y}{z \cdot 3}\right) + t \cdot \frac{1}{\left(z \cdot 3\right) \cdot y}\\ \mathbf{elif}\;z \cdot 3 \le 3.0286975700409486 \cdot 10^{-93}:\\ \;\;\;\;\left(x - \frac{y}{z \cdot 3}\right) + \frac{1}{z \cdot 3} \cdot \frac{t}{y}\\ \mathbf{else}:\\ \;\;\;\;\left(x - \frac{y}{z \cdot 3}\right) + 0.333333333333333315 \cdot \frac{t}{z \cdot y}\\ \end{array}\]

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) <= -1.615601203831001e-15)) {
		VAR = ((x - (y / (z * 3.0))) + (t * (1.0 / ((z * 3.0) * y))));
	} else {
		double VAR_1;
		if (((z * 3.0) <= 3.0286975700409486e-93)) {
			VAR_1 = ((x - (y / (z * 3.0))) + ((1.0 / (z * 3.0)) * (t / y)));
		} else {
			VAR_1 = ((x - (y / (z * 3.0))) + (0.3333333333333333 * (t / (z * 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	3.6
Target	1.8
Herbie	0.6

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

Derivation

1. Split input into 3 regimes

2. if (* z 3.0) < -1.615601203831001e-15

   1. Initial program 0.4

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

   3. Applied div-inv 0.4

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

   if -1.615601203831001e-15 < (* z 3.0) < 3.0286975700409486e-93

   1. Initial program 13.3

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

   3. Applied *-un-lft-identity 13.3

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

   4. Applied times-frac 0.3

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

   if 3.0286975700409486e-93 < (* z 3.0)

   1. Initial program 0.9

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

   2. Taylor expanded around 0 0.9

      \[\leadsto \left(x - \frac{y}{z \cdot 3}\right) + \color{blue}{0.333333333333333315 \cdot \frac{t}{z \cdot y}}\]
3. Recombined 3 regimes into one program.

4. Final simplification 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot 3 \le -1.6156012038310011 \cdot 10^{-15}:\\ \;\;\;\;\left(x - \frac{y}{z \cdot 3}\right) + t \cdot \frac{1}{\left(z \cdot 3\right) \cdot y}\\ \mathbf{elif}\;z \cdot 3 \le 3.0286975700409486 \cdot 10^{-93}:\\ \;\;\;\;\left(x - \frac{y}{z \cdot 3}\right) + \frac{1}{z \cdot 3} \cdot \frac{t}{y}\\ \mathbf{else}:\\ \;\;\;\;\left(x - \frac{y}{z \cdot 3}\right) + 0.333333333333333315 \cdot \frac{t}{z \cdot y}\\ \end{array}\]

Reproduce

herbie shell --seed 2020103 
(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))))