Average Error: 2.1 → 2.5
Time: 4.4s
Precision: binary64
\[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
↓
\[\begin{array}{l} \mathbf{if}\;z \leq 2.204489919394036 \cdot 10^{-232}:\\ \;\;\;\;\mathsf{fma}\left(y, z, \mathsf{fma}\left(a, z \cdot b, a \cdot t + x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x + \mathsf{fma}\left(a, t, z \cdot y\right)\right) + b \cdot \left(z \cdot a\right)\\ \end{array} \]
\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b
↓
\begin{array}{l}
\mathbf{if}\;z \leq 2.204489919394036 \cdot 10^{-232}:\\
\;\;\;\;\mathsf{fma}\left(y, z, \mathsf{fma}\left(a, z \cdot b, a \cdot t + x\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\left(x + \mathsf{fma}\left(a, t, z \cdot y\right)\right) + b \cdot \left(z \cdot a\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 (<= z 2.204489919394036e-232)
   (fma y z (fma a (* z b) (+ (* a t) x)))
   (+ (+ x (fma a t (* z y))) (* b (* z a)))))
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 (z <= 2.204489919394036e-232) {
		tmp = fma(y, z, fma(a, (z * b), ((a * t) + x)));
	} else {
		tmp = (x + fma(a, t, (z * y))) + (b * (z * 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

Original2.1
Target0.4
Herbie2.5
\[\begin{array}{l} \mathbf{if}\;z < -11820553527347888000:\\ \;\;\;\;z \cdot \left(b \cdot a + y\right) + \left(x + t \cdot a\right)\\ \mathbf{elif}\;z < 4.7589743188364287 \cdot 10^{-122}:\\ \;\;\;\;\left(b \cdot z + t\right) \cdot a + \left(z \cdot y + x\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(b \cdot a + y\right) + \left(x + t \cdot a\right)\\ \end{array} \]

Derivation

  1. Split input into 2 regimes

  2. if z < 2.204489919394036e-232

    1. Initial program 1.7

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

    2. Simplified2.4

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

    3. Taylor expanded in a around 0 2.4

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

    4. Applied fma-def_binary642.4

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

    if 2.204489919394036e-232 < z

    1. Initial program 2.6

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

    2. Applied associate-+l+_binary642.6

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

    3. Simplified2.6

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

  4. Final simplification2.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.204489919394036 \cdot 10^{-232}:\\ \;\;\;\;\mathsf{fma}\left(y, z, \mathsf{fma}\left(a, z \cdot b, a \cdot t + x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x + \mathsf{fma}\left(a, t, z \cdot y\right)\right) + b \cdot \left(z \cdot a\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2021313 
(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)))