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

Average Error: 21.2 → 18.6

Time: 15.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y - \frac{z \cdot t}{3} = -\infty \lor \neg \left(y - \frac{z \cdot t}{3} \le 2.470815013403332 \cdot 10^{307}\right):\\ \;\;\;\;\log \left({\left(e^{2 \cdot \sqrt{x}}\right)}^{\left(\mathsf{fma}\left(\cos y, \cos \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right), \sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right)\right)}\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right) + \sin y \cdot \sin \left(\frac{z \cdot \left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}{\frac{3}{\sqrt[3]{t}}}\right)\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

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) (y - ((double) (((double) (z * t)) / 3.0)))) <= -inf.0) || !(((double) (y - ((double) (((double) (z * t)) / 3.0)))) <= 2.470815013403332e+307))) {
		VAR = ((double) (((double) log(((double) pow(((double) exp(((double) (2.0 * ((double) sqrt(x)))))), ((double) fma(((double) cos(y)), ((double) cos(((double) (0.3333333333333333 * ((double) (t * z)))))), ((double) (((double) sin(y)) * ((double) sin(((double) (((double) (z * t)) / 3.0)))))))))))) - ((double) (a / ((double) (b * 3.0))))));
	} else {
		VAR = ((double) (((double) (((double) (2.0 * ((double) sqrt(x)))) * ((double) (((double) (((double) cos(y)) * ((double) cos(((double) (0.3333333333333333 * ((double) (t * z)))))))) + ((double) (((double) sin(y)) * ((double) sin(((double) (((double) (z * ((double) (((double) cbrt(t)) * ((double) cbrt(t)))))) / ((double) (3.0 / ((double) cbrt(t)))))))))))))) - ((double) (a / ((double) (b * 3.0))))));
	}
	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	21.2
Target	19.2
Herbie	18.6

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

Derivation

1. Split input into 2 regimes

2. if (- y (/ (* z t) 3.0)) < -inf.0 or 2.470815013403332e+307 < (- y (/ (* z t) 3.0))

   1. Initial program 63.8

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

   3. Applied cos-diff 63.8

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

   4. Taylor expanded around inf 63.8

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

   6. Applied add-log-exp 63.9

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

   7. Simplified 46.9

      \[\leadsto \log \color{blue}{\left({\left(e^{2 \cdot \sqrt{x}}\right)}^{\left(\mathsf{fma}\left(\cos y, \cos \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right), \sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right)\right)}\right)} - \frac{a}{b \cdot 3}\]

   if -inf.0 < (- y (/ (* z t) 3.0)) < 2.470815013403332e+307

   1. Initial program 15.2

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

   3. Applied cos-diff 14.6

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

   4. Taylor expanded around inf 14.6

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

   6. Applied add-cube-cbrt 14.6

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

   7. Applied associate-*r* 14.6

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

   8. Applied associate-/l* 14.6

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

4. Final simplification 18.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;y - \frac{z \cdot t}{3} = -\infty \lor \neg \left(y - \frac{z \cdot t}{3} \le 2.470815013403332 \cdot 10^{307}\right):\\ \;\;\;\;\log \left({\left(e^{2 \cdot \sqrt{x}}\right)}^{\left(\mathsf{fma}\left(\cos y, \cos \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right), \sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right)\right)}\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right) + \sin y \cdot \sin \left(\frac{z \cdot \left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}{\frac{3}{\sqrt[3]{t}}}\right)\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

Reproduce

herbie shell --seed 2020113 +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))))