Average Error: 20.7 → 15.1
Time: 20.1s
Precision: binary64
\[[z, t] = \mathsf{sort}([z, t]) \\]
\[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3} \]
↓
\[\begin{array}{l} t_1 := 2 \cdot \sqrt{x}\\ \mathbf{if}\;z \cdot t \leq -2.1239002792528875 \cdot 10^{+291}:\\ \;\;\;\;t_1 - \frac{\frac{a}{b}}{3}\\ \mathbf{elif}\;z \cdot t \leq 5.2319661681899375 \cdot 10^{+268}:\\ \;\;\;\;\begin{array}{l} t_2 := z \cdot \frac{t}{3}\\ t_3 := \mathsf{fma}\left(1, y, -t_2\right)\\ t_4 := \mathsf{fma}\left(-\frac{t}{3}, z, t_2\right)\\ t_1 \cdot \left(\cos t_3 \cdot \cos t_4 - \sin t_3 \cdot \sin t_4\right) - \frac{a}{b \cdot 3} \end{array}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \cos y - a \cdot \frac{0.3333333333333333}{b}\\ \end{array} \]
\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3}
↓
\begin{array}{l}
t_1 := 2 \cdot \sqrt{x}\\
\mathbf{if}\;z \cdot t \leq -2.1239002792528875 \cdot 10^{+291}:\\
\;\;\;\;t_1 - \frac{\frac{a}{b}}{3}\\

\mathbf{elif}\;z \cdot t \leq 5.2319661681899375 \cdot 10^{+268}:\\
\;\;\;\;\begin{array}{l}
t_2 := z \cdot \frac{t}{3}\\
t_3 := \mathsf{fma}\left(1, y, -t_2\right)\\
t_4 := \mathsf{fma}\left(-\frac{t}{3}, z, t_2\right)\\
t_1 \cdot \left(\cos t_3 \cdot \cos t_4 - \sin t_3 \cdot \sin t_4\right) - \frac{a}{b \cdot 3}
\end{array}\\

\mathbf{else}:\\
\;\;\;\;t_1 \cdot \cos y - a \cdot \frac{0.3333333333333333}{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
 (let* ((t_1 (* 2.0 (sqrt x))))
   (if (<= (* z t) -2.1239002792528875e+291)
     (- t_1 (/ (/ a b) 3.0))
     (if (<= (* z t) 5.2319661681899375e+268)
       (let* ((t_2 (* z (/ t 3.0)))
              (t_3 (fma 1.0 y (- t_2)))
              (t_4 (fma (- (/ t 3.0)) z t_2)))
         (-
          (* t_1 (- (* (cos t_3) (cos t_4)) (* (sin t_3) (sin t_4))))
          (/ a (* b 3.0))))
       (- (* t_1 (cos y)) (* a (/ 0.3333333333333333 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 t_1 = 2.0 * sqrt(x);
	double tmp;
	if ((z * t) <= -2.1239002792528875e+291) {
		tmp = t_1 - ((a / b) / 3.0);
	} else if ((z * t) <= 5.2319661681899375e+268) {
		double t_2 = z * (t / 3.0);
		double t_3 = fma(1.0, y, -t_2);
		double t_4 = fma(-(t / 3.0), z, t_2);
		tmp = (t_1 * ((cos(t_3) * cos(t_4)) - (sin(t_3) * sin(t_4)))) - (a / (b * 3.0));
	} else {
		tmp = (t_1 * cos(y)) - (a * (0.3333333333333333 / 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

Target

Original20.7
Target18.9
Herbie15.1
\[\begin{array}{l} \mathbf{if}\;z < -1.3793337487235141 \cdot 10^{+129}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\frac{1}{y} - \frac{\frac{0.3333333333333333}{z}}{t}\right) - \frac{\frac{a}{3}}{b}\\ \mathbf{elif}\;z < 3.516290613555987 \cdot 10^{+106}:\\ \;\;\;\;\left(\sqrt{x} \cdot 2\right) \cdot \cos \left(y - \frac{t}{3} \cdot z\right) - \frac{\frac{a}{3}}{b}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(y - \frac{\frac{0.3333333333333333}{z}}{t}\right) \cdot \left(2 \cdot \sqrt{x}\right) - \frac{\frac{a}{b}}{3}\\ \end{array} \]

Derivation

  1. Split input into 3 regimes

  2. if (*.f64 z t) < -2.1239002792528875e291

    1. Initial program 60.7

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

    2. Taylor expanded in z around 0 35.7

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

    3. Applied associate-/r*_binary6435.8

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

    4. Taylor expanded in y around 0 35.8

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

    if -2.1239002792528875e291 < (*.f64 z t) < 5.2319661681899375e268

    1. Initial program 14.0

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

    2. Applied *-un-lft-identity_binary6414.0

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

    3. Applied times-frac_binary6414.0

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

    4. Applied *-un-lft-identity_binary6414.0

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

    5. Applied prod-diff_binary6414.0

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

    6. Applied cos-sum_binary6411.6

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

    if 5.2319661681899375e268 < (*.f64 z t)

    1. Initial program 58.3

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

    2. Taylor expanded in z around 0 34.3

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

    3. Applied div-inv_binary6434.4

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

    4. Simplified34.4

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \cos y - a \cdot \color{blue}{\frac{0.3333333333333333}{b}} \]
  3. Recombined 3 regimes into one program.

  4. Final simplification15.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -2.1239002792528875 \cdot 10^{+291}:\\ \;\;\;\;2 \cdot \sqrt{x} - \frac{\frac{a}{b}}{3}\\ \mathbf{elif}\;z \cdot t \leq 5.2319661681899375 \cdot 10^{+268}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos \left(\mathsf{fma}\left(1, y, -z \cdot \frac{t}{3}\right)\right) \cdot \cos \left(\mathsf{fma}\left(-\frac{t}{3}, z, z \cdot \frac{t}{3}\right)\right) - \sin \left(\mathsf{fma}\left(1, y, -z \cdot \frac{t}{3}\right)\right) \cdot \sin \left(\mathsf{fma}\left(-\frac{t}{3}, z, z \cdot \frac{t}{3}\right)\right)\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \cos y - a \cdot \frac{0.3333333333333333}{b}\\ \end{array} \]

Reproduce

herbie shell --seed 2022077 
(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))))