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

Average Error: 20.4 → 18.8

Time: 20.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \cdot t = -\infty \lor \neg \left(z \cdot t \le 1.9971418824300267 \cdot 10^{142}\right):\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - \frac{1}{2} \cdot {y}^{2}\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(\left(\cos \left(\frac{t \cdot z}{{\left(\sqrt{3}\right)}^{2}}\right) \cdot \cos \left(1 \cdot y\right) - \sin \left(1 \cdot y\right) \cdot \sin \left(-\frac{\sqrt[3]{t}}{\sqrt{3}} \cdot \frac{z \cdot \left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}{\sqrt{3}}\right)\right) \cdot \cos \left(\mathsf{fma}\left(-\frac{\sqrt[3]{t}}{\sqrt{3}}, \frac{z \cdot \left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}{\sqrt{3}}, \frac{\sqrt[3]{t}}{\sqrt{3}} \cdot \frac{z \cdot \left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}{\sqrt{3}}\right)\right) - \sin \left(\mathsf{fma}\left(1, y, -\frac{\sqrt[3]{t}}{\sqrt{3}} \cdot \frac{z \cdot \left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}{\sqrt{3}}\right)\right) \cdot \sin \left(\mathsf{fma}\left(-\frac{\sqrt[3]{t}}{\sqrt{3}}, \frac{z \cdot \left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}{\sqrt{3}}, \frac{\sqrt[3]{t}}{\sqrt{3}} \cdot \frac{z \cdot \left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}{\sqrt{3}}\right)\right)\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

C

double code(double x, double y, double z, double t, double a, double b) {
	return (((2.0 * sqrt(x)) * cos((y - ((z * t) / 3.0)))) - (a / (b * 3.0)));
}

↓

double code(double x, double y, double z, double t, double a, double b) {
	double temp;
	if ((((z * t) <= -inf.0) || !((z * t) <= 1.9971418824300267e+142))) {
		temp = (((2.0 * sqrt(x)) * (1.0 - (0.5 * pow(y, 2.0)))) - (a / (b * 3.0)));
	} else {
		temp = (((2.0 * sqrt(x)) * ((((cos(((t * z) / pow(sqrt(3.0), 2.0))) * cos((1.0 * y))) - (sin((1.0 * y)) * sin(-((cbrt(t) / sqrt(3.0)) * ((z * (cbrt(t) * cbrt(t))) / sqrt(3.0)))))) * cos(fma(-(cbrt(t) / sqrt(3.0)), ((z * (cbrt(t) * cbrt(t))) / sqrt(3.0)), ((cbrt(t) / sqrt(3.0)) * ((z * (cbrt(t) * cbrt(t))) / sqrt(3.0)))))) - (sin(fma(1.0, y, -((cbrt(t) / sqrt(3.0)) * ((z * (cbrt(t) * cbrt(t))) / sqrt(3.0))))) * sin(fma(-(cbrt(t) / sqrt(3.0)), ((z * (cbrt(t) * cbrt(t))) / sqrt(3.0)), ((cbrt(t) / sqrt(3.0)) * ((z * (cbrt(t) * cbrt(t))) / sqrt(3.0)))))))) - (a / (b * 3.0)));
	}
	return temp;
}

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	18.4
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 2 regimes

2. if (* z t) < -inf.0 or 1.9971418824300267e+142 < (* z t)

   1. Initial program 51.1

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

   2. Taylor expanded around 0 44.9

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

   if -inf.0 < (* z t) < 1.9971418824300267e+142

   1. Initial program 12.1

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

   3. Applied add-sqr-sqrt 12.2

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

   4. Applied add-cube-cbrt 12.2

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

   5. Applied associate-*r* 12.2

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

   6. Applied times-frac 12.2

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

   7. Applied add-sqr-sqrt 40.4

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

   8. Applied prod-diff 40.4

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

   9. Applied cos-sum 40.4

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

   10. Simplified 38.0

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

   11. Simplified 12.2

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

   13. Applied fma-udef 12.2

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

   14. Applied cos-sum 11.7

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

   15. Simplified 11.7

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

   16. Taylor expanded around inf 11.8

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

4. Final simplification 18.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t = -\infty \lor \neg \left(z \cdot t \le 1.9971418824300267 \cdot 10^{142}\right):\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - \frac{1}{2} \cdot {y}^{2}\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(\left(\cos \left(\frac{t \cdot z}{{\left(\sqrt{3}\right)}^{2}}\right) \cdot \cos \left(1 \cdot y\right) - \sin \left(1 \cdot y\right) \cdot \sin \left(-\frac{\sqrt[3]{t}}{\sqrt{3}} \cdot \frac{z \cdot \left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}{\sqrt{3}}\right)\right) \cdot \cos \left(\mathsf{fma}\left(-\frac{\sqrt[3]{t}}{\sqrt{3}}, \frac{z \cdot \left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}{\sqrt{3}}, \frac{\sqrt[3]{t}}{\sqrt{3}} \cdot \frac{z \cdot \left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}{\sqrt{3}}\right)\right) - \sin \left(\mathsf{fma}\left(1, y, -\frac{\sqrt[3]{t}}{\sqrt{3}} \cdot \frac{z \cdot \left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}{\sqrt{3}}\right)\right) \cdot \sin \left(\mathsf{fma}\left(-\frac{\sqrt[3]{t}}{\sqrt{3}}, \frac{z \cdot \left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}{\sqrt{3}}, \frac{\sqrt[3]{t}}{\sqrt{3}} \cdot \frac{z \cdot \left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}{\sqrt{3}}\right)\right)\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

Reproduce

herbie shell --seed 2020066 +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))))