Average Error: 3.6 → 2.2
Time: 4.5s
Precision: 64
\[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le 2.02679901355823825 \cdot 10^{-64}:\\ \;\;\;\;\mathsf{fma}\left(a, 27 \cdot b, x \cdot 2 - y \cdot \left(\left(9 \cdot z\right) \cdot t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b\\ \end{array}\]
\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b
↓
\begin{array}{l}
\mathbf{if}\;t \le 2.02679901355823825 \cdot 10^{-64}:\\
\;\;\;\;\mathsf{fma}\left(a, 27 \cdot b, x \cdot 2 - y \cdot \left(\left(9 \cdot z\right) \cdot t\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b\\

\end{array}
double code(double x, double y, double z, double t, double a, double b) {
	return ((double) (((double) (((double) (x * 2.0)) - ((double) (((double) (((double) (y * 9.0)) * z)) * t)))) + ((double) (((double) (a * 27.0)) * b))));
}
↓
double code(double x, double y, double z, double t, double a, double b) {
	double VAR;
	if ((t <= 2.0267990135582383e-64)) {
		VAR = ((double) fma(a, ((double) (27.0 * b)), ((double) (((double) (x * 2.0)) - ((double) (y * ((double) (((double) (9.0 * z)) * t))))))));
	} else {
		VAR = ((double) (((double) (((double) (x * 2.0)) - ((double) (((double) (((double) (y * 9.0)) * z)) * t)))) + ((double) (((double) (a * 27.0)) * b))));
	}
	return VAR;
}

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

Original3.6
Target2.7
Herbie2.2
\[\begin{array}{l} \mathbf{if}\;y \lt 7.590524218811189 \cdot 10^{-161}:\\ \;\;\;\;\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + a \cdot \left(27 \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 2 - 9 \cdot \left(y \cdot \left(t \cdot z\right)\right)\right) + \left(a \cdot 27\right) \cdot b\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if t < 2.0267990135582383e-64

    1. Initial program 4.6

      \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b\]

    2. Simplified4.6

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, 27 \cdot b, x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right)}\]
    3. Using strategy rm

    4. Applied associate-*l*2.7

      \[\leadsto \mathsf{fma}\left(a, 27 \cdot b, x \cdot 2 - \color{blue}{\left(y \cdot 9\right) \cdot \left(z \cdot t\right)}\right)\]
    5. Using strategy rm

    6. Applied associate-*l*2.6

      \[\leadsto \mathsf{fma}\left(a, 27 \cdot b, x \cdot 2 - \color{blue}{y \cdot \left(9 \cdot \left(z \cdot t\right)\right)}\right)\]
    7. Using strategy rm

    8. Applied associate-*r*2.6

      \[\leadsto \mathsf{fma}\left(a, 27 \cdot b, x \cdot 2 - y \cdot \color{blue}{\left(\left(9 \cdot z\right) \cdot t\right)}\right)\]

    if 2.0267990135582383e-64 < t

    1. Initial program 1.0

      \[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b\]
  3. Recombined 2 regimes into one program.

  4. Final simplification2.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le 2.02679901355823825 \cdot 10^{-64}:\\ \;\;\;\;\mathsf{fma}\left(a, 27 \cdot b, x \cdot 2 - y \cdot \left(\left(9 \cdot z\right) \cdot t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b\\ \end{array}\]

Reproduce

herbie shell --seed 2020123 +o rules:numerics
(FPCore (x y z t a b)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, A"
  :precision binary64

  :herbie-target
  (if (< y 7.590524218811189e-161) (+ (- (* x 2) (* (* (* y 9) z) t)) (* a (* 27 b))) (+ (- (* x 2) (* 9 (* y (* t z)))) (* (* a 27) b)))

  (+ (- (* x 2) (* (* (* y 9) z) t)) (* (* a 27) b)))