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

Average Error: 7.2 → 1.4

Time: 3.8s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \cdot y - \left(z \cdot 9\right) \cdot t \le -8.6932731608256353 \cdot 10^{178} \lor \neg \left(x \cdot y - \left(z \cdot 9\right) \cdot t \le 9.7089920573828278 \cdot 10^{301}\right):\\ \;\;\;\;0.5 \cdot \left(x \cdot \frac{y}{a}\right) - 4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\frac{x \cdot y}{\sqrt[3]{a} \cdot \sqrt[3]{a}}}{\sqrt[3]{a}} - 4.5 \cdot \frac{t \cdot z}{a}\\ \end{array}\]

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) - ((z * 9.0) * t)) <= -8.693273160825635e+178) || !(((x * y) - ((z * 9.0) * t)) <= 9.708992057382828e+301))) {
		VAR = ((0.5 * (x * (y / a))) - (4.5 * (t / (a / z))));
	} else {
		VAR = ((0.5 * (((x * y) / (cbrt(a) * cbrt(a))) / cbrt(a))) - (4.5 * ((t * z) / a)));
	}
	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	7.2
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)) < -8.693273160825635e+178 or 9.708992057382828e+301 < (- (* x y) (* (* z 9.0) t))

   1. Initial program 33.7

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

   2. Taylor expanded around 0 33.6

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

   4. Applied associate-/l* 17.1

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

   6. Applied *-un-lft-identity 17.1

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

   7. Applied times-frac 1.5

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

   8. Simplified 1.5

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

   if -8.693273160825635e+178 < (- (* x y) (* (* z 9.0) t)) < 9.708992057382828e+301

   1. Initial program 0.9

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

   2. Taylor expanded around 0 0.9

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

   4. Applied add-cube-cbrt 1.4

      \[\leadsto 0.5 \cdot \frac{x \cdot y}{\color{blue}{\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \sqrt[3]{a}}} - 4.5 \cdot \frac{t \cdot z}{a}\]

   5. Applied associate-/r* 1.4

      \[\leadsto 0.5 \cdot \color{blue}{\frac{\frac{x \cdot y}{\sqrt[3]{a} \cdot \sqrt[3]{a}}}{\sqrt[3]{a}}} - 4.5 \cdot \frac{t \cdot z}{a}\]
3. Recombined 2 regimes into one program.

4. Final simplification 1.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - \left(z \cdot 9\right) \cdot t \le -8.6932731608256353 \cdot 10^{178} \lor \neg \left(x \cdot y - \left(z \cdot 9\right) \cdot t \le 9.7089920573828278 \cdot 10^{301}\right):\\ \;\;\;\;0.5 \cdot \left(x \cdot \frac{y}{a}\right) - 4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\frac{x \cdot y}{\sqrt[3]{a} \cdot \sqrt[3]{a}}}{\sqrt[3]{a}} - 4.5 \cdot \frac{t \cdot z}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020079 
(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)))