Average Error: 19.3 → 14.5
Time: 14.6s
Precision: binary64
\[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3} \]
↓
\[\begin{array}{l} t_1 := \frac{a}{b \cdot 3}\\ t_2 := 2 \cdot \sqrt{x}\\ \mathbf{if}\;z \cdot t \leq -6.753625024129311 \cdot 10^{+193}:\\ \;\;\;\;\begin{array}{l} t_3 := \sqrt[3]{\cos y}\\ t_2 \cdot \left(t_3 \cdot t_3\right) - t_1 \end{array}\\ \mathbf{elif}\;z \cdot t \leq 748913414011201.5:\\ \;\;\;\;\begin{array}{l} t_4 := \left(-z\right) \cdot \frac{t}{3}\\ t_2 \cdot \left(\cos y \cdot \cos t_4 - \sin y \cdot \sin t_4\right) - t_1 \end{array}\\ \mathbf{else}:\\ \;\;\;\;t_2 - t_1\\ \end{array} \]
\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3}
↓
\begin{array}{l}
t_1 := \frac{a}{b \cdot 3}\\
t_2 := 2 \cdot \sqrt{x}\\
\mathbf{if}\;z \cdot t \leq -6.753625024129311 \cdot 10^{+193}:\\
\;\;\;\;\begin{array}{l}
t_3 := \sqrt[3]{\cos y}\\
t_2 \cdot \left(t_3 \cdot t_3\right) - t_1
\end{array}\\

\mathbf{elif}\;z \cdot t \leq 748913414011201.5:\\
\;\;\;\;\begin{array}{l}
t_4 := \left(-z\right) \cdot \frac{t}{3}\\
t_2 \cdot \left(\cos y \cdot \cos t_4 - \sin y \cdot \sin t_4\right) - t_1
\end{array}\\

\mathbf{else}:\\
\;\;\;\;t_2 - t_1\\


\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
 (let* ((t_1 (/ a (* b 3.0))) (t_2 (* 2.0 (sqrt x))))
   (if (<= (* z t) -6.753625024129311e+193)
     (let* ((t_3 (cbrt (cos y)))) (- (* t_2 (* t_3 t_3)) t_1))
     (if (<= (* z t) 748913414011201.5)
       (let* ((t_4 (* (- z) (/ t 3.0))))
         (- (* t_2 (- (* (cos y) (cos t_4)) (* (sin y) (sin t_4)))) t_1))
       (- t_2 t_1)))))
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 t_1 = a / (b * 3.0);
	double t_2 = 2.0 * sqrt(x);
	double tmp;
	if ((z * t) <= -6.753625024129311e+193) {
		double t_3_1 = cbrt(cos(y));
		tmp = (t_2 * (t_3_1 * t_3_1)) - t_1;
	} else if ((z * t) <= 748913414011201.5) {
		double t_4 = -z * (t / 3.0);
		tmp = (t_2 * ((cos(y) * cos(t_4)) - (sin(y) * sin(t_4)))) - t_1;
	} else {
		tmp = t_2 - t_1;
	}
	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

Original19.3
Target16.6
Herbie14.5
\[\begin{array}{l} \mathbf{if}\;z < -1.3793337487235141 \cdot 10^{+129}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\frac{1}{y} - \frac{\frac{0.3333333333333333}{z}}{t}\right) - \frac{\frac{a}{3}}{b}\\ \mathbf{elif}\;z < 3.516290613555987 \cdot 10^{+106}:\\ \;\;\;\;\left(\sqrt{x} \cdot 2\right) \cdot \cos \left(y - \frac{t}{3} \cdot z\right) - \frac{\frac{a}{3}}{b}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(y - \frac{\frac{0.3333333333333333}{z}}{t}\right) \cdot \left(2 \cdot \sqrt{x}\right) - \frac{\frac{a}{b}}{3}\\ \end{array} \]

Derivation

  1. Split input into 3 regimes

  2. if (*.f64 z t) < -6.7536250241293111e193

    1. Initial program 47.6

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

    2. Taylor expanded in z around 0 28.4

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

    3. Applied add-cube-cbrt_binary6428.4

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

    4. Applied associate-*r*_binary6428.4

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

    5. Taylor expanded in y around 0 28.5

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

    if -6.7536250241293111e193 < (*.f64 z t) < 748913414011201.5

    1. Initial program 7.2

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

    2. Applied *-un-lft-identity_binary647.2

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

    3. Applied times-frac_binary647.2

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

    4. Applied cancel-sign-sub-inv_binary647.2

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

    5. Applied cos-sum_binary646.6

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

    if 748913414011201.5 < (*.f64 z t)

    1. Initial program 38.7

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

    2. Taylor expanded in z around 0 29.3

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

    3. Taylor expanded in y around 0 29.3

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

  4. Final simplification14.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -6.753625024129311 \cdot 10^{+193}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(\sqrt[3]{\cos y} \cdot \sqrt[3]{\cos y}\right) - \frac{a}{b \cdot 3}\\ \mathbf{elif}\;z \cdot t \leq 748913414011201.5:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(\left(-z\right) \cdot \frac{t}{3}\right) - \sin y \cdot \sin \left(\left(-z\right) \cdot \frac{t}{3}\right)\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{x} - \frac{a}{b \cdot 3}\\ \end{array} \]

Reproduce

herbie shell --seed 2022088 
(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))))