Average Error: 5.8 → 4.3
Time: 19.6s
Precision: 64
\[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -2.42866793610457989 \cdot 10^{-115}:\\ \;\;\;\;\mathsf{fma}\left(t, \left(\left(x \cdot 18\right) \cdot y\right) \cdot z - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, 27 \cdot \left(j \cdot k\right)\right)\right)\\ \mathbf{elif}\;t \le 1.642560278835955 \cdot 10^{-131}:\\ \;\;\;\;\mathsf{fma}\left(t, 0 - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \left(j \cdot 27\right) \cdot k\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, \left(18 \cdot \left(x \cdot y\right)\right) \cdot z - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \left(j \cdot 27\right) \cdot k\right)\right)\\ \end{array}\]
\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k
↓
\begin{array}{l}
\mathbf{if}\;t \le -2.42866793610457989 \cdot 10^{-115}:\\
\;\;\;\;\mathsf{fma}\left(t, \left(\left(x \cdot 18\right) \cdot y\right) \cdot z - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, 27 \cdot \left(j \cdot k\right)\right)\right)\\

\mathbf{elif}\;t \le 1.642560278835955 \cdot 10^{-131}:\\
\;\;\;\;\mathsf{fma}\left(t, 0 - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \left(j \cdot 27\right) \cdot k\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(t, \left(18 \cdot \left(x \cdot y\right)\right) \cdot z - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \left(j \cdot 27\right) \cdot k\right)\right)\\

\end{array}
double code(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
	return ((((((((x * 18.0) * y) * z) * t) - ((a * 4.0) * t)) + (b * c)) - ((x * 4.0) * i)) - ((j * 27.0) * k));
}
↓
double code(double x, double y, double z, double t, double a, double b, double c, double i, double j, double k) {
	double temp;
	if ((t <= -2.42866793610458e-115)) {
		temp = fma(t, ((((x * 18.0) * y) * z) - (a * 4.0)), ((b * c) - fma(x, (4.0 * i), (27.0 * (j * k)))));
	} else {
		double temp_1;
		if ((t <= 1.6425602788359552e-131)) {
			temp_1 = fma(t, (0.0 - (a * 4.0)), ((b * c) - fma(x, (4.0 * i), ((j * 27.0) * k))));
		} else {
			temp_1 = fma(t, (((18.0 * (x * y)) * z) - (a * 4.0)), ((b * c) - fma(x, (4.0 * i), ((j * 27.0) * k))));
		}
		temp = temp_1;
	}
	return temp;
}

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

Bits error versus c

Bits error versus i

Bits error versus j

Bits error versus k

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if t < -2.42866793610458e-115

    1. Initial program 2.7

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\]

    2. Simplified2.7

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, \left(\left(x \cdot 18\right) \cdot y\right) \cdot z - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \left(j \cdot 27\right) \cdot k\right)\right)}\]
    3. Using strategy rm

    4. Applied *-commutative2.7

      \[\leadsto \mathsf{fma}\left(t, \left(\left(x \cdot 18\right) \cdot y\right) \cdot z - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \color{blue}{\left(27 \cdot j\right)} \cdot k\right)\right)\]

    5. Applied associate-*l*2.7

      \[\leadsto \mathsf{fma}\left(t, \left(\left(x \cdot 18\right) \cdot y\right) \cdot z - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \color{blue}{27 \cdot \left(j \cdot k\right)}\right)\right)\]

    if -2.42866793610458e-115 < t < 1.6425602788359552e-131

    1. Initial program 10.0

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\]

    2. Simplified10.0

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, \left(\left(x \cdot 18\right) \cdot y\right) \cdot z - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \left(j \cdot 27\right) \cdot k\right)\right)}\]

    3. Taylor expanded around 0 6.3

      \[\leadsto \mathsf{fma}\left(t, \color{blue}{0} - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \left(j \cdot 27\right) \cdot k\right)\right)\]

    if 1.6425602788359552e-131 < t

    1. Initial program 3.4

      \[\left(\left(\left(\left(\left(\left(x \cdot 18\right) \cdot y\right) \cdot z\right) \cdot t - \left(a \cdot 4\right) \cdot t\right) + b \cdot c\right) - \left(x \cdot 4\right) \cdot i\right) - \left(j \cdot 27\right) \cdot k\]

    2. Simplified3.4

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, \left(\left(x \cdot 18\right) \cdot y\right) \cdot z - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \left(j \cdot 27\right) \cdot k\right)\right)}\]
    3. Using strategy rm

    4. Applied *-commutative3.4

      \[\leadsto \mathsf{fma}\left(t, \left(\color{blue}{\left(18 \cdot x\right)} \cdot y\right) \cdot z - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \left(j \cdot 27\right) \cdot k\right)\right)\]

    5. Applied associate-*l*3.3

      \[\leadsto \mathsf{fma}\left(t, \color{blue}{\left(18 \cdot \left(x \cdot y\right)\right)} \cdot z - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \left(j \cdot 27\right) \cdot k\right)\right)\]
  3. Recombined 3 regimes into one program.

  4. Final simplification4.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -2.42866793610457989 \cdot 10^{-115}:\\ \;\;\;\;\mathsf{fma}\left(t, \left(\left(x \cdot 18\right) \cdot y\right) \cdot z - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, 27 \cdot \left(j \cdot k\right)\right)\right)\\ \mathbf{elif}\;t \le 1.642560278835955 \cdot 10^{-131}:\\ \;\;\;\;\mathsf{fma}\left(t, 0 - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \left(j \cdot 27\right) \cdot k\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, \left(18 \cdot \left(x \cdot y\right)\right) \cdot z - a \cdot 4, b \cdot c - \mathsf{fma}\left(x, 4 \cdot i, \left(j \cdot 27\right) \cdot k\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020066 +o rules:numerics
(FPCore (x y z t a b c i j k)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1"
  :precision binary64
  (- (- (+ (- (* (* (* (* x 18) y) z) t) (* (* a 4) t)) (* b c)) (* (* x 4) i)) (* (* j 27) k)))