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

Average Error: 7.8 → 1.4

Time: 11.3s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \cdot y - \left(z \cdot 9\right) \cdot t \le -1.9258315324214309 \cdot 10^{239} \lor \neg \left(x \cdot y - \left(z \cdot 9\right) \cdot t \le 2.8811691783977644 \cdot 10^{114}\right):\\ \;\;\;\;\frac{x}{a} \cdot \frac{y}{2} - \frac{z}{1} \cdot \left(\frac{9}{a} \cdot \frac{t}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y - z \cdot \left(9 \cdot t\right)}{a \cdot 2}\\ \end{array}\]

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) (x * y)) - ((double) (((double) (z * 9.0)) * t)))) <= -1.925831532421431e+239) || !(((double) (((double) (x * y)) - ((double) (((double) (z * 9.0)) * t)))) <= 2.8811691783977644e+114))) {
		VAR = ((double) (((double) (((double) (x / a)) * ((double) (y / 2.0)))) - ((double) (((double) (z / 1.0)) * ((double) (((double) (9.0 / a)) * ((double) (t / 2.0))))))));
	} else {
		VAR = ((double) (((double) (((double) (x * y)) - ((double) (z * ((double) (9.0 * t)))))) / ((double) (a * 2.0))));
	}
	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	7.8
Target	5.4
Herbie	1.4

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

Derivation

1. Split input into 2 regimes

2. if (- (* x y) (* (* z 9.0) t)) < -1.9258315324214309e239 or 2.8811691783977644e114 < (- (* x y) (* (* z 9.0) t))

   1. Initial program 24.2

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

   3. Applied div-sub 24.2

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

   5. Applied times-frac 13.7

      \[\leadsto \frac{x \cdot y}{a \cdot 2} - \color{blue}{\frac{z \cdot 9}{a} \cdot \frac{t}{2}}\]
   6. Using strategy rm

   7. Applied times-frac 2.7

      \[\leadsto \color{blue}{\frac{x}{a} \cdot \frac{y}{2}} - \frac{z \cdot 9}{a} \cdot \frac{t}{2}\]
   8. Using strategy rm

   9. Applied *-un-lft-identity 2.7

      \[\leadsto \frac{x}{a} \cdot \frac{y}{2} - \frac{z \cdot 9}{\color{blue}{1 \cdot a}} \cdot \frac{t}{2}\]

   10. Applied times-frac 2.6

       \[\leadsto \frac{x}{a} \cdot \frac{y}{2} - \color{blue}{\left(\frac{z}{1} \cdot \frac{9}{a}\right)} \cdot \frac{t}{2}\]

   11. Applied associate-*l* 2.6

       \[\leadsto \frac{x}{a} \cdot \frac{y}{2} - \color{blue}{\frac{z}{1} \cdot \left(\frac{9}{a} \cdot \frac{t}{2}\right)}\]

   if -1.9258315324214309e239 < (- (* x y) (* (* z 9.0) t)) < 2.8811691783977644e114

   1. Initial program 0.9

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

   3. Applied associate-*l* 0.9

      \[\leadsto \frac{x \cdot y - \color{blue}{z \cdot \left(9 \cdot t\right)}}{a \cdot 2}\]
3. Recombined 2 regimes into one program.

4. Final simplification 1.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - \left(z \cdot 9\right) \cdot t \le -1.9258315324214309 \cdot 10^{239} \lor \neg \left(x \cdot y - \left(z \cdot 9\right) \cdot t \le 2.8811691783977644 \cdot 10^{114}\right):\\ \;\;\;\;\frac{x}{a} \cdot \frac{y}{2} - \frac{z}{1} \cdot \left(\frac{9}{a} \cdot \frac{t}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y - z \cdot \left(9 \cdot t\right)}{a \cdot 2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020162 
(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)))