Average Error: 3.0 → 0.4
Time: 6.6s
Precision: binary64
\[[y, z, t] = \mathsf{sort}([y, z, t]) \[a, b] = \mathsf{sort}([a, b]) \\]
\[\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + \left(a \cdot 27\right) \cdot b \]
↓
\[\begin{array}{l} \mathbf{if}\;z \leq 1.7986634846681185 \cdot 10^{-18}:\\ \;\;\;\;\mathsf{fma}\left(y, \left(z \cdot t\right) \cdot -9, \mathsf{fma}\left(2, x, 27 \cdot \left(a \cdot b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot x - t \cdot \left(z \cdot \left(y \cdot 9\right)\right)\right) + b \cdot \left(27 \cdot a\right)\\ \end{array} \]
\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}\;z \leq 1.7986634846681185 \cdot 10^{-18}:\\
\;\;\;\;\mathsf{fma}\left(y, \left(z \cdot t\right) \cdot -9, \mathsf{fma}\left(2, x, 27 \cdot \left(a \cdot b\right)\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\left(2 \cdot x - t \cdot \left(z \cdot \left(y \cdot 9\right)\right)\right) + b \cdot \left(27 \cdot a\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 (<= z 1.7986634846681185e-18)
   (fma y (* (* z t) -9.0) (fma 2.0 x (* 27.0 (* a b))))
   (+ (- (* 2.0 x) (* t (* z (* y 9.0)))) (* b (* 27.0 a)))))
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 (z <= 1.7986634846681185e-18) {
		tmp = fma(y, ((z * t) * -9.0), fma(2.0, x, (27.0 * (a * b))));
	} else {
		tmp = ((2.0 * x) - (t * (z * (y * 9.0)))) + (b * (27.0 * a));
	}
	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
Herbie0.4
\[\begin{array}{l} \mathbf{if}\;y < 7.590524218811189 \cdot 10^{-161}:\\ \;\;\;\;\left(x \cdot 2 - \left(\left(y \cdot 9\right) \cdot z\right) \cdot t\right) + a \cdot \left(27 \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 2 - 9 \cdot \left(y \cdot \left(t \cdot z\right)\right)\right) + \left(a \cdot 27\right) \cdot b\\ \end{array} \]

Derivation

  1. Split input into 2 regimes

  2. if z < 1.79866348466811854e-18

    1. Initial program 3.4

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

    2. Simplified0.6

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

    3. Taylor expanded in y around 0 0.5

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

    4. Applied associate-*r*_binary640.6

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

    5. Taylor expanded in a around 0 0.5

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

    6. Simplified0.5

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

    if 1.79866348466811854e-18 < z

    1. Initial program 0.3

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.7986634846681185 \cdot 10^{-18}:\\ \;\;\;\;\mathsf{fma}\left(y, \left(z \cdot t\right) \cdot -9, \mathsf{fma}\left(2, x, 27 \cdot \left(a \cdot b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot x - t \cdot \left(z \cdot \left(y \cdot 9\right)\right)\right) + b \cdot \left(27 \cdot a\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022077 
(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)))