Average Error: 20.9 → 16.8
Time: 16.5s
Precision: binary64
\[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \cdot t \leq -9.314420099989661 \cdot 10^{+290}:\\ \;\;\;\;\sqrt[3]{\left(2 \cdot \sqrt{x}\right) \cdot \cos y} \cdot \left(\sqrt[3]{\left(2 \cdot \sqrt{x}\right) \cdot \cos y} \cdot \sqrt[3]{\left(2 \cdot \sqrt{x}\right) \cdot \cos y}\right) - \frac{a}{b \cdot 3}\\ \mathbf{elif}\;z \cdot t \leq 4.3401700005647045 \cdot 10^{+290}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(\left(-\frac{z}{\sqrt{3}}\right) \cdot \frac{t}{\sqrt{3}}\right) - \sin y \cdot \sin \left(\left(-\frac{z}{\sqrt{3}}\right) \cdot \frac{t}{\sqrt{3}}\right)\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{x} - \frac{\frac{a}{b}}{3}\\ \end{array}\]
\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 -9.314420099989661 \cdot 10^{+290}:\\
\;\;\;\;\sqrt[3]{\left(2 \cdot \sqrt{x}\right) \cdot \cos y} \cdot \left(\sqrt[3]{\left(2 \cdot \sqrt{x}\right) \cdot \cos y} \cdot \sqrt[3]{\left(2 \cdot \sqrt{x}\right) \cdot \cos y}\right) - \frac{a}{b \cdot 3}\\

\mathbf{elif}\;z \cdot t \leq 4.3401700005647045 \cdot 10^{+290}:\\
\;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(\left(-\frac{z}{\sqrt{3}}\right) \cdot \frac{t}{\sqrt{3}}\right) - \sin y \cdot \sin \left(\left(-\frac{z}{\sqrt{3}}\right) \cdot \frac{t}{\sqrt{3}}\right)\right) - \frac{a}{b \cdot 3}\\

\mathbf{else}:\\
\;\;\;\;2 \cdot \sqrt{x} - \frac{\frac{a}{b}}{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 (<= (* z t) -9.314420099989661e+290)
   (-
    (*
     (cbrt (* (* 2.0 (sqrt x)) (cos y)))
     (*
      (cbrt (* (* 2.0 (sqrt x)) (cos y)))
      (cbrt (* (* 2.0 (sqrt x)) (cos y)))))
    (/ a (* b 3.0)))
   (if (<= (* z t) 4.3401700005647045e+290)
     (-
      (*
       (* 2.0 (sqrt x))
       (-
        (* (cos y) (cos (* (- (/ z (sqrt 3.0))) (/ t (sqrt 3.0)))))
        (* (sin y) (sin (* (- (/ z (sqrt 3.0))) (/ t (sqrt 3.0)))))))
      (/ a (* b 3.0)))
     (- (* 2.0 (sqrt x)) (/ (/ a b) 3.0)))))
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 ((z * t) <= -9.314420099989661e+290) {
		tmp = (cbrt((2.0 * sqrt(x)) * cos(y)) * (cbrt((2.0 * sqrt(x)) * cos(y)) * cbrt((2.0 * sqrt(x)) * cos(y)))) - (a / (b * 3.0));
	} else if ((z * t) <= 4.3401700005647045e+290) {
		tmp = ((2.0 * sqrt(x)) * ((cos(y) * cos(-(z / sqrt(3.0)) * (t / sqrt(3.0)))) - (sin(y) * sin(-(z / sqrt(3.0)) * (t / sqrt(3.0)))))) - (a / (b * 3.0));
	} else {
		tmp = (2.0 * sqrt(x)) - ((a / 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

Original20.9
Target19.0
Herbie16.8
\[\begin{array}{l} \mathbf{if}\;z < -1.3793337487235141 \cdot 10^{+129}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\frac{1}{y} - \frac{\frac{0.3333333333333333}{z}}{t}\right) - \frac{\frac{a}{3}}{b}\\ \mathbf{elif}\;z < 3.516290613555987 \cdot 10^{+106}:\\ \;\;\;\;\left(\sqrt{x} \cdot 2\right) \cdot \cos \left(y - \frac{t}{3} \cdot z\right) - \frac{\frac{a}{3}}{b}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(y - \frac{\frac{0.3333333333333333}{z}}{t}\right) \cdot \left(2 \cdot \sqrt{x}\right) - \frac{\frac{a}{b}}{3}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (*.f64 z t) < -9.3144200999896612e290

    1. Initial program 61.1

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

    2. Taylor expanded around 0 33.9

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

    4. Applied add-cube-cbrt_binary64_1989133.9

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

    if -9.3144200999896612e290 < (*.f64 z t) < 4.34017000056470449e290

    1. Initial program 14.4

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

    3. Applied add-sqr-sqrt_binary64_1987814.5

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

    4. Applied times-frac_binary64_1986214.5

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

    5. Applied cancel-sign-sub-inv_binary64_1982214.5

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

    6. Applied cos-sum_binary64_1999014.0

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

    if 4.34017000056470449e290 < (*.f64 z t)

    1. Initial program 60.8

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

    2. Taylor expanded around 0 34.4

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

    4. Applied associate-/r*_binary64_1980034.6

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

    5. Taylor expanded around 0 34.6

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

  4. Final simplification16.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -9.314420099989661 \cdot 10^{+290}:\\ \;\;\;\;\sqrt[3]{\left(2 \cdot \sqrt{x}\right) \cdot \cos y} \cdot \left(\sqrt[3]{\left(2 \cdot \sqrt{x}\right) \cdot \cos y} \cdot \sqrt[3]{\left(2 \cdot \sqrt{x}\right) \cdot \cos y}\right) - \frac{a}{b \cdot 3}\\ \mathbf{elif}\;z \cdot t \leq 4.3401700005647045 \cdot 10^{+290}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(\left(-\frac{z}{\sqrt{3}}\right) \cdot \frac{t}{\sqrt{3}}\right) - \sin y \cdot \sin \left(\left(-\frac{z}{\sqrt{3}}\right) \cdot \frac{t}{\sqrt{3}}\right)\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{x} - \frac{\frac{a}{b}}{3}\\ \end{array}\]

Reproduce

herbie shell --seed 2021015 
(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))))