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

Average Error: 3.4 → 0.4

Time: 3.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \cdot 3 \le -310944318954.955505:\\ \;\;\;\;\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{3 \cdot \left(z \cdot y\right)}\\ \mathbf{elif}\;z \cdot 3 \le 2.3366008709042155 \cdot 10^{-37}:\\ \;\;\;\;\left(x - \frac{y}{z \cdot 3}\right) + \frac{1}{z} \cdot \frac{\frac{t}{3}}{y}\\ \mathbf{else}:\\ \;\;\;\;\left(x - \frac{\frac{y}{z}}{3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y}\\ \end{array}\]

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 ((((double) (z * 3.0)) <= -310944318954.9555)) {
		VAR = ((double) (((double) (x - ((double) (y / ((double) (z * 3.0)))))) + ((double) (t / ((double) (3.0 * ((double) (z * y))))))));
	} else {
		double VAR_1;
		if ((((double) (z * 3.0)) <= 2.3366008709042155e-37)) {
			VAR_1 = ((double) (((double) (x - ((double) (y / ((double) (z * 3.0)))))) + ((double) (((double) (1.0 / z)) * ((double) (((double) (t / 3.0)) / 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	3.4
Target	1.8
Herbie	0.4

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

Derivation

1. Split input into 3 regimes

2. if (* z 3.0) < -310944318954.9555

   1. Initial program 0.5

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

   2. Taylor expanded around 0 0.5

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

   if -310944318954.9555 < (* z 3.0) < 2.3366008709042155e-37

   1. Initial program 10.1

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

   3. Applied associate-/r* 3.1

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

   5. Applied *-un-lft-identity 3.1

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

   6. Applied *-un-lft-identity 3.1

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

   7. Applied times-frac 3.2

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

   8. Applied times-frac 0.3

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

   9. Simplified 0.3

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

   if 2.3366008709042155e-37 < (* z 3.0)

   1. Initial program 0.3

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

   3. Applied associate-/r* 0.3

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

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot 3 \le -310944318954.955505:\\ \;\;\;\;\left(x - \frac{y}{z \cdot 3}\right) + \frac{t}{3 \cdot \left(z \cdot y\right)}\\ \mathbf{elif}\;z \cdot 3 \le 2.3366008709042155 \cdot 10^{-37}:\\ \;\;\;\;\left(x - \frac{y}{z \cdot 3}\right) + \frac{1}{z} \cdot \frac{\frac{t}{3}}{y}\\ \mathbf{else}:\\ \;\;\;\;\left(x - \frac{\frac{y}{z}}{3}\right) + \frac{t}{\left(z \cdot 3\right) \cdot y}\\ \end{array}\]

Reproduce

herbie shell --seed 2020130 
(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))))