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

Average Error: 7.5 → 0.9

Time: 4.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \cdot y - \left(z \cdot 9\right) \cdot t \le -4.18413359400388683 \cdot 10^{307} \lor \neg \left(x \cdot y - \left(z \cdot 9\right) \cdot t \le 2.42230842933814323 \cdot 10^{193}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{a}, \frac{y}{2}, -\frac{9 \cdot t}{2} \cdot \frac{z}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x \cdot y}{a} - 4.5 \cdot \frac{t \cdot z}{a}\\ \end{array}\]

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 temp;
	if (((((x * y) - ((z * 9.0) * t)) <= -4.184133594003887e+307) || !(((x * y) - ((z * 9.0) * t)) <= 2.4223084293381432e+193))) {
		temp = fma((x / a), (y / 2.0), -(((9.0 * t) / 2.0) * (z / a)));
	} else {
		temp = ((0.5 * ((x * y) / a)) - (4.5 * ((t * z) / a)));
	}
	return temp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	7.5
Target	5.8
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)) < -4.184133594003887e+307 or 2.4223084293381432e+193 < (- (* x y) (* (* z 9.0) t))

   1. Initial program 37.3

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

   3. Applied associate-*l* 37.1

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

   5. Applied div-sub 37.1

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

   7. Applied times-frac 19.6

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

   8. Applied fma-neg 19.6

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

   9. Simplified 1.2

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

   if -4.184133594003887e+307 < (- (* x y) (* (* z 9.0) t)) < 2.4223084293381432e+193

   1. Initial program 0.8

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

   2. Taylor expanded around 0 0.8

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

4. Final simplification 0.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - \left(z \cdot 9\right) \cdot t \le -4.18413359400388683 \cdot 10^{307} \lor \neg \left(x \cdot y - \left(z \cdot 9\right) \cdot t \le 2.42230842933814323 \cdot 10^{193}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{a}, \frac{y}{2}, -\frac{9 \cdot t}{2} \cdot \frac{z}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x \cdot y}{a} - 4.5 \cdot \frac{t \cdot z}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020058 +o rules:numerics
(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)))