Average Error: 20.1 → 14.9
Time: 19.5s
Precision: binary64
\[[z, t]=\mathsf{sort}([z, t])\]
\[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3} \]
↓
\[\begin{array}{l} t_1 := y - \frac{z \cdot t}{3}\\ t_2 := 2 \cdot \sqrt{x}\\ t_3 := \frac{a}{3 \cdot b}\\ \mathbf{if}\;t_1 \leq -1.834763411526171 \cdot 10^{+260}:\\ \;\;\;\;t_2 \cdot \cos y - t_3\\ \mathbf{elif}\;t_1 \leq 1.4978066105721533 \cdot 10^{+299}:\\ \;\;\;\;\begin{array}{l} t_4 := z \cdot \frac{t}{3}\\ t_5 := \mathsf{fma}\left(1, y, -t_4\right)\\ t_6 := \mathsf{fma}\left(-\frac{t}{3}, z, t_4\right)\\ t_2 \cdot \left(\cos t_5 \cdot \cos t_6 - \sin t_5 \cdot \sin t_6\right) - t_3 \end{array}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, \sqrt{x}, -0.3333333333333333 \cdot \frac{a}{b}\right)\\ \end{array} \]
\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 := y - \frac{z \cdot t}{3}\\
t_2 := 2 \cdot \sqrt{x}\\
t_3 := \frac{a}{3 \cdot b}\\
\mathbf{if}\;t_1 \leq -1.834763411526171 \cdot 10^{+260}:\\
\;\;\;\;t_2 \cdot \cos y - t_3\\

\mathbf{elif}\;t_1 \leq 1.4978066105721533 \cdot 10^{+299}:\\
\;\;\;\;\begin{array}{l}
t_4 := z \cdot \frac{t}{3}\\
t_5 := \mathsf{fma}\left(1, y, -t_4\right)\\
t_6 := \mathsf{fma}\left(-\frac{t}{3}, z, t_4\right)\\
t_2 \cdot \left(\cos t_5 \cdot \cos t_6 - \sin t_5 \cdot \sin t_6\right) - t_3
\end{array}\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(2, \sqrt{x}, -0.3333333333333333 \cdot \frac{a}{b}\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 (- y (/ (* z t) 3.0)))
        (t_2 (* 2.0 (sqrt x)))
        (t_3 (/ a (* 3.0 b))))
   (if (<= t_1 -1.834763411526171e+260)
     (- (* t_2 (cos y)) t_3)
     (if (<= t_1 1.4978066105721533e+299)
       (let* ((t_4 (* z (/ t 3.0)))
              (t_5 (fma 1.0 y (- t_4)))
              (t_6 (fma (- (/ t 3.0)) z t_4)))
         (- (* t_2 (- (* (cos t_5) (cos t_6)) (* (sin t_5) (sin t_6)))) t_3))
       (fma 2.0 (sqrt x) (* -0.3333333333333333 (/ a 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 = y - ((z * t) / 3.0);
	double t_2 = 2.0 * sqrt(x);
	double t_3 = a / (3.0 * b);
	double tmp;
	if (t_1 <= -1.834763411526171e+260) {
		tmp = (t_2 * cos(y)) - t_3;
	} else if (t_1 <= 1.4978066105721533e+299) {
		double t_4 = z * (t / 3.0);
		double t_5 = fma(1.0, y, -t_4);
		double t_6 = fma(-(t / 3.0), z, t_4);
		tmp = (t_2 * ((cos(t_5) * cos(t_6)) - (sin(t_5) * sin(t_6)))) - t_3;
	} else {
		tmp = fma(2.0, sqrt(x), (-0.3333333333333333 * (a / 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

Original20.1
Target18.5
Herbie14.9
\[\begin{array}{l} \mathbf{if}\;z < -1.3793337487235141 \cdot 10^{+129}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\frac{1}{y} - \frac{\frac{0.3333333333333333}{z}}{t}\right) - \frac{\frac{a}{3}}{b}\\ \mathbf{elif}\;z < 3.516290613555987 \cdot 10^{+106}:\\ \;\;\;\;\left(\sqrt{x} \cdot 2\right) \cdot \cos \left(y - \frac{t}{3} \cdot z\right) - \frac{\frac{a}{3}}{b}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(y - \frac{\frac{0.3333333333333333}{z}}{t}\right) \cdot \left(2 \cdot \sqrt{x}\right) - \frac{\frac{a}{b}}{3}\\ \end{array} \]

Derivation

  1. Split input into 3 regimes

  2. if (-.f64 y (/.f64 (*.f64 z t) 3)) < -1.83476341152617101e260

    1. Initial program 45.4

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

    2. Taylor expanded in z around 0 29.7

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

    3. Applied *-un-lft-identity_binary6429.7

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

    4. Applied cancel-sign-sub-inv_binary6429.7

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

    if -1.83476341152617101e260 < (-.f64 y (/.f64 (*.f64 z t) 3)) < 1.49780661057215335e299

    1. Initial program 13.5

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

    2. Applied *-un-lft-identity_binary6413.5

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

    3. Applied times-frac_binary6413.5

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

    4. Applied *-un-lft-identity_binary6413.5

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

    5. Applied prod-diff_binary6413.5

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

    6. Applied cos-sum_binary6411.3

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

    if 1.49780661057215335e299 < (-.f64 y (/.f64 (*.f64 z t) 3))

    1. Initial program 58.4

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

    2. Taylor expanded in z around 0 32.3

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

    3. Taylor expanded in y around 0 33.6

      \[\leadsto \color{blue}{2 \cdot \sqrt{x} - 0.3333333333333333 \cdot \frac{a}{b}} \]

    4. Simplified33.6

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

  4. Final simplification14.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;y - \frac{z \cdot t}{3} \leq -1.834763411526171 \cdot 10^{+260}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \cos y - \frac{a}{3 \cdot b}\\ \mathbf{elif}\;y - \frac{z \cdot t}{3} \leq 1.4978066105721533 \cdot 10^{+299}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos \left(\mathsf{fma}\left(1, y, -z \cdot \frac{t}{3}\right)\right) \cdot \cos \left(\mathsf{fma}\left(-\frac{t}{3}, z, z \cdot \frac{t}{3}\right)\right) - \sin \left(\mathsf{fma}\left(1, y, -z \cdot \frac{t}{3}\right)\right) \cdot \sin \left(\mathsf{fma}\left(-\frac{t}{3}, z, z \cdot \frac{t}{3}\right)\right)\right) - \frac{a}{3 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, \sqrt{x}, -0.3333333333333333 \cdot \frac{a}{b}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2021275 
(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))))