Average Error: 21.1 → 19.6
Time: 7.8s
Precision: 64
\[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \cdot t \le 7.3230218287101624 \cdot 10^{285}:\\ \;\;\;\;2 \cdot \left(\mathsf{fma}\left(\cos y, \cos \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right), \sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right) \cdot \sqrt{x}\right) - \frac{\frac{a}{3}}{b}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - \frac{1}{2} \cdot {y}^{2}\right) - \frac{a}{b \cdot 3}\\ \end{array}\]
\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 \le 7.3230218287101624 \cdot 10^{285}:\\
\;\;\;\;2 \cdot \left(\mathsf{fma}\left(\cos y, \cos \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right), \sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right) \cdot \sqrt{x}\right) - \frac{\frac{a}{3}}{b}\\

\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}
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 VAR;
	if (((z * t) <= 7.323021828710162e+285)) {
		VAR = ((2.0 * (fma(cos(y), cos((0.3333333333333333 * (t * z))), (sin(y) * sin(((z * t) / 3.0)))) * sqrt(x))) - ((a / 3.0) / b));
	} else {
		VAR = (((2.0 * sqrt(x)) * (1.0 - (0.5 * pow(y, 2.0)))) - (a / (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

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original21.1
Target18.9
Herbie19.6
\[\begin{array}{l} \mathbf{if}\;z \lt -1.379333748723514 \cdot 10^{129}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\frac{1}{y} - \frac{\frac{0.333333333333333315}{z}}{t}\right) - \frac{\frac{a}{3}}{b}\\ \mathbf{elif}\;z \lt 3.51629061355598715 \cdot 10^{106}:\\ \;\;\;\;\left(\sqrt{x} \cdot 2\right) \cdot \cos \left(y - \frac{t}{3} \cdot z\right) - \frac{\frac{a}{3}}{b}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(y - \frac{\frac{0.333333333333333315}{z}}{t}\right) \cdot \left(2 \cdot \sqrt{x}\right) - \frac{\frac{a}{b}}{3}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (* z t) < 7.323021828710162e+285

    1. Initial program 18.0

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

    3. Applied cos-diff17.6

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

    5. Applied *-commutative17.6

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

    6. Applied associate-/r*17.6

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

    7. Taylor expanded around inf 17.6

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

    9. Applied associate-*l*17.6

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

    10. Simplified17.6

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

    if 7.323021828710162e+285 < (* 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 44.7

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

  4. Final simplification19.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \le 7.3230218287101624 \cdot 10^{285}:\\ \;\;\;\;2 \cdot \left(\mathsf{fma}\left(\cos y, \cos \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right), \sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right) \cdot \sqrt{x}\right) - \frac{\frac{a}{3}}{b}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - \frac{1}{2} \cdot {y}^{2}\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

Reproduce

herbie shell --seed 2020078 +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))))