Average Error: 21.0 → 17.1
Time: 26.6s
Precision: binary64
\[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3}\]
↓
\[\begin{array}{l} \mathbf{if}\;y - \frac{z \cdot t}{3} \leq -9.841278127820592 \cdot 10^{+303}:\\ \;\;\;\;2 \cdot \sqrt{x} - \frac{0.3333333333333333 \cdot a}{b}\\ \mathbf{elif}\;y - \frac{z \cdot t}{3} \leq 3.6312880224776764 \cdot 10^{+287}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{3 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{x} - \frac{a}{3 \cdot b}\\ \end{array}\]
\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}\;y - \frac{z \cdot t}{3} \leq -9.841278127820592 \cdot 10^{+303}:\\
\;\;\;\;2 \cdot \sqrt{x} - \frac{0.3333333333333333 \cdot a}{b}\\

\mathbf{elif}\;y - \frac{z \cdot t}{3} \leq 3.6312880224776764 \cdot 10^{+287}:\\
\;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{3 \cdot b}\\

\mathbf{else}:\\
\;\;\;\;2 \cdot \sqrt{x} - \frac{a}{3 \cdot b}\\

\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 (<= (- y (/ (* z t) 3.0)) -9.841278127820592e+303)
   (- (* 2.0 (sqrt x)) (/ (* 0.3333333333333333 a) b))
   (if (<= (- y (/ (* z t) 3.0)) 3.6312880224776764e+287)
     (- (* (* 2.0 (sqrt x)) (cos (- y (/ (* z t) 3.0)))) (/ a (* 3.0 b)))
     (- (* 2.0 (sqrt x)) (/ a (* 3.0 b))))))
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 tmp;
	if ((y - ((z * t) / 3.0)) <= -9.841278127820592e+303) {
		tmp = (2.0 * sqrt(x)) - ((0.3333333333333333 * a) / b);
	} else if ((y - ((z * t) / 3.0)) <= 3.6312880224776764e+287) {
		tmp = ((2.0 * sqrt(x)) * cos(y - ((z * t) / 3.0))) - (a / (3.0 * b));
	} else {
		tmp = (2.0 * sqrt(x)) - (a / (3.0 * b));
	}
	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

Original21.0
Target18.8
Herbie17.1
\[\begin{array}{l} \mathbf{if}\;z < -1.3793337487235141 \cdot 10^{+129}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\frac{1}{y} - \frac{\frac{0.3333333333333333}{z}}{t}\right) - \frac{\frac{a}{3}}{b}\\ \mathbf{elif}\;z < 3.516290613555987 \cdot 10^{+106}:\\ \;\;\;\;\left(\sqrt{x} \cdot 2\right) \cdot \cos \left(y - \frac{t}{3} \cdot z\right) - \frac{\frac{a}{3}}{b}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(y - \frac{\frac{0.3333333333333333}{z}}{t}\right) \cdot \left(2 \cdot \sqrt{x}\right) - \frac{\frac{a}{b}}{3}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (-.f64 y (/.f64 (*.f64 z t) 3)) < -9.84127812782059189e303

    1. Initial program 61.2

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

    2. Taylor expanded around 0 29.4

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

    4. Applied *-un-lft-identity_binary64_1712829.4

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

    5. Applied times-frac_binary64_1713429.5

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos y - \color{blue}{\frac{1}{b} \cdot \frac{a}{3}}\]
    6. Using strategy rm

    7. Applied associate-*l/_binary64_1707129.4

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

    8. Simplified29.5

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos y - \frac{\color{blue}{0.3333333333333333 \cdot a}}{b}\]

    9. Taylor expanded around 0 30.1

      \[\leadsto \color{blue}{2 \cdot \sqrt{x}} - \frac{0.3333333333333333 \cdot a}{b}\]

    if -9.84127812782059189e303 < (-.f64 y (/.f64 (*.f64 z t) 3)) < 3.63128802247767639e287

    1. Initial program 14.4

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

    if 3.63128802247767639e287 < (-.f64 y (/.f64 (*.f64 z t) 3))

    1. Initial program 52.5

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

    2. Taylor expanded around 0 28.8

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

    3. Taylor expanded around 0 31.9

      \[\leadsto \color{blue}{2 \cdot \sqrt{x}} - \frac{a}{b \cdot 3}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification17.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;y - \frac{z \cdot t}{3} \leq -9.841278127820592 \cdot 10^{+303}:\\ \;\;\;\;2 \cdot \sqrt{x} - \frac{0.3333333333333333 \cdot a}{b}\\ \mathbf{elif}\;y - \frac{z \cdot t}{3} \leq 3.6312880224776764 \cdot 10^{+287}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{3 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{x} - \frac{a}{3 \cdot b}\\ \end{array}\]

Reproduce

herbie shell --seed 2021023 
(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))))