Average Error: 2.1 → 0.6
Time: 4.9s
Precision: binary64
\[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
↓
\[\begin{array}{l} t_1 := \mathsf{fma}\left(y, z, \mathsf{fma}\left(a, \mathsf{fma}\left(z, b, t\right), x\right)\right)\\ \mathbf{if}\;a \leq -1.1990245494804759 \cdot 10^{-54}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 1.6657956137560575 \cdot 10^{-137}:\\ \;\;\;\;\left(\left(x + y \cdot z\right) + a \cdot t\right) + b \cdot \left(a \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b
↓
\begin{array}{l}
t_1 := \mathsf{fma}\left(y, z, \mathsf{fma}\left(a, \mathsf{fma}\left(z, b, t\right), x\right)\right)\\
\mathbf{if}\;a \leq -1.1990245494804759 \cdot 10^{-54}:\\
\;\;\;\;t_1\\

\mathbf{elif}\;a \leq 1.6657956137560575 \cdot 10^{-137}:\\
\;\;\;\;\left(\left(x + y \cdot z\right) + a \cdot t\right) + b \cdot \left(a \cdot z\right)\\

\mathbf{else}:\\
\;\;\;\;t_1\\


\end{array}

(FPCore (x y z t a b)
 :precision binary64
 (+ (+ (+ x (* y z)) (* t a)) (* (* a z) b)))
↓
(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma y z (fma a (fma z b t) x))))
   (if (<= a -1.1990245494804759e-54)
     t_1
     (if (<= a 1.6657956137560575e-137)
       (+ (+ (+ x (* y z)) (* a t)) (* b (* a z)))
       t_1))))
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 t_1 = fma(y, z, fma(a, fma(z, b, t), x));
	double tmp;
	if (a <= -1.1990245494804759e-54) {
		tmp = t_1;
	} else if (a <= 1.6657956137560575e-137) {
		tmp = ((x + (y * z)) + (a * t)) + (b * (a * z));
	} else {
		tmp = t_1;
	}
	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

Original2.1
Target0.4
Herbie0.6
\[\begin{array}{l} \mathbf{if}\;z < -11820553527347888000:\\ \;\;\;\;z \cdot \left(b \cdot a + y\right) + \left(x + t \cdot a\right)\\ \mathbf{elif}\;z < 4.7589743188364287 \cdot 10^{-122}:\\ \;\;\;\;\left(b \cdot z + t\right) \cdot a + \left(z \cdot y + x\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(b \cdot a + y\right) + \left(x + t \cdot a\right)\\ \end{array} \]

Derivation

  1. Split input into 2 regimes

  2. if a < -1.1990245494804759e-54 or 1.66579561375605755e-137 < a

    1. Initial program 3.4

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

    2. Simplified0.8

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

    if -1.1990245494804759e-54 < a < 1.66579561375605755e-137

    1. Initial program 0.4

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

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.1990245494804759 \cdot 10^{-54}:\\ \;\;\;\;\mathsf{fma}\left(y, z, \mathsf{fma}\left(a, \mathsf{fma}\left(z, b, t\right), x\right)\right)\\ \mathbf{elif}\;a \leq 1.6657956137560575 \cdot 10^{-137}:\\ \;\;\;\;\left(\left(x + y \cdot z\right) + a \cdot t\right) + b \cdot \left(a \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, z, \mathsf{fma}\left(a, \mathsf{fma}\left(z, b, t\right), x\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022076 
(FPCore (x y z t a b)
  :name "Graphics.Rasterific.CubicBezier:cachedBezierAt from Rasterific-0.6.1"
  :precision binary64

  :herbie-target
  (if (< z -11820553527347888000.0) (+ (* 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)))