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

Average Error: 7.9 → 4.3

Time: 3.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \cdot y \le -1.7792708608241411 \cdot 10^{256}:\\ \;\;\;\;0.5 \cdot \left(x \cdot \frac{y}{a}\right) - 4.5 \cdot \frac{t \cdot z}{a}\\ \mathbf{elif}\;x \cdot y \le 3.7534540503272961 \cdot 10^{247}:\\ \;\;\;\;0.5 \cdot \frac{x \cdot y}{a} - \frac{4.5 \cdot \left(t \cdot z\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{a}{y}} - 4.5 \cdot \frac{t \cdot z}{a}\\ \end{array}\]

C

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

↓

double code(double x, double y, double z, double t, double a) {
	double VAR;
	if (((x * y) <= -1.779270860824141e+256)) {
		VAR = ((0.5 * (x * (y / a))) - (4.5 * ((t * z) / a)));
	} else {
		double VAR_1;
		if (((x * y) <= 3.753454050327296e+247)) {
			VAR_1 = ((0.5 * ((x * y) / a)) - ((4.5 * (t * z)) / a));
		} else {
			VAR_1 = ((0.5 * (x / (a / y))) - (4.5 * ((t * z) / a)));
		}
		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	7.9
Target	6.0
Herbie	4.3

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

Derivation

1. Split input into 3 regimes

2. if (* x y) < -1.779270860824141e+256

   1. Initial program 42.6

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

   2. Taylor expanded around 0 42.5

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

   4. Applied associate-/l* 5.4

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

   6. Applied div-inv 5.5

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

   7. Simplified 5.4

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

   if -1.779270860824141e+256 < (* x y) < 3.753454050327296e+247

   1. Initial program 4.3

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

   2. Taylor expanded around 0 4.2

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

   4. Applied associate-*r/ 4.2

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

   if 3.753454050327296e+247 < (* x y)

   1. Initial program 38.6

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

   2. Taylor expanded around 0 38.4

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

   4. Applied associate-/l* 5.5

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

4. Final simplification 4.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \le -1.7792708608241411 \cdot 10^{256}:\\ \;\;\;\;0.5 \cdot \left(x \cdot \frac{y}{a}\right) - 4.5 \cdot \frac{t \cdot z}{a}\\ \mathbf{elif}\;x \cdot y \le 3.7534540503272961 \cdot 10^{247}:\\ \;\;\;\;0.5 \cdot \frac{x \cdot y}{a} - \frac{4.5 \cdot \left(t \cdot z\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{a}{y}} - 4.5 \cdot \frac{t \cdot z}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020078 
(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 t))) (* a 2)) (- (* (/ y a) (* x 0.5)) (* (/ t a) (* z 4.5)))))

  (/ (- (* x y) (* (* z 9) t)) (* a 2)))