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

Average Error: 18.8 → 14.4

Time: 16.2s

Precision: binary64

Math

\[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3} \]

↓

\[\begin{array}{l} t_1 := 2 \cdot \sqrt{x}\\ t_2 := y - \frac{z \cdot t}{3}\\ \mathbf{if}\;t_2 \leq -1.7729749144956428 \cdot 10^{+308} \lor \neg \left(t_2 \leq 1.7727566483509542 \cdot 10^{+308}\right):\\ \;\;\;\;\mathsf{fma}\left(a, \frac{-0.3333333333333333}{b}, t_1\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \cos t_2 - \frac{a}{3 \cdot b}\\ \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (- (* (* 2.0 (sqrt x)) (cos (- y (/ (* z t) 3.0)))) (/ a (* b 3.0))))

↓

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* 2.0 (sqrt x))) (t_2 (- y (/ (* z t) 3.0))))
   (if (or (<= t_2 -1.7729749144956428e+308)
           (not (<= t_2 1.7727566483509542e+308)))
     (fma a (/ -0.3333333333333333 b) t_1)
     (- (* t_1 (cos t_2)) (/ a (* 3.0 b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	return ((2.0 * sqrt(x)) * cos(y - ((z * t) / 3.0))) - (a / (b * 3.0));
}

↓

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = 2.0 * sqrt(x);
	double t_2 = y - ((z * t) / 3.0);
	double tmp;
	if ((t_2 <= -1.7729749144956428e+308) || !(t_2 <= 1.7727566483509542e+308)) {
		tmp = fma(a, (-0.3333333333333333 / b), t_1);
	} else {
		tmp = (t_1 * cos(t_2)) - (a / (3.0 * b));
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Bits error versus b

Target

Original	18.8
Target	16.3
Herbie	14.4

\[\begin{array}{l} \mathbf{if}\;z < -1.3793337487235141 \cdot 10^{+129}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\frac{1}{y} - \frac{\frac{0.3333333333333333}{z}}{t}\right) - \frac{\frac{a}{3}}{b}\\ \mathbf{elif}\;z < 3.516290613555987 \cdot 10^{+106}:\\ \;\;\;\;\left(\sqrt{x} \cdot 2\right) \cdot \cos \left(y - \frac{t}{3} \cdot z\right) - \frac{\frac{a}{3}}{b}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(y - \frac{\frac{0.3333333333333333}{z}}{t}\right) \cdot \left(2 \cdot \sqrt{x}\right) - \frac{\frac{a}{b}}{3}\\ \end{array} \]

Derivation

1. Split input into 2 regimes

2. if (-.f64 y (/.f64 (*.f64 z t) 3)) < -1.77297491449564282e308 or 1.7727566483509542e308 < (-.f64 y (/.f64 (*.f64 z t) 3))

   1. Initial program 63.9

      \[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3} \]

   2. Simplified 63.9

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, \frac{-0.3333333333333333}{b}, \left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\mathsf{fma}\left(z, t \cdot -0.3333333333333333, y\right)\right)\right)} \]

   3. Taylor expanded in z around 0 29.7

      \[\leadsto \mathsf{fma}\left(a, \frac{-0.3333333333333333}{b}, \left(2 \cdot \sqrt{x}\right) \cdot \color{blue}{\cos y}\right) \]

   4. Taylor expanded in y around 0 29.7

      \[\leadsto \mathsf{fma}\left(a, \frac{-0.3333333333333333}{b}, \color{blue}{2 \cdot \sqrt{x}}\right) \]

   if -1.77297491449564282e308 < (-.f64 y (/.f64 (*.f64 z t) 3)) < 1.7727566483509542e308

   1. Initial program 12.1

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

4. Final simplification 14.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;y - \frac{z \cdot t}{3} \leq -1.7729749144956428 \cdot 10^{+308} \lor \neg \left(y - \frac{z \cdot t}{3} \leq 1.7727566483509542 \cdot 10^{+308}\right):\\ \;\;\;\;\mathsf{fma}\left(a, \frac{-0.3333333333333333}{b}, 2 \cdot \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{3 \cdot b}\\ \end{array} \]

Reproduce

herbie shell --seed 2022068 
(FPCore (x y z t a b)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, K"
  :precision binary64

  :herbie-target
  (if (< z -1.3793337487235141e+129) (- (* (* 2.0 (sqrt x)) (cos (- (/ 1.0 y) (/ (/ 0.3333333333333333 z) t)))) (/ (/ a 3.0) b)) (if (< z 3.516290613555987e+106) (- (* (* (sqrt x) 2.0) (cos (- y (* (/ t 3.0) z)))) (/ (/ a 3.0) b)) (- (* (cos (- y (/ (/ 0.3333333333333333 z) t))) (* 2.0 (sqrt x))) (/ (/ a b) 3.0))))

  (- (* (* 2.0 (sqrt x)) (cos (- y (/ (* z t) 3.0)))) (/ a (* b 3.0))))