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

Average Error: 20.3 → 19.0

Time: 13.9s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \le -5.14492741003823326 \cdot 10^{-29}:\\ \;\;\;\;2 \cdot \left(\sqrt{x} \cdot \cos \left(y - z \cdot \left(t \cdot 0.333333333333333315\right)\right)\right) - \frac{a}{3 \cdot b}\\ \mathbf{elif}\;y \le 9.8173097746395403 \cdot 10^{-115}:\\ \;\;\;\;2 \cdot \left(\sqrt{x} \cdot \left(1 + y \cdot \left(y \cdot \frac{-1}{2}\right)\right)\right) - \frac{a}{3 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\cos y \cdot \left(\sqrt{x} \cdot \left(\left(\sqrt[3]{\cos \left(z \cdot \frac{t}{3}\right)} \cdot \sqrt[3]{\cos \left(z \cdot \frac{t}{3}\right)}\right) \cdot \sqrt[3]{\cos \left(z \cdot \left(t \cdot 0.333333333333333315\right)\right)}\right)\right) + \sin y \cdot \left(\sqrt{x} \cdot \sin \left(z \cdot \frac{t}{3}\right)\right)\right) - \frac{a}{3 \cdot b}\\ \end{array}\]

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 <= -5.144927410038233e-29)) {
		VAR = ((double) (((double) (2.0 * ((double) (((double) sqrt(x)) * ((double) cos(((double) (y - ((double) (z * ((double) (t * 0.3333333333333333)))))))))))) - ((double) (a / ((double) (3.0 * b))))));
	} else {
		double VAR_1;
		if ((y <= 9.81730977463954e-115)) {
			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) (((double) cos(y)) * ((double) (((double) sqrt(x)) * ((double) (((double) (((double) cbrt(((double) cos(((double) (z * ((double) (t / 3.0)))))))) * ((double) cbrt(((double) cos(((double) (z * ((double) (t / 3.0)))))))))) * ((double) cbrt(((double) cos(((double) (z * ((double) (t * 0.3333333333333333)))))))))))))) + ((double) (((double) sin(y)) * ((double) (((double) sqrt(x)) * ((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	20.3
Target	18.6
Herbie	19.0

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

Derivation

1. Split input into 3 regimes

2. if y < -5.14492741003823326e-29

   1. Initial program 20.8

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

   2. Simplified 20.9

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

   3. Taylor expanded around inf 20.9

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

   4. Simplified 20.8

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

   if -5.14492741003823326e-29 < y < 9.8173097746395403e-115

   1. Initial program 20.2

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

   2. Simplified 20.1

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

   3. Taylor expanded around 0 17.1

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

   4. Simplified 17.1

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

   if 9.8173097746395403e-115 < y

   1. Initial program 20.1

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

   2. Simplified 20.1

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

   4. Applied cos-diff 19.5

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

   5. Applied distribute-lft-in 19.5

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

   6. Simplified 19.5

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

   7. Simplified 19.5

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

   9. Applied add-cube-cbrt 19.5

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

   10. Taylor expanded around inf 19.5

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

   11. Simplified 19.6

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

4. Final simplification 19.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -5.14492741003823326 \cdot 10^{-29}:\\ \;\;\;\;2 \cdot \left(\sqrt{x} \cdot \cos \left(y - z \cdot \left(t \cdot 0.333333333333333315\right)\right)\right) - \frac{a}{3 \cdot b}\\ \mathbf{elif}\;y \le 9.8173097746395403 \cdot 10^{-115}:\\ \;\;\;\;2 \cdot \left(\sqrt{x} \cdot \left(1 + y \cdot \left(y \cdot \frac{-1}{2}\right)\right)\right) - \frac{a}{3 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\cos y \cdot \left(\sqrt{x} \cdot \left(\left(\sqrt[3]{\cos \left(z \cdot \frac{t}{3}\right)} \cdot \sqrt[3]{\cos \left(z \cdot \frac{t}{3}\right)}\right) \cdot \sqrt[3]{\cos \left(z \cdot \left(t \cdot 0.333333333333333315\right)\right)}\right)\right) + \sin y \cdot \left(\sqrt{x} \cdot \sin \left(z \cdot \frac{t}{3}\right)\right)\right) - \frac{a}{3 \cdot b}\\ \end{array}\]

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))))