Average Error: 20.0 → 15.9
Time: 21.8s
Precision: binary64
\[[z, t]=\mathsf{sort}([z, t])\]
\[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \cdot t \leq -1.178476436106435 \cdot 10^{+192}:\\ \;\;\;\;2 \cdot \sqrt{x} - \frac{1}{b} \cdot \frac{a}{3}\\ \mathbf{elif}\;z \cdot t \leq 3.3700040350114406 \cdot 10^{+180}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos \left(\left(z \cdot t\right) \cdot 0.3333333333333333\right) \cdot \cos y - \sin \left(z \cdot \left(t \cdot -0.3333333333333333\right)\right) \cdot \sin y\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{x} - \frac{a}{b \cdot 3}\\ \end{array}\]
\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 \leq -1.178476436106435 \cdot 10^{+192}:\\
\;\;\;\;2 \cdot \sqrt{x} - \frac{1}{b} \cdot \frac{a}{3}\\

\mathbf{elif}\;z \cdot t \leq 3.3700040350114406 \cdot 10^{+180}:\\
\;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos \left(\left(z \cdot t\right) \cdot 0.3333333333333333\right) \cdot \cos y - \sin \left(z \cdot \left(t \cdot -0.3333333333333333\right)\right) \cdot \sin y\right) - \frac{a}{b \cdot 3}\\

\mathbf{else}:\\
\;\;\;\;2 \cdot \sqrt{x} - \frac{a}{b \cdot 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) -1.178476436106435e+192)
   (- (* 2.0 (sqrt x)) (* (/ 1.0 b) (/ a 3.0)))
   (if (<= (* z t) 3.3700040350114406e+180)
     (-
      (*
       (* 2.0 (sqrt x))
       (-
        (* (cos (* (* z t) 0.3333333333333333)) (cos y))
        (* (sin (* z (* t -0.3333333333333333))) (sin y))))
      (/ 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) <= -1.178476436106435e+192) {
		tmp = (2.0 * sqrt(x)) - ((1.0 / b) * (a / 3.0));
	} else if ((z * t) <= 3.3700040350114406e+180) {
		tmp = ((2.0 * sqrt(x)) * ((cos((z * t) * 0.3333333333333333) * cos(y)) - (sin(z * (t * -0.3333333333333333)) * sin(y)))) - (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

Original20.0
Target18.2
Herbie15.9
\[\begin{array}{l} \mathbf{if}\;z < -1.3793337487235141 \cdot 10^{+129}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\frac{1}{y} - \frac{\frac{0.3333333333333333}{z}}{t}\right) - \frac{\frac{a}{3}}{b}\\ \mathbf{elif}\;z < 3.516290613555987 \cdot 10^{+106}:\\ \;\;\;\;\left(\sqrt{x} \cdot 2\right) \cdot \cos \left(y - \frac{t}{3} \cdot z\right) - \frac{\frac{a}{3}}{b}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(y - \frac{\frac{0.3333333333333333}{z}}{t}\right) \cdot \left(2 \cdot \sqrt{x}\right) - \frac{\frac{a}{b}}{3}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (*.f64 z t) < -1.178476436106435e192

    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 around 0 31.7

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

    4. Applied *-un-lft-identity_binary64_1815131.7

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

    5. Applied times-frac_binary64_1815731.8

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

    6. Taylor expanded around 0 31.8

      \[\leadsto \color{blue}{2 \cdot \sqrt{x}} - \frac{1}{b} \cdot \frac{a}{3}\]

    if -1.178476436106435e192 < (*.f64 z t) < 3.3700040350114406e180

    1. Initial program 11.6

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

    3. Applied sub-neg_binary64_1814411.6

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

    4. Applied cos-sum_binary64_1828511.0

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

    5. Simplified11.0

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \left(\color{blue}{\cos \left(0.3333333333333333 \cdot \left(t \cdot z\right)\right) \cdot \cos y} - \sin y \cdot \sin \left(-\frac{z \cdot t}{3}\right)\right) - \frac{a}{b \cdot 3}\]

    6. Simplified11.0

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \left(\cos \left(0.3333333333333333 \cdot \left(t \cdot z\right)\right) \cdot \cos y - \color{blue}{\sin \left(z \cdot \left(t \cdot -0.3333333333333333\right)\right) \cdot \sin y}\right) - \frac{a}{b \cdot 3}\]

    if 3.3700040350114406e180 < (*.f64 z t)

    1. Initial program 48.8

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

    2. Taylor expanded around 0 33.0

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

    3. Taylor expanded around 0 32.7

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

  4. Final simplification15.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -1.178476436106435 \cdot 10^{+192}:\\ \;\;\;\;2 \cdot \sqrt{x} - \frac{1}{b} \cdot \frac{a}{3}\\ \mathbf{elif}\;z \cdot t \leq 3.3700040350114406 \cdot 10^{+180}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos \left(\left(z \cdot t\right) \cdot 0.3333333333333333\right) \cdot \cos y - \sin \left(z \cdot \left(t \cdot -0.3333333333333333\right)\right) \cdot \sin y\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{x} - \frac{a}{b \cdot 3}\\ \end{array}\]

Reproduce

herbie shell --seed 2021044 
(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))))