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

Average Error: 21.1 → 18.4

Time: 11.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \cdot t = -\infty:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - \frac{1}{2} \cdot {y}^{2}\right) - \frac{a}{b \cdot 3}\\ \mathbf{elif}\;z \cdot t \le 5.41910633754577338 \cdot 10^{303}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \left(\left(\sqrt[3]{\cos \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right)} \cdot \sqrt[3]{\cos \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right)}\right) \cdot \sqrt[3]{\cos \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right)}\right) + \sin y \cdot \left(\left(\sqrt[3]{\sin \left(\frac{z \cdot t}{3}\right)} \cdot \sqrt[3]{\sin \left(\frac{z \cdot t}{3}\right)}\right) \cdot \sqrt[3]{\sin \left(\frac{z \cdot t}{3}\right)}\right)\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\log \left({\left(e^{2 \cdot \sqrt{x}}\right)}^{\left(\mathsf{fma}\left(\cos y, \cos \left(\frac{z \cdot t}{3}\right), \sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right)\right)}\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

C

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

↓

double code(double x, double y, double z, double t, double a, double b) {
	double VAR;
	if ((((double) (z * t)) <= -inf.0)) {
		VAR = ((double) (((double) (((double) (2.0 * ((double) sqrt(x)))) * ((double) (1.0 - ((double) (0.5 * ((double) pow(y, 2.0)))))))) - ((double) (a / ((double) (b * 3.0))))));
	} else {
		double VAR_1;
		if ((((double) (z * t)) <= 5.4191063375457734e+303)) {
			VAR_1 = ((double) (((double) (((double) (2.0 * ((double) sqrt(x)))) * ((double) (((double) (((double) cos(y)) * ((double) (((double) (((double) cbrt(((double) cos(((double) (0.3333333333333333 * ((double) (t * z)))))))) * ((double) cbrt(((double) cos(((double) (0.3333333333333333 * ((double) (t * z)))))))))) * ((double) cbrt(((double) cos(((double) (0.3333333333333333 * ((double) (t * z)))))))))))) + ((double) (((double) sin(y)) * ((double) (((double) (((double) cbrt(((double) sin(((double) (((double) (z * t)) / 3.0)))))) * ((double) cbrt(((double) sin(((double) (((double) (z * t)) / 3.0)))))))) * ((double) cbrt(((double) sin(((double) (((double) (z * t)) / 3.0)))))))))))))) - ((double) (a / ((double) (b * 3.0))))));
		} else {
			VAR_1 = ((double) (((double) log(((double) pow(((double) exp(((double) (2.0 * ((double) sqrt(x)))))), ((double) fma(((double) cos(y)), ((double) cos(((double) (((double) (z * t)) / 3.0)))), ((double) (((double) sin(y)) * ((double) sin(((double) (((double) (z * t)) / 3.0)))))))))))) - ((double) (a / ((double) (b * 3.0))))));
		}
		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

Bits error versus b

Target

Original	21.1
Target	18.8
Herbie	18.4

\[\begin{array}{l} \mathbf{if}\;z \lt -1.379333748723514 \cdot 10^{129}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\frac{1}{y} - \frac{\frac{0.333333333333333315}{z}}{t}\right) - \frac{\frac{a}{3}}{b}\\ \mathbf{elif}\;z \lt 3.51629061355598715 \cdot 10^{106}:\\ \;\;\;\;\left(\sqrt{x} \cdot 2\right) \cdot \cos \left(y - \frac{t}{3} \cdot z\right) - \frac{\frac{a}{3}}{b}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(y - \frac{\frac{0.333333333333333315}{z}}{t}\right) \cdot \left(2 \cdot \sqrt{x}\right) - \frac{\frac{a}{b}}{3}\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if (* z t) < -inf.0

   1. Initial program 64.0

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

   2. Taylor expanded around 0 45.2

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \color{blue}{\left(1 - \frac{1}{2} \cdot {y}^{2}\right)} - \frac{a}{b \cdot 3}\]

   if -inf.0 < (* z t) < 5.4191063375457734e+303

   1. Initial program 14.8

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

   3. Applied cos-diff 14.2

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

   4. Taylor expanded around inf 14.2

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \color{blue}{\cos \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right)} + \sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right) - \frac{a}{b \cdot 3}\]
   5. Using strategy rm

   6. Applied add-cube-cbrt 14.3

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \color{blue}{\left(\left(\sqrt[3]{\cos \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right)} \cdot \sqrt[3]{\cos \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right)}\right) \cdot \sqrt[3]{\cos \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right)}\right)} + \sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right) - \frac{a}{b \cdot 3}\]
   7. Using strategy rm

   8. Applied add-cube-cbrt 14.3

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \left(\left(\sqrt[3]{\cos \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right)} \cdot \sqrt[3]{\cos \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right)}\right) \cdot \sqrt[3]{\cos \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right)}\right) + \sin y \cdot \color{blue}{\left(\left(\sqrt[3]{\sin \left(\frac{z \cdot t}{3}\right)} \cdot \sqrt[3]{\sin \left(\frac{z \cdot t}{3}\right)}\right) \cdot \sqrt[3]{\sin \left(\frac{z \cdot t}{3}\right)}\right)}\right) - \frac{a}{b \cdot 3}\]

   if 5.4191063375457734e+303 < (* z t)

   1. Initial program 63.1

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

   3. Applied cos-diff 63.1

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \color{blue}{\left(\cos y \cdot \cos \left(\frac{z \cdot t}{3}\right) + \sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right)} - \frac{a}{b \cdot 3}\]
   4. Using strategy rm

   5. Applied add-log-exp 63.7

      \[\leadsto \color{blue}{\log \left(e^{\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(\frac{z \cdot t}{3}\right) + \sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right)}\right)} - \frac{a}{b \cdot 3}\]

   6. Simplified 47.1

      \[\leadsto \log \color{blue}{\left({\left(e^{2 \cdot \sqrt{x}}\right)}^{\left(\mathsf{fma}\left(\cos y, \cos \left(\frac{z \cdot t}{3}\right), \sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right)\right)}\right)} - \frac{a}{b \cdot 3}\]
3. Recombined 3 regimes into one program.

4. Final simplification 18.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t = -\infty:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - \frac{1}{2} \cdot {y}^{2}\right) - \frac{a}{b \cdot 3}\\ \mathbf{elif}\;z \cdot t \le 5.41910633754577338 \cdot 10^{303}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \left(\left(\sqrt[3]{\cos \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right)} \cdot \sqrt[3]{\cos \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right)}\right) \cdot \sqrt[3]{\cos \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right)}\right) + \sin y \cdot \left(\left(\sqrt[3]{\sin \left(\frac{z \cdot t}{3}\right)} \cdot \sqrt[3]{\sin \left(\frac{z \cdot t}{3}\right)}\right) \cdot \sqrt[3]{\sin \left(\frac{z \cdot t}{3}\right)}\right)\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\log \left({\left(e^{2 \cdot \sqrt{x}}\right)}^{\left(\mathsf{fma}\left(\cos y, \cos \left(\frac{z \cdot t}{3}\right), \sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right)\right)}\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

Reproduce

herbie shell --seed 2020120 +o rules:numerics
(FPCore (x y z t a b)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, K"
  :precision binary64

  :herbie-target
  (if (< z -1.379333748723514e+129) (- (* (* 2 (sqrt x)) (cos (- (/ 1 y) (/ (/ 0.3333333333333333 z) t)))) (/ (/ a 3) b)) (if (< z 3.516290613555987e+106) (- (* (* (sqrt x) 2) (cos (- y (* (/ t 3) z)))) (/ (/ a 3) b)) (- (* (cos (- y (/ (/ 0.3333333333333333 z) t))) (* 2 (sqrt x))) (/ (/ a b) 3))))

  (- (* (* 2 (sqrt x)) (cos (- y (/ (* z t) 3)))) (/ a (* b 3))))