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

Average Error: 20.4 → 18.8

Time: 14.0s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \le -1.80267380308005252 \cdot 10^{-128}:\\ \;\;\;\;2 \cdot \left(\sqrt{x} \cdot \left(\cos y \cdot \sqrt[3]{{\left(\cos \left(z \cdot \frac{t}{3}\right)\right)}^{3}} + \sin y \cdot \log \left(e^{\sin \left(z \cdot \left(t \cdot 0.333333333333333315\right)\right)}\right)\right)\right) - \frac{a}{3 \cdot b}\\ \mathbf{elif}\;y \le 27.71990394146246:\\ \;\;\;\;2 \cdot \left(\sqrt{x} \cdot \left(1 + y \cdot \left(y \cdot \frac{-1}{2}\right)\right)\right) - \frac{a}{3 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{x} \cdot \left(\cos y \cdot \left(\sqrt[3]{\cos \left(z \cdot \frac{t}{3}\right)} \cdot \left(\sqrt[3]{\cos \left(z \cdot \frac{t}{3}\right)} \cdot \sqrt[3]{\cos \left(z \cdot \frac{t}{3}\right)}\right)\right) + \sin y \cdot \sin \left(z \cdot \frac{t}{3}\right)\right)\right) - \frac{a}{3 \cdot b}\\ \end{array}\]

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) (((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 ((y <= -1.8026738030800525e-128)) {
		VAR = ((double) (((double) (2.0 * ((double) (((double) sqrt(x)) * ((double) (((double) (((double) cos(y)) * ((double) cbrt(((double) pow(((double) cos(((double) (z * ((double) (t / 3.0)))))), 3.0)))))) + ((double) (((double) sin(y)) * ((double) log(((double) exp(((double) sin(((double) (z * ((double) (t * 0.3333333333333333)))))))))))))))))) - ((double) (a / ((double) (3.0 * b))))));
	} else {
		double VAR_1;
		if ((y <= 27.719903941462462)) {
			VAR_1 = ((double) (((double) (2.0 * ((double) (((double) sqrt(x)) * ((double) (1.0 + ((double) (y * ((double) (y * -0.5)))))))))) - ((double) (a / ((double) (3.0 * b))))));
		} else {
			VAR_1 = ((double) (((double) (2.0 * ((double) (((double) sqrt(x)) * ((double) (((double) (((double) cos(y)) * ((double) (((double) cbrt(((double) cos(((double) (z * ((double) (t / 3.0)))))))) * ((double) (((double) cbrt(((double) cos(((double) (z * ((double) (t / 3.0)))))))) * ((double) cbrt(((double) cos(((double) (z * ((double) (t / 3.0)))))))))))))) + ((double) (((double) sin(y)) * ((double) sin(((double) (z * ((double) (t / 3.0)))))))))))))) - ((double) (a / ((double) (3.0 * b))))));
		}
		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	20.4
Target	18.5
Herbie	18.8

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

Derivation

1. Split input into 3 regimes

2. if y < -1.80267380308005252e-128

   1. Initial program 21.5

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

   2. Simplified 21.5

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

   4. Applied cos-diff 21.0

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

   5. Taylor expanded around inf 20.9

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

   6. Simplified 21.0

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

   8. Applied add-log-exp 21.0

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

   10. Applied add-cbrt-cube 21.0

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

   11. Simplified 21.0

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

   if -1.80267380308005252e-128 < y < 27.71990394146246

   1. Initial program 19.3

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

   2. Simplified 19.2

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

   3. Taylor expanded around 0 16.5

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

   4. Simplified 16.5

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

   if 27.71990394146246 < y

   1. Initial program 20.3

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

   2. Simplified 20.3

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

   4. Applied cos-diff 19.4

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

   6. Applied add-cube-cbrt 19.4

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

4. Final simplification 18.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -1.80267380308005252 \cdot 10^{-128}:\\ \;\;\;\;2 \cdot \left(\sqrt{x} \cdot \left(\cos y \cdot \sqrt[3]{{\left(\cos \left(z \cdot \frac{t}{3}\right)\right)}^{3}} + \sin y \cdot \log \left(e^{\sin \left(z \cdot \left(t \cdot 0.333333333333333315\right)\right)}\right)\right)\right) - \frac{a}{3 \cdot b}\\ \mathbf{elif}\;y \le 27.71990394146246:\\ \;\;\;\;2 \cdot \left(\sqrt{x} \cdot \left(1 + y \cdot \left(y \cdot \frac{-1}{2}\right)\right)\right) - \frac{a}{3 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sqrt{x} \cdot \left(\cos y \cdot \left(\sqrt[3]{\cos \left(z \cdot \frac{t}{3}\right)} \cdot \left(\sqrt[3]{\cos \left(z \cdot \frac{t}{3}\right)} \cdot \sqrt[3]{\cos \left(z \cdot \frac{t}{3}\right)}\right)\right) + \sin y \cdot \sin \left(z \cdot \frac{t}{3}\right)\right)\right) - \frac{a}{3 \cdot b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020184 
(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))))