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

Average Error: 8.0 → 1.2

Time: 12.7s

Precision: binary64

\[[x, y]=\mathsf{sort}([x, y])\]

Math

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

↓

\[\begin{array}{l} t_1 := x \cdot y - \left(z \cdot 9\right) \cdot t\\ \mathbf{if}\;t_1 \leq -2.3640912854159315 \cdot 10^{+152} \lor \neg \left(t_1 \leq 1.407837306476548 \cdot 10^{+202}\right):\\ \;\;\;\;0.5 \cdot \frac{y}{\frac{a}{x}} - 4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{a \cdot 2}\\ \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (/ (- (* x y) (* (* z 9.0) t)) (* a 2.0)))

↓

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- (* x y) (* (* z 9.0) t))))
   (if (or (<= t_1 -2.3640912854159315e+152)
           (not (<= t_1 1.407837306476548e+202)))
     (- (* 0.5 (/ y (/ a x))) (* 4.5 (/ t (/ a z))))
     (/ t_1 (* a 2.0)))))

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 t_1 = (x * y) - ((z * 9.0) * t);
	double tmp;
	if ((t_1 <= -2.3640912854159315e+152) || !(t_1 <= 1.407837306476548e+202)) {
		tmp = (0.5 * (y / (a / x))) - (4.5 * (t / (a / z)));
	} else {
		tmp = t_1 / (a * 2.0);
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	8.0
Target	5.6
Herbie	1.2

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

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 x y) (*.f64 (*.f64 z 9) t)) < -2.36409128541593147e152 or 1.407837306476548e202 < (-.f64 (*.f64 x y) (*.f64 (*.f64 z 9) t))

   1. Initial program 25.3

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

   2. Simplified 25.1

      \[\leadsto \color{blue}{\mathsf{fma}\left(-9, z \cdot t, x \cdot y\right) \cdot \frac{0.5}{a}} \]

   3. Taylor expanded in z around 0 25.0

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

   4. Applied associate-/l*_binary64 14.2

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

   5. Applied associate-/l*_binary64 2.0

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

   if -2.36409128541593147e152 < (-.f64 (*.f64 x y) (*.f64 (*.f64 z 9) t)) < 1.407837306476548e202

   1. Initial program 0.9

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

4. Final simplification 1.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - \left(z \cdot 9\right) \cdot t \leq -2.3640912854159315 \cdot 10^{+152} \lor \neg \left(x \cdot y - \left(z \cdot 9\right) \cdot t \leq 1.407837306476548 \cdot 10^{+202}\right):\\ \;\;\;\;0.5 \cdot \frac{y}{\frac{a}{x}} - 4.5 \cdot \frac{t}{\frac{a}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y - \left(z \cdot 9\right) \cdot t}{a \cdot 2}\\ \end{array} \]

Reproduce

herbie shell --seed 2021275 
(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)))