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

Average Error: 7.6 → 5.8

Time: 5.3s

Precision: binary64

Math

\[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}\]

↓

\[\begin{array}{l} \mathbf{if}\;\left(z \cdot 9\right) \cdot t \le -2.8676201329546916 \cdot 10^{53}:\\ \;\;\;\;0.5 \cdot \frac{x \cdot y}{a} - 4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{elif}\;\left(z \cdot 9\right) \cdot t \le -8.7773831707727847 \cdot 10^{-296}:\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{a}{y}} - 4.5 \cdot \frac{t \cdot z}{a}\\ \mathbf{elif}\;\left(z \cdot 9\right) \cdot t \le 9.45536168987990705 \cdot 10^{196}:\\ \;\;\;\;\frac{x \cdot y - 9 \cdot \left(t \cdot z\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x \cdot y}{a} - \left(t \cdot 4.5\right) \cdot \frac{z}{a}\\ \end{array}\]

C

double code(double x, double y, double z, double t, double a) {
	return ((double) (((double) (((double) (x * y)) - ((double) (((double) (z * 9.0)) * t)))) / ((double) (a * 2.0))));
}

↓

double code(double x, double y, double z, double t, double a) {
	double VAR;
	if ((((double) (((double) (z * 9.0)) * t)) <= -2.8676201329546916e+53)) {
		VAR = ((double) (((double) (0.5 * ((double) (((double) (x * y)) / a)))) - ((double) (4.5 * ((double) (t / ((double) (a / z))))))));
	} else {
		double VAR_1;
		if ((((double) (((double) (z * 9.0)) * t)) <= -8.777383170772785e-296)) {
			VAR_1 = ((double) (((double) (0.5 * ((double) (x / ((double) (a / y)))))) - ((double) (4.5 * ((double) (((double) (t * z)) / a))))));
		} else {
			double VAR_2;
			if ((((double) (((double) (z * 9.0)) * t)) <= 9.455361689879907e+196)) {
				VAR_2 = ((double) (((double) (((double) (x * y)) - ((double) (9.0 * ((double) (t * z)))))) / ((double) (a * 2.0))));
			} else {
				VAR_2 = ((double) (((double) (0.5 * ((double) (((double) (x * y)) / a)))) - ((double) (((double) (t * 4.5)) * ((double) (z / a))))));
			}
			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

Bits error versus a

Target

Original	7.6
Target	5.3
Herbie	5.8

\[\begin{array}{l} \mathbf{if}\;a \lt -2.090464557976709 \cdot 10^{86}:\\ \;\;\;\;0.5 \cdot \frac{y \cdot x}{a} - 4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{elif}\;a \lt 2.14403070783397609 \cdot 10^{99}:\\ \;\;\;\;\frac{x \cdot y - z \cdot \left(9 \cdot t\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} \cdot \left(x \cdot 0.5\right) - \frac{t}{a} \cdot \left(z \cdot 4.5\right)\\ \end{array}\]

Derivation

1. Split input into 4 regimes

2. if (* (* z 9.0) t) < -2.8676201329546916e53

   1. Initial program 15.2

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}\]

   2. Taylor expanded around 0 14.6

      \[\leadsto \color{blue}{0.5 \cdot \frac{x \cdot y}{a} - 4.5 \cdot \frac{t \cdot z}{a}}\]
   3. Using strategy rm

   4. Applied associate-/l* 8.5

      \[\leadsto 0.5 \cdot \frac{x \cdot y}{a} - 4.5 \cdot \color{blue}{\frac{t}{\frac{a}{z}}}\]

   if -2.8676201329546916e53 < (* (* z 9.0) t) < -8.7773831707727847e-296

   1. Initial program 3.2

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}\]

   2. Taylor expanded around 0 3.2

      \[\leadsto \color{blue}{0.5 \cdot \frac{x \cdot y}{a} - 4.5 \cdot \frac{t \cdot z}{a}}\]
   3. Using strategy rm

   4. Applied associate-/l* 6.5

      \[\leadsto 0.5 \cdot \color{blue}{\frac{x}{\frac{a}{y}}} - 4.5 \cdot \frac{t \cdot z}{a}\]

   if -8.7773831707727847e-296 < (* (* z 9.0) t) < 9.45536168987990705e196

   1. Initial program 4.5

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}\]

   2. Taylor expanded around inf 4.5

      \[\leadsto \frac{\color{blue}{x \cdot y - 9 \cdot \left(t \cdot z\right)}}{a \cdot 2}\]

   if 9.45536168987990705e196 < (* (* z 9.0) t)

   1. Initial program 29.7

      \[\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}\]

   2. Taylor expanded around 0 29.4

      \[\leadsto \color{blue}{0.5 \cdot \frac{x \cdot y}{a} - 4.5 \cdot \frac{t \cdot z}{a}}\]
   3. Using strategy rm

   4. Applied *-un-lft-identity 29.4

      \[\leadsto 0.5 \cdot \frac{x \cdot y}{a} - 4.5 \cdot \frac{t \cdot z}{\color{blue}{1 \cdot a}}\]

   5. Applied times-frac 5.4

      \[\leadsto 0.5 \cdot \frac{x \cdot y}{a} - 4.5 \cdot \color{blue}{\left(\frac{t}{1} \cdot \frac{z}{a}\right)}\]

   6. Applied associate-*r* 5.8

      \[\leadsto 0.5 \cdot \frac{x \cdot y}{a} - \color{blue}{\left(4.5 \cdot \frac{t}{1}\right) \cdot \frac{z}{a}}\]

   7. Simplified 5.8

      \[\leadsto 0.5 \cdot \frac{x \cdot y}{a} - \color{blue}{\left(t \cdot 4.5\right)} \cdot \frac{z}{a}\]
3. Recombined 4 regimes into one program.

4. Final simplification 5.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z \cdot 9\right) \cdot t \le -2.8676201329546916 \cdot 10^{53}:\\ \;\;\;\;0.5 \cdot \frac{x \cdot y}{a} - 4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{elif}\;\left(z \cdot 9\right) \cdot t \le -8.7773831707727847 \cdot 10^{-296}:\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{a}{y}} - 4.5 \cdot \frac{t \cdot z}{a}\\ \mathbf{elif}\;\left(z \cdot 9\right) \cdot t \le 9.45536168987990705 \cdot 10^{196}:\\ \;\;\;\;\frac{x \cdot y - 9 \cdot \left(t \cdot z\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x \cdot y}{a} - \left(t \cdot 4.5\right) \cdot \frac{z}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020161 
(FPCore (x y z t a)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, I"
  :precision binary64

  :herbie-target
  (if (< a -2.090464557976709e+86) (- (* 0.5 (/ (* y x) a)) (* 4.5 (/ t (/ a z)))) (if (< a 2.144030707833976e+99) (/ (- (* x y) (* z (* 9.0 t))) (* a 2.0)) (- (* (/ y a) (* x 0.5)) (* (/ t a) (* z 4.5)))))

  (/ (- (* x y) (* (* z 9.0) t)) (* a 2.0)))