Average Error: 3.1 → 0.6
Time: 7.7s
Precision: binary64
\[[y, z, t] = \mathsf{sort}([y, z, t]) \[a, b] = \mathsf{sort}([a, b]) \\]
\[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]
↓
\[\begin{array}{l} t_1 := a \cdot \left(27 \cdot b\right)\\ t_2 := \left(y \cdot 9\right) \cdot z\\ \mathbf{if}\;t_2 \leq 4.5310325608367633 \cdot 10^{+285}:\\ \;\;\;\;\left(x \cdot 2 - t_2 \cdot t\right) + t_1\\ \mathbf{else}:\\ \;\;\;\;t_1 + \mathsf{fma}\left(x, 2, \left(z \cdot \left(y \cdot t\right)\right) \cdot -9\right)\\ \end{array} \]
\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}
t_1 := a \cdot \left(27 \cdot b\right)\\
t_2 := \left(y \cdot 9\right) \cdot z\\
\mathbf{if}\;t_2 \leq 4.5310325608367633 \cdot 10^{+285}:\\
\;\;\;\;\left(x \cdot 2 - t_2 \cdot t\right) + t_1\\

\mathbf{else}:\\
\;\;\;\;t_1 + \mathsf{fma}\left(x, 2, \left(z \cdot \left(y \cdot t\right)\right) \cdot -9\right)\\


\end{array}

(FPCore (x y z t a b)
 :precision binary64
 (+ (- (* x 2.0) (* (* (* y 9.0) z) t)) (* (* a 27.0) b)))
↓
(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* a (* 27.0 b))) (t_2 (* (* y 9.0) z)))
   (if (<= t_2 4.5310325608367633e+285)
     (+ (- (* x 2.0) (* t_2 t)) t_1)
     (+ t_1 (fma x 2.0 (* (* z (* y t)) -9.0))))))
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 t_1 = a * (27.0 * b);
	double t_2 = (y * 9.0) * z;
	double tmp;
	if (t_2 <= 4.5310325608367633e+285) {
		tmp = ((x * 2.0) - (t_2 * t)) + t_1;
	} else {
		tmp = t_1 + fma(x, 2.0, ((z * (y * t)) * -9.0));
	}
	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

Original3.1
Target3.5
Herbie0.6
\[\begin{array}{l} \mathbf{if}\;y < 7.590524218811189 \cdot 10^{-161}:\\ \;\;\;\;\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + a \cdot \left(27 \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 2 - 9 \cdot \left(y \cdot \left(t \cdot z\right)\right)\right) + \left(a \cdot 27\right) \cdot b\\ \end{array} \]

Derivation

  1. Split input into 2 regimes

  2. if (*.f64 (*.f64 y 9) z) < 4.5310325608367633e285

    1. Initial program 0.6

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

    2. Applied associate-*l*_binary640.6

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

    if 4.5310325608367633e285 < (*.f64 (*.f64 y 9) z)

    1. Initial program 52.7

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

    2. Applied associate-*l*_binary6452.7

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

    3. Applied fma-neg_binary6452.7

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

    4. Simplified0.3

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

    5. Applied associate-*r*_binary640.9

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

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(y \cdot 9\right) \cdot z \leq 4.5310325608367633 \cdot 10^{+285}:\\ \;\;\;\;\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + a \cdot \left(27 \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(27 \cdot b\right) + \mathsf{fma}\left(x, 2, \left(z \cdot \left(y \cdot t\right)\right) \cdot -9\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022125 
(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 2.0) (* (* (* y 9.0) z) t)) (* a (* 27.0 b))) (+ (- (* x 2.0) (* 9.0 (* y (* t z)))) (* (* a 27.0) b)))

  (+ (- (* x 2.0) (* (* (* y 9.0) z) t)) (* (* a 27.0) b)))