Average Error: 3.8 → 1.1
Time: 3.8s
Precision: 64
\[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -3.79473196254615423 \cdot 10^{-41} \lor \neg \left(t \le 2.1735761525819559 \cdot 10^{-141}\right):\\ \;\;\;\;\mathsf{fma}\left(a, 27 \cdot b, x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, 2, 27 \cdot \left(a \cdot b\right) - \left(9 \cdot \left(t \cdot z\right)\right) \cdot y\right)\\ \end{array}\]
\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 -3.79473196254615423 \cdot 10^{-41} \lor \neg \left(t \le 2.1735761525819559 \cdot 10^{-141}\right):\\
\;\;\;\;\mathsf{fma}\left(a, 27 \cdot b, x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right)\\

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

\end{array}
double code(double x, double y, double z, double t, double a, double b) {
	return (((x * 2.0) - (((y * 9.0) * z) * t)) + ((a * 27.0) * b));
}
↓
double code(double x, double y, double z, double t, double a, double b) {
	double VAR;
	if (((t <= -3.794731962546154e-41) || !(t <= 2.173576152581956e-141))) {
		VAR = fma(a, (27.0 * b), ((x * 2.0) - (((y * 9.0) * z) * t)));
	} else {
		VAR = fma(x, 2.0, ((27.0 * (a * b)) - ((9.0 * (t * z)) * y)));
	}
	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

Original3.8
Target2.7
Herbie1.1
\[\begin{array}{l} \mathbf{if}\;y \lt 7.590524218811189 \cdot 10^{-161}:\\ \;\;\;\;\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + a \cdot \left(27 \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 2 - 9 \cdot \left(y \cdot \left(t \cdot z\right)\right)\right) + \left(a \cdot 27\right) \cdot b\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if t < -3.794731962546154e-41 or 2.173576152581956e-141 < t

    1. Initial program 1.6

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

    2. Simplified1.5

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

    if -3.794731962546154e-41 < t < 2.173576152581956e-141

    1. Initial program 6.9

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

    3. Applied *-commutative6.9

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

    4. Applied associate-*l*6.8

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

    5. Applied associate-*l*6.8

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

    6. Simplified6.8

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

    8. Applied *-commutative6.8

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

    9. Applied fma-neg6.8

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

    10. Simplified6.8

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

    11. Taylor expanded around inf 6.7

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

    12. Simplified6.7

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

    14. Applied associate-*r*0.5

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

    15. Applied associate-*r*0.5

      \[\leadsto \mathsf{fma}\left(x, 2, 27 \cdot \left(a \cdot b\right) - \color{blue}{\left(9 \cdot \left(t \cdot z\right)\right) \cdot y}\right)\]
  3. Recombined 2 regimes into one program.

  4. Final simplification1.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -3.79473196254615423 \cdot 10^{-41} \lor \neg \left(t \le 2.1735761525819559 \cdot 10^{-141}\right):\\ \;\;\;\;\mathsf{fma}\left(a, 27 \cdot b, x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, 2, 27 \cdot \left(a \cdot b\right) - \left(9 \cdot \left(t \cdot z\right)\right) \cdot y\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020078 +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)))