Average Error: 3.0 → 0.6
Time: 6.7s
Precision: binary64
\[[y, z, t] = \mathsf{sort}([y, z, t]) \[a, b] = \mathsf{sort}([a, b]) \\]
\[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]
↓
\[\begin{array}{l} \mathbf{if}\;y \cdot 9 \leq -2.2241331749014985 \cdot 10^{-49}:\\ \;\;\;\;\mathsf{fma}\left(y, \left(z \cdot t\right) \cdot -9, \mathsf{fma}\left(x, 2, a \cdot \left(27 \cdot b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(27 \cdot \left(a \cdot b\right) + x \cdot 2\right) - 9 \cdot \left(z \cdot \left(y \cdot t\right)\right)\\ \end{array} \]
\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}\;y \cdot 9 \leq -2.2241331749014985 \cdot 10^{-49}:\\
\;\;\;\;\mathsf{fma}\left(y, \left(z \cdot t\right) \cdot -9, \mathsf{fma}\left(x, 2, a \cdot \left(27 \cdot b\right)\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\left(27 \cdot \left(a \cdot b\right) + x \cdot 2\right) - 9 \cdot \left(z \cdot \left(y \cdot t\right)\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
 (if (<= (* y 9.0) -2.2241331749014985e-49)
   (fma y (* (* z t) -9.0) (fma x 2.0 (* a (* 27.0 b))))
   (- (+ (* 27.0 (* a b)) (* x 2.0)) (* 9.0 (* z (* y t))))))
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 tmp;
	if ((y * 9.0) <= -2.2241331749014985e-49) {
		tmp = fma(y, ((z * t) * -9.0), fma(x, 2.0, (a * (27.0 * b))));
	} else {
		tmp = ((27.0 * (a * b)) + (x * 2.0)) - (9.0 * (z * (y * t)));
	}
	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

Original3.0
Target3.4
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 y 9) < -2.2241331749014985e-49

    1. Initial program 4.3

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

    2. Simplified0.8

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

    3. Applied associate-*l*_binary640.8

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

    if -2.2241331749014985e-49 < (*.f64 y 9)

    1. Initial program 0.9

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

    2. Simplified6.2

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

    3. Taylor expanded in y around 0 6.0

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

    4. Applied associate-*r*_binary640.2

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

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot 9 \leq -2.2241331749014985 \cdot 10^{-49}:\\ \;\;\;\;\mathsf{fma}\left(y, \left(z \cdot t\right) \cdot -9, \mathsf{fma}\left(x, 2, a \cdot \left(27 \cdot b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(27 \cdot \left(a \cdot b\right) + x \cdot 2\right) - 9 \cdot \left(z \cdot \left(y \cdot t\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022096 
(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)))