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

Average Error: 20.4 → 17.8

Time: 14.2s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) \leq 4.296115373454275 \cdot 10^{+153}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(t \cdot \frac{z}{3}\right) + \left(\sqrt[3]{\sin \left(t \cdot \frac{z}{3}\right)} \cdot \left(\sqrt[3]{\sin \left(t \cdot \frac{z}{3}\right)} \cdot \sin y\right)\right) \cdot \sqrt[3]{\sqrt[3]{\sin \left(t \cdot \frac{z}{3}\right)} \cdot \left(\sqrt[3]{\sin \left(t \cdot \frac{z}{3}\right)} \cdot \sqrt[3]{\sin \left(t \cdot \frac{z}{3}\right)}\right)}\right) - \frac{a}{3 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 + y \cdot \left(y \cdot -0.5\right)\right) - \frac{a}{3 \cdot b}\\ \end{array}\]

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) (z * t)) / 3.0))))))) - (a / ((double) (b * 3.0)))));
}

↓

double code(double x, double y, double z, double t, double a, double b) {
	double VAR;
	if ((((double) (((double) (2.0 * ((double) sqrt(x)))) * ((double) cos(((double) (y - (((double) (z * t)) / 3.0))))))) <= 4.296115373454275e+153)) {
		VAR = ((double) (((double) (((double) (2.0 * ((double) sqrt(x)))) * ((double) (((double) (((double) cos(y)) * ((double) cos(((double) (t * (z / 3.0))))))) + ((double) (((double) (((double) cbrt(((double) sin(((double) (t * (z / 3.0))))))) * ((double) (((double) cbrt(((double) sin(((double) (t * (z / 3.0))))))) * ((double) sin(y)))))) * ((double) cbrt(((double) (((double) cbrt(((double) sin(((double) (t * (z / 3.0))))))) * ((double) (((double) cbrt(((double) sin(((double) (t * (z / 3.0))))))) * ((double) cbrt(((double) sin(((double) (t * (z / 3.0))))))))))))))))))) - (a / ((double) (3.0 * b)))));
	} else {
		VAR = ((double) (((double) (((double) (2.0 * ((double) sqrt(x)))) * ((double) (1.0 + ((double) (y * ((double) (y * -0.5)))))))) - (a / ((double) (3.0 * b)))));
	}
	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	20.4
Target	18.7
Herbie	17.8

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

Derivation

1. Split input into 2 regimes

2. if (* (* 2.0 (sqrt x)) (cos (- y (/ (* z t) 3.0)))) < 4.29611537345427503e153

   1. Initial program 14.3

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

   3. Applied cos-diff 13.8

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

   4. Simplified 13.9

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

   5. Simplified 13.9

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

   7. Applied add-cube-cbrt 13.9

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

   8. Applied associate-*r* 13.9

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

   9. Simplified 13.9

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

   11. Applied add-cube-cbrt 13.9

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

   if 4.29611537345427503e153 < (* (* 2.0 (sqrt x)) (cos (- y (/ (* z t) 3.0))))

   1. Initial program 63.7

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

   2. Taylor expanded around 0 45.7

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \color{blue}{\left(1 - 0.5 \cdot {y}^{2}\right)} - \frac{a}{b \cdot 3}\]

   3. Simplified 45.7

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \color{blue}{\left(1 + y \cdot \left(y \cdot -0.5\right)\right)} - \frac{a}{b \cdot 3}\]
3. Recombined 2 regimes into one program.

4. Final simplification 17.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) \leq 4.296115373454275 \cdot 10^{+153}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(t \cdot \frac{z}{3}\right) + \left(\sqrt[3]{\sin \left(t \cdot \frac{z}{3}\right)} \cdot \left(\sqrt[3]{\sin \left(t \cdot \frac{z}{3}\right)} \cdot \sin y\right)\right) \cdot \sqrt[3]{\sqrt[3]{\sin \left(t \cdot \frac{z}{3}\right)} \cdot \left(\sqrt[3]{\sin \left(t \cdot \frac{z}{3}\right)} \cdot \sqrt[3]{\sin \left(t \cdot \frac{z}{3}\right)}\right)}\right) - \frac{a}{3 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 + y \cdot \left(y \cdot -0.5\right)\right) - \frac{a}{3 \cdot b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020199 
(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))))