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

Average Error: 20.7 → 18.1

Time: 17.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \le 0.999877438093644932:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(\left(\left(\left(\sqrt[3]{\cos \left(\frac{t}{\sqrt[3]{3}} \cdot \frac{z}{\sqrt[3]{3} \cdot \sqrt[3]{3}}\right)} \cdot \sqrt[3]{\cos \left(\frac{t}{\sqrt[3]{3}} \cdot \frac{z}{\sqrt[3]{3} \cdot \sqrt[3]{3}}\right)}\right) \cdot \sqrt[3]{\cos \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right)}\right) \cdot \cos \left(1 \cdot y\right) - \sin \left(1 \cdot y\right) \cdot \sin \left(-\frac{t}{\sqrt[3]{3}} \cdot \frac{z}{\sqrt[3]{3} \cdot \sqrt[3]{3}}\right)\right) \cdot \cos \left(\mathsf{fma}\left(-\frac{t}{\sqrt[3]{3}}, \frac{z}{\sqrt[3]{3} \cdot \sqrt[3]{3}}, \frac{t}{\sqrt[3]{3}} \cdot \frac{z}{\sqrt[3]{3} \cdot \sqrt[3]{3}}\right)\right) - \sin \left(\mathsf{fma}\left(1, y, -\frac{t}{\sqrt[3]{3}} \cdot \frac{z}{\sqrt[3]{3} \cdot \sqrt[3]{3}}\right)\right) \cdot \sin \left(\mathsf{fma}\left(-\frac{t}{\sqrt[3]{3}}, \frac{z}{\sqrt[3]{3} \cdot \sqrt[3]{3}}, \frac{t}{\sqrt[3]{3}} \cdot \frac{z}{\sqrt[3]{3} \cdot \sqrt[3]{3}}\right)\right)\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - \frac{1}{2} \cdot {y}^{2}\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

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) cos(((double) (y - ((double) (((double) (z * t)) / 3.0)))))) <= 0.9998774380936449)) {
		VAR = ((double) (((double) (((double) (2.0 * ((double) sqrt(x)))) * ((double) (((double) (((double) (((double) (((double) (((double) (((double) cbrt(((double) cos(((double) (((double) (t / ((double) cbrt(3.0)))) * ((double) (z / ((double) (((double) cbrt(3.0)) * ((double) cbrt(3.0)))))))))))) * ((double) cbrt(((double) cos(((double) (((double) (t / ((double) cbrt(3.0)))) * ((double) (z / ((double) (((double) cbrt(3.0)) * ((double) cbrt(3.0)))))))))))))) * ((double) cbrt(((double) cos(((double) (0.3333333333333333 * ((double) (t * z)))))))))) * ((double) cos(((double) (1.0 * y)))))) - ((double) (((double) sin(((double) (1.0 * y)))) * ((double) sin(((double) -(((double) (((double) (t / ((double) cbrt(3.0)))) * ((double) (z / ((double) (((double) cbrt(3.0)) * ((double) cbrt(3.0)))))))))))))))) * ((double) cos(((double) fma(((double) -(((double) (t / ((double) cbrt(3.0)))))), ((double) (z / ((double) (((double) cbrt(3.0)) * ((double) cbrt(3.0)))))), ((double) (((double) (t / ((double) cbrt(3.0)))) * ((double) (z / ((double) (((double) cbrt(3.0)) * ((double) cbrt(3.0)))))))))))))) - ((double) (((double) sin(((double) fma(1.0, y, ((double) -(((double) (((double) (t / ((double) cbrt(3.0)))) * ((double) (z / ((double) (((double) cbrt(3.0)) * ((double) cbrt(3.0)))))))))))))) * ((double) sin(((double) fma(((double) -(((double) (t / ((double) cbrt(3.0)))))), ((double) (z / ((double) (((double) cbrt(3.0)) * ((double) cbrt(3.0)))))), ((double) (((double) (t / ((double) cbrt(3.0)))) * ((double) (z / ((double) (((double) cbrt(3.0)) * ((double) cbrt(3.0)))))))))))))))))) - ((double) (a / ((double) (b * 3.0))))));
	} else {
		VAR = ((double) (((double) (((double) (2.0 * ((double) sqrt(x)))) * ((double) (1.0 - ((double) (0.5 * ((double) pow(y, 2.0)))))))) - ((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	20.7
Target	18.8
Herbie	18.1

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

Derivation

1. Split input into 2 regimes

2. if (cos (- y (/ (* z t) 3.0))) < 0.9998774380936449

   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. Using strategy rm

   3. Applied add-cube-cbrt 20.1

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

   4. Applied times-frac 20.1

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

   5. Applied add-sqr-sqrt 44.6

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

   6. Applied prod-diff 44.6

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

   7. Applied cos-sum 44.6

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

   8. Simplified 41.7

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \left(\color{blue}{\cos \left(\mathsf{fma}\left(1, y, -\frac{t}{\sqrt[3]{3}} \cdot \frac{z}{\sqrt[3]{3} \cdot \sqrt[3]{3}}\right)\right) \cdot \cos \left(\mathsf{fma}\left(-\frac{t}{\sqrt[3]{3}}, \frac{z}{\sqrt[3]{3} \cdot \sqrt[3]{3}}, \frac{t}{\sqrt[3]{3}} \cdot \frac{z}{\sqrt[3]{3} \cdot \sqrt[3]{3}}\right)\right)} - \sin \left(\mathsf{fma}\left(\sqrt{y}, \sqrt{y}, -\frac{t}{\sqrt[3]{3}} \cdot \frac{z}{\sqrt[3]{3} \cdot \sqrt[3]{3}}\right)\right) \cdot \sin \left(\mathsf{fma}\left(-\frac{t}{\sqrt[3]{3}}, \frac{z}{\sqrt[3]{3} \cdot \sqrt[3]{3}}, \frac{t}{\sqrt[3]{3}} \cdot \frac{z}{\sqrt[3]{3} \cdot \sqrt[3]{3}}\right)\right)\right) - \frac{a}{b \cdot 3}\]

   9. Simplified 20.1

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \left(\cos \left(\mathsf{fma}\left(1, y, -\frac{t}{\sqrt[3]{3}} \cdot \frac{z}{\sqrt[3]{3} \cdot \sqrt[3]{3}}\right)\right) \cdot \cos \left(\mathsf{fma}\left(-\frac{t}{\sqrt[3]{3}}, \frac{z}{\sqrt[3]{3} \cdot \sqrt[3]{3}}, \frac{t}{\sqrt[3]{3}} \cdot \frac{z}{\sqrt[3]{3} \cdot \sqrt[3]{3}}\right)\right) - \color{blue}{\sin \left(\mathsf{fma}\left(1, y, -\frac{t}{\sqrt[3]{3}} \cdot \frac{z}{\sqrt[3]{3} \cdot \sqrt[3]{3}}\right)\right) \cdot \sin \left(\mathsf{fma}\left(-\frac{t}{\sqrt[3]{3}}, \frac{z}{\sqrt[3]{3} \cdot \sqrt[3]{3}}, \frac{t}{\sqrt[3]{3}} \cdot \frac{z}{\sqrt[3]{3} \cdot \sqrt[3]{3}}\right)\right)}\right) - \frac{a}{b \cdot 3}\]
   10. Using strategy rm

   11. Applied fma-udef 20.1

       \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \left(\cos \color{blue}{\left(1 \cdot y + \left(-\frac{t}{\sqrt[3]{3}} \cdot \frac{z}{\sqrt[3]{3} \cdot \sqrt[3]{3}}\right)\right)} \cdot \cos \left(\mathsf{fma}\left(-\frac{t}{\sqrt[3]{3}}, \frac{z}{\sqrt[3]{3} \cdot \sqrt[3]{3}}, \frac{t}{\sqrt[3]{3}} \cdot \frac{z}{\sqrt[3]{3} \cdot \sqrt[3]{3}}\right)\right) - \sin \left(\mathsf{fma}\left(1, y, -\frac{t}{\sqrt[3]{3}} \cdot \frac{z}{\sqrt[3]{3} \cdot \sqrt[3]{3}}\right)\right) \cdot \sin \left(\mathsf{fma}\left(-\frac{t}{\sqrt[3]{3}}, \frac{z}{\sqrt[3]{3} \cdot \sqrt[3]{3}}, \frac{t}{\sqrt[3]{3}} \cdot \frac{z}{\sqrt[3]{3} \cdot \sqrt[3]{3}}\right)\right)\right) - \frac{a}{b \cdot 3}\]

   12. Applied cos-sum 19.5

       \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \left(\color{blue}{\left(\cos \left(1 \cdot y\right) \cdot \cos \left(-\frac{t}{\sqrt[3]{3}} \cdot \frac{z}{\sqrt[3]{3} \cdot \sqrt[3]{3}}\right) - \sin \left(1 \cdot y\right) \cdot \sin \left(-\frac{t}{\sqrt[3]{3}} \cdot \frac{z}{\sqrt[3]{3} \cdot \sqrt[3]{3}}\right)\right)} \cdot \cos \left(\mathsf{fma}\left(-\frac{t}{\sqrt[3]{3}}, \frac{z}{\sqrt[3]{3} \cdot \sqrt[3]{3}}, \frac{t}{\sqrt[3]{3}} \cdot \frac{z}{\sqrt[3]{3} \cdot \sqrt[3]{3}}\right)\right) - \sin \left(\mathsf{fma}\left(1, y, -\frac{t}{\sqrt[3]{3}} \cdot \frac{z}{\sqrt[3]{3} \cdot \sqrt[3]{3}}\right)\right) \cdot \sin \left(\mathsf{fma}\left(-\frac{t}{\sqrt[3]{3}}, \frac{z}{\sqrt[3]{3} \cdot \sqrt[3]{3}}, \frac{t}{\sqrt[3]{3}} \cdot \frac{z}{\sqrt[3]{3} \cdot \sqrt[3]{3}}\right)\right)\right) - \frac{a}{b \cdot 3}\]

   13. Simplified 19.5

       \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \left(\left(\color{blue}{\cos \left(\frac{t}{\sqrt[3]{3}} \cdot \frac{z}{\sqrt[3]{3} \cdot \sqrt[3]{3}}\right) \cdot \cos \left(1 \cdot y\right)} - \sin \left(1 \cdot y\right) \cdot \sin \left(-\frac{t}{\sqrt[3]{3}} \cdot \frac{z}{\sqrt[3]{3} \cdot \sqrt[3]{3}}\right)\right) \cdot \cos \left(\mathsf{fma}\left(-\frac{t}{\sqrt[3]{3}}, \frac{z}{\sqrt[3]{3} \cdot \sqrt[3]{3}}, \frac{t}{\sqrt[3]{3}} \cdot \frac{z}{\sqrt[3]{3} \cdot \sqrt[3]{3}}\right)\right) - \sin \left(\mathsf{fma}\left(1, y, -\frac{t}{\sqrt[3]{3}} \cdot \frac{z}{\sqrt[3]{3} \cdot \sqrt[3]{3}}\right)\right) \cdot \sin \left(\mathsf{fma}\left(-\frac{t}{\sqrt[3]{3}}, \frac{z}{\sqrt[3]{3} \cdot \sqrt[3]{3}}, \frac{t}{\sqrt[3]{3}} \cdot \frac{z}{\sqrt[3]{3} \cdot \sqrt[3]{3}}\right)\right)\right) - \frac{a}{b \cdot 3}\]
   14. Using strategy rm

   15. Applied add-cube-cbrt 19.5

       \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \left(\left(\color{blue}{\left(\left(\sqrt[3]{\cos \left(\frac{t}{\sqrt[3]{3}} \cdot \frac{z}{\sqrt[3]{3} \cdot \sqrt[3]{3}}\right)} \cdot \sqrt[3]{\cos \left(\frac{t}{\sqrt[3]{3}} \cdot \frac{z}{\sqrt[3]{3} \cdot \sqrt[3]{3}}\right)}\right) \cdot \sqrt[3]{\cos \left(\frac{t}{\sqrt[3]{3}} \cdot \frac{z}{\sqrt[3]{3} \cdot \sqrt[3]{3}}\right)}\right)} \cdot \cos \left(1 \cdot y\right) - \sin \left(1 \cdot y\right) \cdot \sin \left(-\frac{t}{\sqrt[3]{3}} \cdot \frac{z}{\sqrt[3]{3} \cdot \sqrt[3]{3}}\right)\right) \cdot \cos \left(\mathsf{fma}\left(-\frac{t}{\sqrt[3]{3}}, \frac{z}{\sqrt[3]{3} \cdot \sqrt[3]{3}}, \frac{t}{\sqrt[3]{3}} \cdot \frac{z}{\sqrt[3]{3} \cdot \sqrt[3]{3}}\right)\right) - \sin \left(\mathsf{fma}\left(1, y, -\frac{t}{\sqrt[3]{3}} \cdot \frac{z}{\sqrt[3]{3} \cdot \sqrt[3]{3}}\right)\right) \cdot \sin \left(\mathsf{fma}\left(-\frac{t}{\sqrt[3]{3}}, \frac{z}{\sqrt[3]{3} \cdot \sqrt[3]{3}}, \frac{t}{\sqrt[3]{3}} \cdot \frac{z}{\sqrt[3]{3} \cdot \sqrt[3]{3}}\right)\right)\right) - \frac{a}{b \cdot 3}\]

   16. Taylor expanded around inf 19.4

       \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \left(\left(\left(\left(\sqrt[3]{\cos \left(\frac{t}{\sqrt[3]{3}} \cdot \frac{z}{\sqrt[3]{3} \cdot \sqrt[3]{3}}\right)} \cdot \sqrt[3]{\cos \left(\frac{t}{\sqrt[3]{3}} \cdot \frac{z}{\sqrt[3]{3} \cdot \sqrt[3]{3}}\right)}\right) \cdot \sqrt[3]{\color{blue}{\cos \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right)}}\right) \cdot \cos \left(1 \cdot y\right) - \sin \left(1 \cdot y\right) \cdot \sin \left(-\frac{t}{\sqrt[3]{3}} \cdot \frac{z}{\sqrt[3]{3} \cdot \sqrt[3]{3}}\right)\right) \cdot \cos \left(\mathsf{fma}\left(-\frac{t}{\sqrt[3]{3}}, \frac{z}{\sqrt[3]{3} \cdot \sqrt[3]{3}}, \frac{t}{\sqrt[3]{3}} \cdot \frac{z}{\sqrt[3]{3} \cdot \sqrt[3]{3}}\right)\right) - \sin \left(\mathsf{fma}\left(1, y, -\frac{t}{\sqrt[3]{3}} \cdot \frac{z}{\sqrt[3]{3} \cdot \sqrt[3]{3}}\right)\right) \cdot \sin \left(\mathsf{fma}\left(-\frac{t}{\sqrt[3]{3}}, \frac{z}{\sqrt[3]{3} \cdot \sqrt[3]{3}}, \frac{t}{\sqrt[3]{3}} \cdot \frac{z}{\sqrt[3]{3} \cdot \sqrt[3]{3}}\right)\right)\right) - \frac{a}{b \cdot 3}\]

   if 0.9998774380936449 < (cos (- y (/ (* z t) 3.0)))

   1. Initial program 21.7

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

   2. Taylor expanded around 0 16.0

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

4. Final simplification 18.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \le 0.999877438093644932:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(\left(\left(\left(\sqrt[3]{\cos \left(\frac{t}{\sqrt[3]{3}} \cdot \frac{z}{\sqrt[3]{3} \cdot \sqrt[3]{3}}\right)} \cdot \sqrt[3]{\cos \left(\frac{t}{\sqrt[3]{3}} \cdot \frac{z}{\sqrt[3]{3} \cdot \sqrt[3]{3}}\right)}\right) \cdot \sqrt[3]{\cos \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right)}\right) \cdot \cos \left(1 \cdot y\right) - \sin \left(1 \cdot y\right) \cdot \sin \left(-\frac{t}{\sqrt[3]{3}} \cdot \frac{z}{\sqrt[3]{3} \cdot \sqrt[3]{3}}\right)\right) \cdot \cos \left(\mathsf{fma}\left(-\frac{t}{\sqrt[3]{3}}, \frac{z}{\sqrt[3]{3} \cdot \sqrt[3]{3}}, \frac{t}{\sqrt[3]{3}} \cdot \frac{z}{\sqrt[3]{3} \cdot \sqrt[3]{3}}\right)\right) - \sin \left(\mathsf{fma}\left(1, y, -\frac{t}{\sqrt[3]{3}} \cdot \frac{z}{\sqrt[3]{3} \cdot \sqrt[3]{3}}\right)\right) \cdot \sin \left(\mathsf{fma}\left(-\frac{t}{\sqrt[3]{3}}, \frac{z}{\sqrt[3]{3} \cdot \sqrt[3]{3}}, \frac{t}{\sqrt[3]{3}} \cdot \frac{z}{\sqrt[3]{3} \cdot \sqrt[3]{3}}\right)\right)\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - \frac{1}{2} \cdot {y}^{2}\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

Reproduce

herbie shell --seed 2020121 +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))))