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

Average Error: 7.9 → 1.6

Time: 7.1s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \cdot y - \left(z \cdot 9\right) \cdot t \leq -4.6149648683498855 \cdot 10^{+202}:\\ \;\;\;\;\frac{y}{a} \cdot \frac{x}{2} - t \cdot \left(\frac{z}{a} \cdot \frac{9}{2}\right)\\ \mathbf{elif}\;x \cdot y - \left(z \cdot 9\right) \cdot t \leq 2.4060341540881886 \cdot 10^{+130}:\\ \;\;\;\;\frac{1}{a} \cdot \frac{x \cdot y - z \cdot \left(9 \cdot t\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{a \cdot 2} - \frac{9}{2} \cdot \left(t \cdot \frac{z}{a}\right)\\ \end{array}\]

C

double code(double x, double y, double z, double t, double a) {
	return (((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)))) <= -4.6149648683498855e+202)) {
		VAR = ((double) (((double) ((y / a) * (x / 2.0))) - ((double) (t * ((double) ((z / a) * (9.0 / 2.0)))))));
	} else {
		double VAR_1;
		if ((((double) (((double) (x * y)) - ((double) (((double) (z * 9.0)) * t)))) <= 2.4060341540881886e+130)) {
			VAR_1 = ((double) ((1.0 / a) * (((double) (((double) (x * y)) - ((double) (z * ((double) (9.0 * t)))))) / 2.0)));
		} else {
			VAR_1 = ((double) (((double) (y * (x / ((double) (a * 2.0))))) - ((double) ((9.0 / 2.0) * ((double) (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	5.6
Herbie	1.6

\[\begin{array}{l} \mathbf{if}\;a < -2.090464557976709 \cdot 10^{+86}:\\ \;\;\;\;0.5 \cdot \frac{y \cdot x}{a} - 4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{elif}\;a < 2.144030707833976 \cdot 10^{+99}:\\ \;\;\;\;\frac{x \cdot y - z \cdot \left(9 \cdot t\right)}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} \cdot \left(x \cdot 0.5\right) - \frac{t}{a} \cdot \left(z \cdot 4.5\right)\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if (- (* x y) (* (* z 9.0) t)) < -4.6149648683498855e202

   1. Initial program 27.1

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

   3. Applied div-sub 27.1

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

   4. Simplified 15.5

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

   5. Simplified 1.2

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

   7. Applied *-un-lft-identity 1.2

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

   8. Applied times-frac 1.1

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

   9. Applied associate-*r* 1.4

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

   10. Simplified 1.4

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

   if -4.6149648683498855e202 < (- (* x y) (* (* z 9.0) t)) < 2.4060341540881886e130

   1. Initial program 1.0

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

   3. Applied *-un-lft-identity 1.0

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

   4. Applied times-frac 1.1

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

   5. Simplified 1.1

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

   if 2.4060341540881886e130 < (- (* x y) (* (* z 9.0) t))

   1. Initial program 19.0

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

   3. Applied div-sub 19.0

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

   4. Simplified 12.3

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

   5. Simplified 3.2

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

   7. Applied associate-*r* 3.2

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

4. Final simplification 1.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - \left(z \cdot 9\right) \cdot t \leq -4.6149648683498855 \cdot 10^{+202}:\\ \;\;\;\;\frac{y}{a} \cdot \frac{x}{2} - t \cdot \left(\frac{z}{a} \cdot \frac{9}{2}\right)\\ \mathbf{elif}\;x \cdot y - \left(z \cdot 9\right) \cdot t \leq 2.4060341540881886 \cdot 10^{+130}:\\ \;\;\;\;\frac{1}{a} \cdot \frac{x \cdot y - z \cdot \left(9 \cdot t\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{a \cdot 2} - \frac{9}{2} \cdot \left(t \cdot \frac{z}{a}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020196 
(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)))