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

Average Error: 3.5 → 0.6

Time: 4.1s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le -1.85718037152604949 \cdot 10^{-125}:\\ \;\;\;\;\left(x - y \cdot \frac{1}{z \cdot 3}\right) + \frac{t}{y \cdot \left(z \cdot 3\right)}\\ \mathbf{elif}\;t \le 6.76851578320920031 \cdot 10^{-15}:\\ \;\;\;\;\left(x - \frac{y}{z \cdot 3}\right) + \frac{1}{z \cdot 3} \cdot \frac{t}{y}\\ \mathbf{else}:\\ \;\;\;\;\left(x - \frac{y}{z \cdot 3}\right) + \frac{1}{3 \cdot \frac{y \cdot z}{t}}\\ \end{array}\]

C

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

↓

double code(double x, double y, double z, double t) {
	double VAR;
	if ((t <= -1.8571803715260495e-125)) {
		VAR = ((double) (((double) (x - ((double) (y * (1.0 / ((double) (z * 3.0))))))) + (t / ((double) (y * ((double) (z * 3.0)))))));
	} else {
		double VAR_1;
		if ((t <= 6.7685157832092e-15)) {
			VAR_1 = ((double) (((double) (x - (y / ((double) (z * 3.0))))) + ((double) ((1.0 / ((double) (z * 3.0))) * (t / y)))));
		} else {
			VAR_1 = ((double) (((double) (x - (y / ((double) (z * 3.0))))) + (1.0 / ((double) (3.0 * (((double) (y * z)) / t))))));
		}
		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	2.0
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 t < -1.85718037152604949e-125

   1. Initial program 1.3

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

   3. Applied div-inv 1.3

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

   if -1.85718037152604949e-125 < t < 6.76851578320920031e-15

   1. Initial program 6.6

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

   3. Applied *-un-lft-identity 6.6

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

   4. Applied times-frac 0.2

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

   if 6.76851578320920031e-15 < t

   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 clear-num 0.6

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

   4. Simplified 0.5

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

4. Final simplification 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -1.85718037152604949 \cdot 10^{-125}:\\ \;\;\;\;\left(x - y \cdot \frac{1}{z \cdot 3}\right) + \frac{t}{y \cdot \left(z \cdot 3\right)}\\ \mathbf{elif}\;t \le 6.76851578320920031 \cdot 10^{-15}:\\ \;\;\;\;\left(x - \frac{y}{z \cdot 3}\right) + \frac{1}{z \cdot 3} \cdot \frac{t}{y}\\ \mathbf{else}:\\ \;\;\;\;\left(x - \frac{y}{z \cdot 3}\right) + \frac{1}{3 \cdot \frac{y \cdot z}{t}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020182 
(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))))