Average Error: 20.4 → 16.2
Time: 17.3s
Precision: binary64
\[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3} \]
↓
\[\begin{array}{l} t_1 := 2 \cdot \sqrt{x}\\ t_2 := \frac{a}{b \cdot 3}\\ t_3 := \frac{z \cdot t}{3}\\ \mathbf{if}\;z \cdot t \leq -2.1488931442503473 \cdot 10^{+34}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{-0.3333333333333333}{b}, t_1\right)\\ \mathbf{elif}\;z \cdot t \leq 4.50876227333934 \cdot 10^{+210}:\\ \;\;\;\;t_1 \cdot \mathsf{fma}\left(\cos y, \cos t_3, \sin y \cdot \sin t_3\right) - t_2\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt[3]{x \cdot 4}, \cos y \cdot \sqrt[3]{t_1}, -t_2\right)\\ \end{array} \]
\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}\\
t_2 := \frac{a}{b \cdot 3}\\
t_3 := \frac{z \cdot t}{3}\\
\mathbf{if}\;z \cdot t \leq -2.1488931442503473 \cdot 10^{+34}:\\
\;\;\;\;\mathsf{fma}\left(a, \frac{-0.3333333333333333}{b}, t_1\right)\\

\mathbf{elif}\;z \cdot t \leq 4.50876227333934 \cdot 10^{+210}:\\
\;\;\;\;t_1 \cdot \mathsf{fma}\left(\cos y, \cos t_3, \sin y \cdot \sin t_3\right) - t_2\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\sqrt[3]{x \cdot 4}, \cos y \cdot \sqrt[3]{t_1}, -t_2\right)\\


\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))) (t_2 (/ a (* b 3.0))) (t_3 (/ (* z t) 3.0)))
   (if (<= (* z t) -2.1488931442503473e+34)
     (fma a (/ -0.3333333333333333 b) t_1)
     (if (<= (* z t) 4.50876227333934e+210)
       (- (* t_1 (fma (cos y) (cos t_3) (* (sin y) (sin t_3)))) t_2)
       (fma (cbrt (* x 4.0)) (* (cos y) (cbrt t_1)) (- t_2))))))
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 t_2 = a / (b * 3.0);
	double t_3 = (z * t) / 3.0;
	double tmp;
	if ((z * t) <= -2.1488931442503473e+34) {
		tmp = fma(a, (-0.3333333333333333 / b), t_1);
	} else if ((z * t) <= 4.50876227333934e+210) {
		tmp = (t_1 * fma(cos(y), cos(t_3), (sin(y) * sin(t_3)))) - t_2;
	} else {
		tmp = fma(cbrt((x * 4.0)), (cos(y) * cbrt(t_1)), -t_2);
	}
	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

Original20.4
Target18.5
Herbie16.2
\[\begin{array}{l} \mathbf{if}\;z < -1.3793337487235141 \cdot 10^{+129}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\frac{1}{y} - \frac{\frac{0.3333333333333333}{z}}{t}\right) - \frac{\frac{a}{3}}{b}\\ \mathbf{elif}\;z < 3.516290613555987 \cdot 10^{+106}:\\ \;\;\;\;\left(\sqrt{x} \cdot 2\right) \cdot \cos \left(y - \frac{t}{3} \cdot z\right) - \frac{\frac{a}{3}}{b}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(y - \frac{\frac{0.3333333333333333}{z}}{t}\right) \cdot \left(2 \cdot \sqrt{x}\right) - \frac{\frac{a}{b}}{3}\\ \end{array} \]

Derivation

  1. Split input into 3 regimes

  2. if (*.f64 z t) < -2.1488931442503473e34

    1. Initial program 41.9

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

    2. Taylor expanded in z around 0 33.6

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

    3. Applied egg-rr33.6

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, -\frac{0.3333333333333333}{b}, \left(2 \cdot \sqrt{x}\right) \cdot \cos y\right)} \]

    4. Taylor expanded in y around 0 33.3

      \[\leadsto \mathsf{fma}\left(a, -\frac{0.3333333333333333}{b}, \color{blue}{2 \cdot \sqrt{x}}\right) \]

    if -2.1488931442503473e34 < (*.f64 z t) < 4.50876227333934e210

    1. Initial program 8.3

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

    2. Applied egg-rr7.7

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

    if 4.50876227333934e210 < (*.f64 z t)

    1. Initial program 51.2

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

    2. Taylor expanded in z around 0 33.8

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

    3. Applied egg-rr33.8

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt[3]{x \cdot 4}, \sqrt[3]{2 \cdot \sqrt{x}} \cdot \cos y, \left(-\frac{a}{b \cdot 3}\right) \cdot 1\right)} \]
  3. Recombined 3 regimes into one program.

  4. Final simplification16.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -2.1488931442503473 \cdot 10^{+34}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{-0.3333333333333333}{b}, 2 \cdot \sqrt{x}\right)\\ \mathbf{elif}\;z \cdot t \leq 4.50876227333934 \cdot 10^{+210}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \mathsf{fma}\left(\cos y, \cos \left(\frac{z \cdot t}{3}\right), \sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt[3]{x \cdot 4}, \cos y \cdot \sqrt[3]{2 \cdot \sqrt{x}}, -\frac{a}{b \cdot 3}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022130 
(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))))