Average Error: 20.5 → 18.0
Time: 15.2s
Precision: binary64
\[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \cdot t \leq -3.2693904690009335 \cdot 10^{+294} \lor \neg \left(z \cdot t \leq 2.4914869409948083 \cdot 10^{+287}\right):\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - 0.5 \cdot \left(y \cdot y\right)\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(\left(\sqrt[3]{t \cdot \frac{z}{3}} \cdot \sqrt[3]{t \cdot \frac{z}{3}}\right) \cdot \left(\sqrt[3]{z \cdot t} \cdot \sqrt[3]{0.3333333333333333}\right)\right)\right) + \left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \sin \left(\frac{t \cdot \frac{z}{\sqrt{3}}}{\sqrt{3}}\right)\right)\right) - \frac{a}{b \cdot 3}\\ \end{array}\]
\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 \leq -3.2693904690009335 \cdot 10^{+294} \lor \neg \left(z \cdot t \leq 2.4914869409948083 \cdot 10^{+287}\right):\\
\;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - 0.5 \cdot \left(y \cdot y\right)\right) - \frac{a}{b \cdot 3}\\

\mathbf{else}:\\
\;\;\;\;\left(\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(\left(\sqrt[3]{t \cdot \frac{z}{3}} \cdot \sqrt[3]{t \cdot \frac{z}{3}}\right) \cdot \left(\sqrt[3]{z \cdot t} \cdot \sqrt[3]{0.3333333333333333}\right)\right)\right) + \left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \sin \left(\frac{t \cdot \frac{z}{\sqrt{3}}}{\sqrt{3}}\right)\right)\right) - \frac{a}{b \cdot 3}\\

\end{array}
(FPCore (x y z t a b)
 :precision binary64
 (- (* (* 2.0 (sqrt x)) (cos (- y (/ (* z t) 3.0)))) (/ a (* b 3.0))))
↓
(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= (* z t) -3.2693904690009335e+294)
         (not (<= (* z t) 2.4914869409948083e+287)))
   (- (* (* 2.0 (sqrt x)) (- 1.0 (* 0.5 (* y y)))) (/ a (* b 3.0)))
   (-
    (+
     (*
      (* 2.0 (sqrt x))
      (*
       (cos y)
       (cos
        (*
         (* (cbrt (* t (/ z 3.0))) (cbrt (* t (/ z 3.0))))
         (* (cbrt (* z t)) (cbrt 0.3333333333333333))))))
     (*
      (* 2.0 (sqrt x))
      (* (sin y) (sin (/ (* t (/ z (sqrt 3.0))) (sqrt 3.0))))))
    (/ a (* b 3.0)))))
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) (z * t)) / 3.0))))))) - (a / ((double) (b * 3.0)))));
}
↓
double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (((((double) (z * t)) <= -3.2693904690009335e+294) || !(((double) (z * t)) <= 2.4914869409948083e+287))) {
		tmp = ((double) (((double) (((double) (2.0 * ((double) sqrt(x)))) * ((double) (1.0 - ((double) (0.5 * ((double) (y * y)))))))) - (a / ((double) (b * 3.0)))));
	} else {
		tmp = ((double) (((double) (((double) (((double) (2.0 * ((double) sqrt(x)))) * ((double) (((double) cos(y)) * ((double) cos(((double) (((double) (((double) cbrt(((double) (t * (z / 3.0))))) * ((double) cbrt(((double) (t * (z / 3.0))))))) * ((double) (((double) cbrt(((double) (z * t)))) * ((double) cbrt(0.3333333333333333)))))))))))) + ((double) (((double) (2.0 * ((double) sqrt(x)))) * ((double) (((double) sin(y)) * ((double) sin((((double) (t * (z / ((double) sqrt(3.0))))) / ((double) sqrt(3.0))))))))))) - (a / ((double) (b * 3.0)))));
	}
	return tmp;
}

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

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original20.5
Target18.6
Herbie18.0
\[\begin{array}{l} \mathbf{if}\;z < -1.3793337487235141 \cdot 10^{+129}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\frac{1}{y} - \frac{\frac{0.3333333333333333}{z}}{t}\right) - \frac{\frac{a}{3}}{b}\\ \mathbf{elif}\;z < 3.516290613555987 \cdot 10^{+106}:\\ \;\;\;\;\left(\sqrt{x} \cdot 2\right) \cdot \cos \left(y - \frac{t}{3} \cdot z\right) - \frac{\frac{a}{3}}{b}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(y - \frac{\frac{0.3333333333333333}{z}}{t}\right) \cdot \left(2 \cdot \sqrt{x}\right) - \frac{\frac{a}{b}}{3}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (*.f64 z t) < -3.26939046900093348e294 or 2.49148694099480831e287 < (*.f64 z t)

    1. Initial program 60.6

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

    2. Taylor expanded around 0 45.0

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \color{blue}{\left(1 - 0.5 \cdot {y}^{2}\right)} - \frac{a}{b \cdot 3}\]

    3. Simplified45.0

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \color{blue}{\left(1 - 0.5 \cdot \left(y \cdot y\right)\right)} - \frac{a}{b \cdot 3}\]

    if -3.26939046900093348e294 < (*.f64 z t) < 2.49148694099480831e287

    1. Initial program 14.1

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

    3. Applied cos-diff_binary6413.6

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

    4. Applied distribute-lft-in_binary6413.6

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

    5. Simplified13.6

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

    6. Simplified13.6

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

    8. Applied add-sqr-sqrt_binary6413.6

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

    9. Applied associate-/r*_binary6413.6

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

    10. Simplified13.6

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

    12. Applied add-cube-cbrt_binary6413.7

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

    13. Simplified13.6

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

    14. Simplified13.6

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

    15. Taylor expanded around 0 51.0

      \[\leadsto \left(\left(\cos y \cdot \cos \left(\left(\sqrt[3]{t \cdot \frac{z}{3}} \cdot \sqrt[3]{t \cdot \frac{z}{3}}\right) \cdot \color{blue}{\left(e^{0.3333333333333333 \cdot \left(\log t + \log z\right)} \cdot \sqrt[3]{0.3333333333333333}\right)}\right)\right) \cdot \left(2 \cdot \sqrt{x}\right) + \left(\sin y \cdot \sin \left(\frac{t \cdot \frac{z}{\sqrt{3}}}{\sqrt{3}}\right)\right) \cdot \left(2 \cdot \sqrt{x}\right)\right) - \frac{a}{b \cdot 3}\]

    16. Simplified13.6

      \[\leadsto \left(\left(\cos y \cdot \cos \left(\left(\sqrt[3]{t \cdot \frac{z}{3}} \cdot \sqrt[3]{t \cdot \frac{z}{3}}\right) \cdot \color{blue}{\left(\sqrt[3]{t \cdot z} \cdot \sqrt[3]{0.3333333333333333}\right)}\right)\right) \cdot \left(2 \cdot \sqrt{x}\right) + \left(\sin y \cdot \sin \left(\frac{t \cdot \frac{z}{\sqrt{3}}}{\sqrt{3}}\right)\right) \cdot \left(2 \cdot \sqrt{x}\right)\right) - \frac{a}{b \cdot 3}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification18.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -3.2693904690009335 \cdot 10^{+294} \lor \neg \left(z \cdot t \leq 2.4914869409948083 \cdot 10^{+287}\right):\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - 0.5 \cdot \left(y \cdot y\right)\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(\left(\sqrt[3]{t \cdot \frac{z}{3}} \cdot \sqrt[3]{t \cdot \frac{z}{3}}\right) \cdot \left(\sqrt[3]{z \cdot t} \cdot \sqrt[3]{0.3333333333333333}\right)\right)\right) + \left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \sin \left(\frac{t \cdot \frac{z}{\sqrt{3}}}{\sqrt{3}}\right)\right)\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

Reproduce

herbie shell --seed 2020210 
(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))))