Average Error: 20.4 → 17.7
Time: 11.1s
Precision: binary64
\[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3}\]
↓
\[\begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \leq 0.9999999999999947:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(\frac{z \cdot t}{3}\right) + \sqrt[3]{\sin \left(\frac{z \cdot t}{3}\right)} \cdot \left(\left(\sqrt[3]{\sin \left(\frac{z \cdot t}{3}\right)} \cdot \sqrt[3]{\sin \left(\frac{z \cdot t}{3}\right)}\right) \cdot \sin y\right)\right) - \frac{a}{3 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - 0.5 \cdot \left(y \cdot y\right)\right) - \frac{a}{3 \cdot b}\\ \end{array}\]
\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}\;\cos \left(y - \frac{z \cdot t}{3}\right) \leq 0.9999999999999947:\\
\;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(\frac{z \cdot t}{3}\right) + \sqrt[3]{\sin \left(\frac{z \cdot t}{3}\right)} \cdot \left(\left(\sqrt[3]{\sin \left(\frac{z \cdot t}{3}\right)} \cdot \sqrt[3]{\sin \left(\frac{z \cdot t}{3}\right)}\right) \cdot \sin y\right)\right) - \frac{a}{3 \cdot b}\\

\mathbf{else}:\\
\;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - 0.5 \cdot \left(y \cdot y\right)\right) - \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 (<= (cos (- y (/ (* z t) 3.0))) 0.9999999999999947)
   (-
    (*
     (* 2.0 (sqrt x))
     (+
      (* (cos y) (cos (/ (* z t) 3.0)))
      (*
       (cbrt (sin (/ (* z t) 3.0)))
       (*
        (* (cbrt (sin (/ (* z t) 3.0))) (cbrt (sin (/ (* z t) 3.0))))
        (sin y)))))
    (/ a (* 3.0 b)))
   (- (* (* 2.0 (sqrt x)) (- 1.0 (* 0.5 (* y y)))) (/ 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 (cos(y - ((z * t) / 3.0)) <= 0.9999999999999947) {
		tmp = ((2.0 * sqrt(x)) * ((cos(y) * cos((z * t) / 3.0)) + (cbrt(sin((z * t) / 3.0)) * ((cbrt(sin((z * t) / 3.0)) * cbrt(sin((z * t) / 3.0))) * sin(y))))) - (a / (3.0 * b));
	} else {
		tmp = ((2.0 * sqrt(x)) * (1.0 - (0.5 * (y * y)))) - (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

Original20.4
Target18.4
Herbie17.7
\[\begin{array}{l} \mathbf{if}\;z < -1.3793337487235141 \cdot 10^{+129}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\frac{1}{y} - \frac{\frac{0.3333333333333333}{z}}{t}\right) - \frac{\frac{a}{3}}{b}\\ \mathbf{elif}\;z < 3.516290613555987 \cdot 10^{+106}:\\ \;\;\;\;\left(\sqrt{x} \cdot 2\right) \cdot \cos \left(y - \frac{t}{3} \cdot z\right) - \frac{\frac{a}{3}}{b}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(y - \frac{\frac{0.3333333333333333}{z}}{t}\right) \cdot \left(2 \cdot \sqrt{x}\right) - \frac{\frac{a}{b}}{3}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (cos.f64 (-.f64 y (/.f64 (*.f64 z t) 3))) < 0.99999999999999467

    1. Initial program 19.6

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

    3. Applied cos-diff_binary6418.9

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

    5. Applied add-cube-cbrt_binary6418.9

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

    6. Applied associate-*r*_binary6418.9

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

    7. Simplified18.9

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

    if 0.99999999999999467 < (cos.f64 (-.f64 y (/.f64 (*.f64 z t) 3)))

    1. Initial program 21.9

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

    2. Taylor expanded around 0 15.5

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

    3. Simplified15.5

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

  4. Final simplification17.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \leq 0.9999999999999947:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(\frac{z \cdot t}{3}\right) + \sqrt[3]{\sin \left(\frac{z \cdot t}{3}\right)} \cdot \left(\left(\sqrt[3]{\sin \left(\frac{z \cdot t}{3}\right)} \cdot \sqrt[3]{\sin \left(\frac{z \cdot t}{3}\right)}\right) \cdot \sin y\right)\right) - \frac{a}{3 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - 0.5 \cdot \left(y \cdot y\right)\right) - \frac{a}{3 \cdot b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020224 
(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))))