Average Error: 20.8 → 18.3
Time: 18.0s
Precision: binary64
\[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) \leq 1.3726967043725497 \cdot 10^{+147}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(z \cdot \frac{t}{3}\right) + \sin y \cdot \sin \left(\left(z \cdot t\right) \cdot 0.3333333333333333\right)\right) - \frac{a}{3 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - 0.5 \cdot \left(y \cdot y\right)\right) - \frac{a}{3 \cdot b}\\ \end{array}\]
\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}\;\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) \leq 1.3726967043725497 \cdot 10^{+147}:\\
\;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(z \cdot \frac{t}{3}\right) + \sin y \cdot \sin \left(\left(z \cdot t\right) \cdot 0.3333333333333333\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 (<=
      (* (* 2.0 (sqrt x)) (cos (- y (/ (* z t) 3.0))))
      1.3726967043725497e+147)
   (-
    (*
     (* 2.0 (sqrt x))
     (+
      (* (cos y) (cos (* z (/ t 3.0))))
      (* (sin y) (sin (* (* z t) 0.3333333333333333)))))
    (/ 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 ((double) (((double) (((double) (2.0 * ((double) sqrt(x)))) * ((double) cos(((double) (y - (((double) (z * t)) / 3.0))))))) - (a / ((double) (b * 3.0)))));
}
↓
double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((((double) (((double) (2.0 * ((double) sqrt(x)))) * ((double) cos(((double) (y - (((double) (z * t)) / 3.0))))))) <= 1.3726967043725497e+147)) {
		tmp = ((double) (((double) (((double) (2.0 * ((double) sqrt(x)))) * ((double) (((double) (((double) cos(y)) * ((double) cos(((double) (z * (t / 3.0))))))) + ((double) (((double) sin(y)) * ((double) sin(((double) (((double) (z * t)) * 0.3333333333333333)))))))))) - (a / ((double) (3.0 * b)))));
	} else {
		tmp = ((double) (((double) (((double) (2.0 * ((double) sqrt(x)))) * ((double) (1.0 - ((double) (0.5 * ((double) (y * y)))))))) - (a / ((double) (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

Original20.8
Target18.9
Herbie18.3
\[\begin{array}{l} \mathbf{if}\;z < -1.3793337487235141 \cdot 10^{+129}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\frac{1}{y} - \frac{\frac{0.3333333333333333}{z}}{t}\right) - \frac{\frac{a}{3}}{b}\\ \mathbf{elif}\;z < 3.516290613555987 \cdot 10^{+106}:\\ \;\;\;\;\left(\sqrt{x} \cdot 2\right) \cdot \cos \left(y - \frac{t}{3} \cdot z\right) - \frac{\frac{a}{3}}{b}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(y - \frac{\frac{0.3333333333333333}{z}}{t}\right) \cdot \left(2 \cdot \sqrt{x}\right) - \frac{\frac{a}{b}}{3}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (* (* 2.0 (sqrt x)) (cos (- y (/ (* z t) 3.0)))) < 1.3726967043725497e147

    1. Initial program Error: 14.7 bits

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

    3. Applied add-sqr-sqrtError: 14.7 bits

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

    4. Applied times-fracError: 14.7 bits

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

    6. Applied cos-diffError: 14.3 bits

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

    7. SimplifiedError: 14.3 bits

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

    8. SimplifiedError: 14.3 bits

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

    9. Taylor expanded around inf Error: 14.3 bits

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

    10. SimplifiedError: 14.3 bits

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

    if 1.3726967043725497e147 < (* (* 2.0 (sqrt x)) (cos (- y (/ (* z t) 3.0))))

    1. Initial program Error: 60.3 bits

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

    2. Taylor expanded around 0 Error: 44.4 bits

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

    3. SimplifiedError: 44.4 bits

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

  4. Final simplificationError: 18.3 bits

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) \leq 1.3726967043725497 \cdot 10^{+147}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(z \cdot \frac{t}{3}\right) + \sin y \cdot \sin \left(\left(z \cdot t\right) \cdot 0.3333333333333333\right)\right) - \frac{a}{3 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - 0.5 \cdot \left(y \cdot y\right)\right) - \frac{a}{3 \cdot b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020205 
(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))))