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

Average Error: 7.9 → 0.9

Time: 22.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \cdot y - \left(z \cdot 9\right) \cdot t = -\infty \lor \neg \left(x \cdot y - \left(z \cdot 9\right) \cdot t \le 2.8728438966309182 \cdot 10^{213}\right):\\ \;\;\;\;\frac{y}{\frac{a \cdot 2}{x}} - \frac{9 \cdot z}{2} \cdot \frac{t}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{a \cdot 2} - \frac{z \cdot \frac{t}{1}}{\frac{2}{9} \cdot a}\\ \end{array}\]

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) || !(((double) (((double) (x * y)) - ((double) (((double) (z * 9.0)) * t)))) <= 2.872843896630918e+213))) {
		VAR = ((double) (((double) (y / ((double) (((double) (a * 2.0)) / x)))) - ((double) (((double) (((double) (9.0 * z)) / 2.0)) * ((double) (t / a))))));
	} else {
		VAR = ((double) (((double) (((double) (x * y)) / ((double) (a * 2.0)))) - ((double) (((double) (z * ((double) (t / 1.0)))) / ((double) (((double) (2.0 / 9.0)) * 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	7.9
Target	5.5
Herbie	0.9

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

Derivation

1. Split input into 2 regimes

2. if (- (* x y) (* (* z 9.0) t)) < -inf.0 or 2.872843896630918e+213 < (- (* x y) (* (* z 9.0) t))

   1. Initial program 42.2

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

   3. Applied associate-*l* 41.9

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

   5. Applied add-sqr-sqrt 41.9

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

   6. Applied associate-*l* 42.0

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

   7. Applied associate-*r* 42.2

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

   9. Applied div-sub 42.2

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

   10. Simplified 22.3

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

   12. Applied *-commutative 22.3

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

   13. Applied associate-/l* 1.3

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

   if -inf.0 < (- (* x y) (* (* z 9.0) t)) < 2.872843896630918e+213

   1. Initial program 0.8

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

   3. Applied associate-*l* 0.9

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

   5. Applied add-sqr-sqrt 0.9

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

   6. Applied associate-*l* 0.9

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

   7. Applied associate-*r* 0.9

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

   9. Applied div-sub 0.9

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

   10. Simplified 5.5

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

   12. Applied *-un-lft-identity 5.5

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

   13. Applied associate-/r* 5.5

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

   14. Applied *-commutative 5.5

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

   15. Applied associate-/l* 5.5

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

   16. Applied frac-times 0.9

       \[\leadsto \frac{x \cdot y}{a \cdot 2} - \color{blue}{\frac{z \cdot \frac{t}{1}}{\frac{2}{9} \cdot a}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - \left(z \cdot 9\right) \cdot t = -\infty \lor \neg \left(x \cdot y - \left(z \cdot 9\right) \cdot t \le 2.8728438966309182 \cdot 10^{213}\right):\\ \;\;\;\;\frac{y}{\frac{a \cdot 2}{x}} - \frac{9 \cdot z}{2} \cdot \frac{t}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{a \cdot 2} - \frac{z \cdot \frac{t}{1}}{\frac{2}{9} \cdot a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020114 
(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)))