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

Average Error: 20.4 → 18.9

Time: 13.4s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \le -4725053.9243208431:\\ \;\;\;\;2 \cdot \left(\sqrt{x} \cdot \cos \left(y - \frac{\sqrt[3]{t}}{\sqrt{3}} \cdot \left(z \cdot \left(\sqrt[3]{t} \cdot \frac{\sqrt[3]{t}}{\sqrt{3}}\right)\right)\right)\right) - \frac{a}{3 \cdot b}\\ \mathbf{elif}\;y \le 8.7758221551366551 \cdot 10^{-86}:\\ \;\;\;\;2 \cdot \left(\sqrt{x} \cdot \left(1 + y \cdot \left(y \cdot \frac{-1}{2}\right)\right)\right) - \frac{a}{3 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\sqrt{x} \cdot \left(\cos \left(z \cdot \frac{t}{3}\right) \cdot \left(\cos \left(z \cdot \frac{t}{3}\right) \cdot \left(\cos y \cdot \cos y\right)\right) - \sin \left(z \cdot \frac{t}{3}\right) \cdot \left(\sin \left(z \cdot \frac{t}{3}\right) \cdot \left(\sin y \cdot \sin y\right)\right)\right)}{\cos \left(z \cdot \frac{t}{3}\right) \cdot \cos y - \sin y \cdot \sin \left(z \cdot \left(t \cdot 0.333333333333333315\right)\right)} - \frac{a}{3 \cdot b}\\ \end{array}\]

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 <= -4725053.924320843)) {
		VAR = ((double) (((double) (2.0 * ((double) (((double) sqrt(x)) * ((double) cos(((double) (y - ((double) (((double) (((double) cbrt(t)) / ((double) sqrt(3.0)))) * ((double) (z * ((double) (((double) cbrt(t)) * ((double) (((double) cbrt(t)) / ((double) sqrt(3.0)))))))))))))))))) - ((double) (a / ((double) (3.0 * b))))));
	} else {
		double VAR_1;
		if ((y <= 8.775822155136655e-86)) {
			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) sqrt(x)) * ((double) (((double) (((double) cos(((double) (z * ((double) (t / 3.0)))))) * ((double) (((double) cos(((double) (z * ((double) (t / 3.0)))))) * ((double) (((double) cos(y)) * ((double) cos(y)))))))) - ((double) (((double) sin(((double) (z * ((double) (t / 3.0)))))) * ((double) (((double) sin(((double) (z * ((double) (t / 3.0)))))) * ((double) (((double) sin(y)) * ((double) sin(y)))))))))))) / ((double) (((double) (((double) cos(((double) (z * ((double) (t / 3.0)))))) * ((double) cos(y)))) - ((double) (((double) sin(y)) * ((double) sin(((double) (z * ((double) (t * 0.3333333333333333)))))))))))))) - ((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	18.3
Herbie	18.9

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

Derivation

1. Split input into 3 regimes

2. if y < -4725053.9243208431

   1. Initial program 19.9

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

   2. Simplified 20.0

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

   4. Applied add-sqr-sqrt 20.0

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

   5. Applied add-cube-cbrt 20.0

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

   6. Applied times-frac 20.0

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

   7. Applied associate-*r* 20.0

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

   8. Simplified 20.0

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

   if -4725053.9243208431 < y < 8.7758221551366551e-86

   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 20.0

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

   3. Taylor expanded around 0 16.8

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

   4. Simplified 16.8

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

   if 8.7758221551366551e-86 < y

   1. Initial program 21.1

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

   2. Simplified 21.1

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

   4. Applied cos-diff 20.9

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

   6. Applied flip-+ 20.9

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

   7. Applied associate-*r/ 20.9

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

   8. Simplified 20.9

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

   9. Taylor expanded around inf 20.9

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

   10. Simplified 20.9

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

4. Final simplification 18.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -4725053.9243208431:\\ \;\;\;\;2 \cdot \left(\sqrt{x} \cdot \cos \left(y - \frac{\sqrt[3]{t}}{\sqrt{3}} \cdot \left(z \cdot \left(\sqrt[3]{t} \cdot \frac{\sqrt[3]{t}}{\sqrt{3}}\right)\right)\right)\right) - \frac{a}{3 \cdot b}\\ \mathbf{elif}\;y \le 8.7758221551366551 \cdot 10^{-86}:\\ \;\;\;\;2 \cdot \left(\sqrt{x} \cdot \left(1 + y \cdot \left(y \cdot \frac{-1}{2}\right)\right)\right) - \frac{a}{3 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\sqrt{x} \cdot \left(\cos \left(z \cdot \frac{t}{3}\right) \cdot \left(\cos \left(z \cdot \frac{t}{3}\right) \cdot \left(\cos y \cdot \cos y\right)\right) - \sin \left(z \cdot \frac{t}{3}\right) \cdot \left(\sin \left(z \cdot \frac{t}{3}\right) \cdot \left(\sin y \cdot \sin y\right)\right)\right)}{\cos \left(z \cdot \frac{t}{3}\right) \cdot \cos y - \sin y \cdot \sin \left(z \cdot \left(t \cdot 0.333333333333333315\right)\right)} - \frac{a}{3 \cdot b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020185 
(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))))