Average Error: 3.2 → 1.2
Time: 4.5s
Precision: 64
\[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -4.2584306516392116 \cdot 10^{70}:\\ \;\;\;\;\mathsf{fma}\left(2, x, \sqrt{27} \cdot \left(\sqrt{27} \cdot \left(a \cdot b\right)\right) - 9 \cdot \left(t \cdot \left(z \cdot y\right)\right)\right)\\ \mathbf{elif}\;t \le 4.10590766711501 \cdot 10^{-174}:\\ \;\;\;\;\mathsf{fma}\left(a, 27 \cdot b, x \cdot 2 - \left(y \cdot 9\right) \cdot \left(z \cdot t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b\\ \end{array}\]
\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 -4.2584306516392116 \cdot 10^{70}:\\
\;\;\;\;\mathsf{fma}\left(2, x, \sqrt{27} \cdot \left(\sqrt{27} \cdot \left(a \cdot b\right)\right) - 9 \cdot \left(t \cdot \left(z \cdot y\right)\right)\right)\\

\mathbf{elif}\;t \le 4.10590766711501 \cdot 10^{-174}:\\
\;\;\;\;\mathsf{fma}\left(a, 27 \cdot b, x \cdot 2 - \left(y \cdot 9\right) \cdot \left(z \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 (((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 <= -4.2584306516392116e+70)) {
		VAR = fma(2.0, x, ((sqrt(27.0) * (sqrt(27.0) * (a * b))) - (9.0 * (t * (z * y)))));
	} else {
		double VAR_1;
		if ((t <= 4.1059076671150105e-174)) {
			VAR_1 = fma(a, (27.0 * b), ((x * 2.0) - ((y * 9.0) * (z * t))));
		} else {
			VAR_1 = (((x * 2.0) - (((y * 9.0) * z) * t)) + ((a * 27.0) * b));
		}
		VAR = VAR_1;
	}
	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

Original3.2
Target2.3
Herbie1.2
\[\begin{array}{l} \mathbf{if}\;y \lt 7.590524218811189 \cdot 10^{-161}:\\ \;\;\;\;\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + a \cdot \left(27 \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 2 - 9 \cdot \left(y \cdot \left(t \cdot z\right)\right)\right) + \left(a \cdot 27\right) \cdot b\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if t < -4.2584306516392116e+70

    1. Initial program 0.8

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

    2. Simplified0.7

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

    3. Taylor expanded around inf 0.7

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

    4. Simplified0.7

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

    6. Applied add-sqr-sqrt0.7

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

    7. Applied associate-*l*0.8

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

    if -4.2584306516392116e+70 < t < 4.1059076671150105e-174

    1. Initial program 5.0

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

    2. Simplified5.1

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

    4. Applied associate-*l*0.7

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

    if 4.1059076671150105e-174 < t

    1. Initial program 1.9

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

  4. Final simplification1.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -4.2584306516392116 \cdot 10^{70}:\\ \;\;\;\;\mathsf{fma}\left(2, x, \sqrt{27} \cdot \left(\sqrt{27} \cdot \left(a \cdot b\right)\right) - 9 \cdot \left(t \cdot \left(z \cdot y\right)\right)\right)\\ \mathbf{elif}\;t \le 4.10590766711501 \cdot 10^{-174}:\\ \;\;\;\;\mathsf{fma}\left(a, 27 \cdot b, x \cdot 2 - \left(y \cdot 9\right) \cdot \left(z \cdot t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b\\ \end{array}\]

Reproduce

herbie shell --seed 2020092 +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)))