Average Error: 20.8 → 15.2
Time: 23.1s
Precision: binary64
\[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3} \]
↓
\[\begin{array}{l} t_1 := 2 \cdot \sqrt{x}\\ t_2 := t_1 \cdot \cos y - \frac{\frac{a}{b}}{3}\\ t_3 := z \cdot \frac{t}{3}\\ t_4 := \mathsf{fma}\left(1, y, -t_3\right)\\ t_5 := y - \frac{z \cdot t}{3}\\ t_6 := \mathsf{fma}\left(-\frac{t}{3}, z, t_3\right)\\ \mathbf{if}\;t_5 \leq -2.980195530851706 \cdot 10^{+190}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_5 \leq 3.298944504388582 \cdot 10^{+244}:\\ \;\;\;\;t_1 \cdot \left(\cos t_4 \cdot \cos t_6 - \sin t_4 \cdot \sin t_6\right) - \frac{a}{3 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]
\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 := t_1 \cdot \cos y - \frac{\frac{a}{b}}{3}\\
t_3 := z \cdot \frac{t}{3}\\
t_4 := \mathsf{fma}\left(1, y, -t_3\right)\\
t_5 := y - \frac{z \cdot t}{3}\\
t_6 := \mathsf{fma}\left(-\frac{t}{3}, z, t_3\right)\\
\mathbf{if}\;t_5 \leq -2.980195530851706 \cdot 10^{+190}:\\
\;\;\;\;t_2\\

\mathbf{elif}\;t_5 \leq 3.298944504388582 \cdot 10^{+244}:\\
\;\;\;\;t_1 \cdot \left(\cos t_4 \cdot \cos t_6 - \sin t_4 \cdot \sin t_6\right) - \frac{a}{3 \cdot b}\\

\mathbf{else}:\\
\;\;\;\;t_2\\


\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 (- (* t_1 (cos y)) (/ (/ a b) 3.0)))
        (t_3 (* z (/ t 3.0)))
        (t_4 (fma 1.0 y (- t_3)))
        (t_5 (- y (/ (* z t) 3.0)))
        (t_6 (fma (- (/ t 3.0)) z t_3)))
   (if (<= t_5 -2.980195530851706e+190)
     t_2
     (if (<= t_5 3.298944504388582e+244)
       (-
        (* t_1 (- (* (cos t_4) (cos t_6)) (* (sin t_4) (sin t_6))))
        (/ a (* 3.0 b)))
       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 = (t_1 * cos(y)) - ((a / b) / 3.0);
	double t_3 = z * (t / 3.0);
	double t_4 = fma(1.0, y, -t_3);
	double t_5 = y - ((z * t) / 3.0);
	double t_6 = fma(-(t / 3.0), z, t_3);
	double tmp;
	if (t_5 <= -2.980195530851706e+190) {
		tmp = t_2;
	} else if (t_5 <= 3.298944504388582e+244) {
		tmp = (t_1 * ((cos(t_4) * cos(t_6)) - (sin(t_4) * sin(t_6)))) - (a / (3.0 * b));
	} else {
		tmp = 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

Original20.8
Target18.6
Herbie15.2
\[\begin{array}{l} \mathbf{if}\;z < -1.3793337487235141 \cdot 10^{+129}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\frac{1}{y} - \frac{\frac{0.3333333333333333}{z}}{t}\right) - \frac{\frac{a}{3}}{b}\\ \mathbf{elif}\;z < 3.516290613555987 \cdot 10^{+106}:\\ \;\;\;\;\left(\sqrt{x} \cdot 2\right) \cdot \cos \left(y - \frac{t}{3} \cdot z\right) - \frac{\frac{a}{3}}{b}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(y - \frac{\frac{0.3333333333333333}{z}}{t}\right) \cdot \left(2 \cdot \sqrt{x}\right) - \frac{\frac{a}{b}}{3}\\ \end{array} \]

Derivation

  1. Split input into 2 regimes

  2. if (-.f64 y (/.f64 (*.f64 z t) 3)) < -2.9801955308517058e190 or 3.29894450438858178e244 < (-.f64 y (/.f64 (*.f64 z t) 3))

    1. Initial program 38.8

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

    2. Taylor expanded in z around 0 26.1

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

    3. Applied associate-/r*_binary6426.1

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

    if -2.9801955308517058e190 < (-.f64 y (/.f64 (*.f64 z t) 3)) < 3.29894450438858178e244

    1. Initial program 12.4

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

    2. Applied *-un-lft-identity_binary6412.4

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

    3. Applied times-frac_binary6412.3

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

    4. Applied *-un-lft-identity_binary6412.3

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

    5. Applied prod-diff_binary6412.3

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

    6. Applied cos-sum_binary6410.2

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

  4. Final simplification15.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;y - \frac{z \cdot t}{3} \leq -2.980195530851706 \cdot 10^{+190}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \cos y - \frac{\frac{a}{b}}{3}\\ \mathbf{elif}\;y - \frac{z \cdot t}{3} \leq 3.298944504388582 \cdot 10^{+244}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos \left(\mathsf{fma}\left(1, y, -z \cdot \frac{t}{3}\right)\right) \cdot \cos \left(\mathsf{fma}\left(-\frac{t}{3}, z, z \cdot \frac{t}{3}\right)\right) - \sin \left(\mathsf{fma}\left(1, y, -z \cdot \frac{t}{3}\right)\right) \cdot \sin \left(\mathsf{fma}\left(-\frac{t}{3}, z, z \cdot \frac{t}{3}\right)\right)\right) - \frac{a}{3 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \cos y - \frac{\frac{a}{b}}{3}\\ \end{array} \]

Reproduce

herbie shell --seed 2022125 
(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))))