Average Error: 20.7 → 20.3
Time: 14.4s
Precision: binary64
\[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3}\]
↓
\[2 \cdot \left(\cos y \cdot \left(\left(\sqrt[3]{\cos \left(z \cdot \frac{t}{3}\right)} \cdot \left(\sqrt[3]{\cos \left(z \cdot \frac{t}{3}\right)} \cdot \sqrt[3]{\cos \left(z \cdot \left(t \cdot 0.333333333333333315\right)\right)}\right)\right) \cdot \sqrt{x}\right) + \sin y \cdot \left(\sqrt{x} \cdot \left(\sqrt[3]{\sin \left(z \cdot \frac{t}{3}\right)} \cdot \left(\sqrt[3]{\sin \left(z \cdot \frac{t}{3}\right)} \cdot \sqrt[3]{\sin \left(z \cdot \frac{t}{3}\right)}\right)\right)\right)\right) - \frac{a}{3 \cdot b}\]
\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3}
↓
2 \cdot \left(\cos y \cdot \left(\left(\sqrt[3]{\cos \left(z \cdot \frac{t}{3}\right)} \cdot \left(\sqrt[3]{\cos \left(z \cdot \frac{t}{3}\right)} \cdot \sqrt[3]{\cos \left(z \cdot \left(t \cdot 0.333333333333333315\right)\right)}\right)\right) \cdot \sqrt{x}\right) + \sin y \cdot \left(\sqrt{x} \cdot \left(\sqrt[3]{\sin \left(z \cdot \frac{t}{3}\right)} \cdot \left(\sqrt[3]{\sin \left(z \cdot \frac{t}{3}\right)} \cdot \sqrt[3]{\sin \left(z \cdot \frac{t}{3}\right)}\right)\right)\right)\right) - \frac{a}{3 \cdot b}
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) {
	return ((double) (((double) (2.0 * ((double) (((double) (((double) cos(y)) * ((double) (((double) (((double) cbrt(((double) cos(((double) (z * ((double) (t / 3.0)))))))) * ((double) (((double) cbrt(((double) cos(((double) (z * ((double) (t / 3.0)))))))) * ((double) cbrt(((double) cos(((double) (z * ((double) (t * 0.3333333333333333)))))))))))) * ((double) sqrt(x)))))) + ((double) (((double) sin(y)) * ((double) (((double) sqrt(x)) * ((double) (((double) cbrt(((double) sin(((double) (z * ((double) (t / 3.0)))))))) * ((double) (((double) cbrt(((double) sin(((double) (z * ((double) (t / 3.0)))))))) * ((double) cbrt(((double) sin(((double) (z * ((double) (t / 3.0)))))))))))))))))))) - ((double) (a / ((double) (3.0 * b))))));
}

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

Original20.7
Target18.7
Herbie20.3
\[\begin{array}{l} \mathbf{if}\;z \lt -1.379333748723514 \cdot 10^{129}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\frac{1}{y} - \frac{\frac{0.333333333333333315}{z}}{t}\right) - \frac{\frac{a}{3}}{b}\\ \mathbf{elif}\;z \lt 3.51629061355598715 \cdot 10^{106}:\\ \;\;\;\;\left(\sqrt{x} \cdot 2\right) \cdot \cos \left(y - \frac{t}{3} \cdot z\right) - \frac{\frac{a}{3}}{b}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(y - \frac{\frac{0.333333333333333315}{z}}{t}\right) \cdot \left(2 \cdot \sqrt{x}\right) - \frac{\frac{a}{b}}{3}\\ \end{array}\]

Derivation

  1. Initial program 20.7

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

  2. Simplified20.7

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

  4. Applied cos-diff20.3

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

  5. Applied distribute-lft-in20.3

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

  6. Simplified20.3

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

  7. Simplified20.3

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

  9. Applied add-cube-cbrt20.3

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

  10. Taylor expanded around inf 20.3

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

  11. Simplified20.3

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

  13. Applied add-cube-cbrt20.3

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

  14. Final simplification20.3

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

Reproduce

herbie shell --seed 2020181 
(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))))