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

Average Error: 7.6 → 0.4

Time: 6.9s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \cdot y - \left(z \cdot 9\right) \cdot t = -inf.0:\\ \;\;\;\;\sqrt{0.5} \cdot \left(x \cdot \left(\sqrt{0.5} \cdot \frac{y}{a}\right)\right) - 4.5 \cdot \left(z \cdot \frac{t}{a}\right)\\ \mathbf{elif}\;x \cdot y - \left(z \cdot 9\right) \cdot t \le -3.5205740236497639 \cdot 10^{-220} \lor \neg \left(x \cdot y - \left(z \cdot 9\right) \cdot t \le 6.14121788209426953 \cdot 10^{-185}\right) \land x \cdot y - \left(z \cdot 9\right) \cdot t \le 2.5285528398041918 \cdot 10^{255}:\\ \;\;\;\;\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{a \cdot 2} - z \cdot \frac{9 \cdot t}{a \cdot 2}\\ \end{array}\]

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)))) <= -inf.0)) {
		VAR = ((double) (((double) (((double) sqrt(0.5)) * ((double) (x * ((double) (((double) sqrt(0.5)) * ((double) (y / a)))))))) - ((double) (4.5 * ((double) (z * ((double) (t / a))))))));
	} else {
		double VAR_1;
		if (((((double) (((double) (x * y)) - ((double) (((double) (z * 9.0)) * t)))) <= -3.520574023649764e-220) || (!(((double) (((double) (x * y)) - ((double) (((double) (z * 9.0)) * t)))) <= 6.14121788209427e-185) && (((double) (((double) (x * y)) - ((double) (((double) (z * 9.0)) * t)))) <= 2.528552839804192e+255)))) {
			VAR_1 = ((double) (((double) (((double) (x * y)) - ((double) (((double) (z * 9.0)) * t)))) / ((double) (a * 2.0))));
		} else {
			VAR_1 = ((double) (((double) (x * ((double) (y / ((double) (a * 2.0)))))) - ((double) (z * ((double) (((double) (9.0 * t)) / ((double) (a * 2.0))))))));
		}
		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	5.4
Herbie	0.4

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

Derivation

1. Split input into 3 regimes

2. if (- (* x y) (* (* z 9.0) t)) < -inf.0

   1. Initial program 64.0

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

   2. Taylor expanded around 0 63.4

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

   3. Simplified 0.5

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

   5. Applied add-sqr-sqrt 1.0

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

   6. Applied associate-*l* 0.8

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

   7. Simplified 0.8

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

   if -inf.0 < (- (* x y) (* (* z 9.0) t)) < -3.5205740236497639e-220 or 6.14121788209426953e-185 < (- (* x y) (* (* z 9.0) t)) < 2.5285528398041918e255

   1. Initial program 0.3

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

   if -3.5205740236497639e-220 < (- (* x y) (* (* z 9.0) t)) < 6.14121788209426953e-185 or 2.5285528398041918e255 < (- (* x y) (* (* z 9.0) t))

   1. Initial program 25.5

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

   3. Applied div-sub 25.5

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

   4. Simplified 14.7

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

   5. Simplified 0.8

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

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - \left(z \cdot 9\right) \cdot t = -inf.0:\\ \;\;\;\;\sqrt{0.5} \cdot \left(x \cdot \left(\sqrt{0.5} \cdot \frac{y}{a}\right)\right) - 4.5 \cdot \left(z \cdot \frac{t}{a}\right)\\ \mathbf{elif}\;x \cdot y - \left(z \cdot 9\right) \cdot t \le -3.5205740236497639 \cdot 10^{-220} \lor \neg \left(x \cdot y - \left(z \cdot 9\right) \cdot t \le 6.14121788209426953 \cdot 10^{-185}\right) \land x \cdot y - \left(z \cdot 9\right) \cdot t \le 2.5285528398041918 \cdot 10^{255}:\\ \;\;\;\;\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{a \cdot 2} - z \cdot \frac{9 \cdot t}{a \cdot 2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020184 
(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)))