Average Error: 2.1 → 0.5
Time: 4.2s
Precision: binary64
\[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
↓
\[\begin{array}{l} \mathbf{if}\;a \leq -1.8881264951566855 \cdot 10^{+48} \lor \neg \left(a \leq 1.845139882369355 \cdot 10^{-143}\right):\\ \;\;\;\;\mathsf{fma}\left(y, z, a \cdot \mathsf{fma}\left(b, z, t\right) + x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, t, \mathsf{fma}\left(z, \mathsf{fma}\left(a, b, y\right), x\right)\right)\\ \end{array} \]
\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b
↓
\begin{array}{l}
\mathbf{if}\;a \leq -1.8881264951566855 \cdot 10^{+48} \lor \neg \left(a \leq 1.845139882369355 \cdot 10^{-143}\right):\\
\;\;\;\;\mathsf{fma}\left(y, z, a \cdot \mathsf{fma}\left(b, z, t\right) + x\right)\\

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


\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
 (if (or (<= a -1.8881264951566855e+48) (not (<= a 1.845139882369355e-143)))
   (fma y z (+ (* a (fma b z t)) x))
   (fma a t (fma z (fma a b y) x))))
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 tmp;
	if ((a <= -1.8881264951566855e+48) || !(a <= 1.845139882369355e-143)) {
		tmp = fma(y, z, ((a * fma(b, z, t)) + x));
	} else {
		tmp = fma(a, t, fma(z, fma(a, b, y), x));
	}
	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
Herbie0.5
\[\begin{array}{l} \mathbf{if}\;z < -11820553527347888000:\\ \;\;\;\;z \cdot \left(b \cdot a + y\right) + \left(x + t \cdot a\right)\\ \mathbf{elif}\;z < 4.7589743188364287 \cdot 10^{-122}:\\ \;\;\;\;\left(b \cdot z + t\right) \cdot a + \left(z \cdot y + x\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(b \cdot a + y\right) + \left(x + t \cdot a\right)\\ \end{array} \]

Derivation

  1. Split input into 2 regimes

  2. if a < -1.88812649515668551e48 or 1.84513988236935508e-143 < a

    1. Initial program 4.0

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

    2. Simplified0.9

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

    3. Applied fma-udef_binary640.9

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

    4. Simplified0.9

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

    if -1.88812649515668551e48 < a < 1.84513988236935508e-143

    1. Initial program 0.4

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

    2. Simplified4.5

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

    3. Taylor expanded in y around 0 4.5

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

    4. Simplified0.2

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.8881264951566855 \cdot 10^{+48} \lor \neg \left(a \leq 1.845139882369355 \cdot 10^{-143}\right):\\ \;\;\;\;\mathsf{fma}\left(y, z, a \cdot \mathsf{fma}\left(b, z, t\right) + x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, t, \mathsf{fma}\left(z, \mathsf{fma}\left(a, b, y\right), x\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2021307 
(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)))