Average Error: 2.1 → 0.3
Time: 4.0s
Precision: 64
\[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -770672845832513782000 \lor \neg \left(z \le 2.8821032236672868 \cdot 10^{-106}\right):\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(a, b, y\right), z, \mathsf{fma}\left(a, t, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + \mathsf{fma}\left(a, z \cdot b, \mathsf{fma}\left(z, y, a \cdot t\right)\right)\\ \end{array}\]
\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b
↓
\begin{array}{l}
\mathbf{if}\;z \le -770672845832513782000 \lor \neg \left(z \le 2.8821032236672868 \cdot 10^{-106}\right):\\
\;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(a, b, y\right), z, \mathsf{fma}\left(a, t, x\right)\right)\\

\mathbf{else}:\\
\;\;\;\;x + \mathsf{fma}\left(a, z \cdot b, \mathsf{fma}\left(z, y, a \cdot t\right)\right)\\

\end{array}
double code(double x, double y, double z, double t, double a, double b) {
	return (((x + (y * z)) + (t * a)) + ((a * z) * b));
}
↓
double code(double x, double y, double z, double t, double a, double b) {
	double temp;
	if (((z <= -7.706728458325138e+20) || !(z <= 2.882103223667287e-106))) {
		temp = fma(fma(a, b, y), z, fma(a, t, x));
	} else {
		temp = (x + fma(a, (z * b), fma(z, y, (a * t))));
	}
	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

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original2.1
Target0.4
Herbie0.3
\[\begin{array}{l} \mathbf{if}\;z \lt -11820553527347888000:\\ \;\;\;\;z \cdot \left(b \cdot a + y\right) + \left(x + t \cdot a\right)\\ \mathbf{elif}\;z \lt 4.75897431883642871 \cdot 10^{-122}:\\ \;\;\;\;\left(b \cdot z + t\right) \cdot a + \left(z \cdot y + x\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(b \cdot a + y\right) + \left(x + t \cdot a\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -7.706728458325138e+20 or 2.882103223667287e-106 < z

    1. Initial program 4.1

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b\]

    2. Simplified0.6

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

    if -7.706728458325138e+20 < z < 2.882103223667287e-106

    1. Initial program 0.4

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b\]
    2. Using strategy rm

    3. Applied associate-+l+0.4

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

    4. Applied associate-+l+0.4

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

    5. Simplified0.1

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -770672845832513782000 \lor \neg \left(z \le 2.8821032236672868 \cdot 10^{-106}\right):\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(a, b, y\right), z, \mathsf{fma}\left(a, t, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + \mathsf{fma}\left(a, z \cdot b, \mathsf{fma}\left(z, y, a \cdot t\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020066 +o rules:numerics
(FPCore (x y z t a b)
  :name "Graphics.Rasterific.CubicBezier:cachedBezierAt from Rasterific-0.6.1"
  :precision binary64

  :herbie-target
  (if (< z -11820553527347888000) (+ (* z (+ (* b a) y)) (+ x (* t a))) (if (< z 4.7589743188364287e-122) (+ (* (+ (* b z) t) a) (+ (* z y) x)) (+ (* z (+ (* b a) y)) (+ x (* t a)))))

  (+ (+ (+ x (* y z)) (* t a)) (* (* a z) b)))