Average Error: 1.9 → 0.2
Time: 6.7s
Precision: binary64
\[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
↓
\[\begin{array}{l} \mathbf{if}\;a \leq -5.4202676746054004 \cdot 10^{+26}:\\ \;\;\;\;\mathsf{fma}\left(y, z, a \cdot \left(z \cdot b\right) + \left(a \cdot t + x\right)\right)\\ \mathbf{elif}\;a \leq 6.248196026292757 \cdot 10^{+70}:\\ \;\;\;\;\mathsf{fma}\left(a, t, x + z \cdot \mathsf{fma}\left(a, b, y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, z, \mathsf{fma}\left(a, \mathsf{fma}\left(z, b, t\right), x\right)\right)\\ \end{array} \]
\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 -5.4202676746054004 \cdot 10^{+26}:\\
\;\;\;\;\mathsf{fma}\left(y, z, a \cdot \left(z \cdot b\right) + \left(a \cdot t + x\right)\right)\\

\mathbf{elif}\;a \leq 6.248196026292757 \cdot 10^{+70}:\\
\;\;\;\;\mathsf{fma}\left(a, t, x + z \cdot \mathsf{fma}\left(a, b, y\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(y, z, \mathsf{fma}\left(a, \mathsf{fma}\left(z, b, t\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 (<= a -5.4202676746054004e+26)
   (fma y z (+ (* a (* z b)) (+ (* a t) x)))
   (if (<= a 6.248196026292757e+70)
     (fma a t (+ x (* z (fma a b y))))
     (fma y z (fma a (fma z b t) 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 <= -5.4202676746054004e+26) {
		tmp = fma(y, z, ((a * (z * b)) + ((a * t) + x)));
	} else if (a <= 6.248196026292757e+70) {
		tmp = fma(a, t, (x + (z * fma(a, b, y))));
	} else {
		tmp = fma(y, z, fma(a, fma(z, b, t), 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

Original1.9
Target0.3
Herbie0.2
\[\begin{array}{l} \mathbf{if}\;z < -11820553527347888000:\\ \;\;\;\;z \cdot \left(b \cdot a + y\right) + \left(x + t \cdot a\right)\\ \mathbf{elif}\;z < 4.7589743188364287 \cdot 10^{-122}:\\ \;\;\;\;\left(b \cdot z + t\right) \cdot a + \left(z \cdot y + x\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(b \cdot a + y\right) + \left(x + t \cdot a\right)\\ \end{array} \]

Derivation

  1. Split input into 3 regimes

  2. if a < -5.4202676746054004e26

    1. Initial program 5.6

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

    2. Simplified0.1

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

    3. Taylor expanded in a around 0 0.1

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

    if -5.4202676746054004e26 < a < 6.2481960262927567e70

    1. Initial program 0.4

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

    2. Simplified3.2

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

    3. Taylor expanded in y around 0 3.2

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

    4. Simplified0.2

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

    5. Applied fma-udef_binary640.2

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

    if 6.2481960262927567e70 < a

    1. Initial program 5.8

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

    2. Simplified0.1

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

    3. Applied *-un-lft-identity_binary640.1

      \[\leadsto \mathsf{fma}\left(y, z, \color{blue}{1 \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(z, b, t\right), x\right)}\right) \]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -5.4202676746054004 \cdot 10^{+26}:\\ \;\;\;\;\mathsf{fma}\left(y, z, a \cdot \left(z \cdot b\right) + \left(a \cdot t + x\right)\right)\\ \mathbf{elif}\;a \leq 6.248196026292757 \cdot 10^{+70}:\\ \;\;\;\;\mathsf{fma}\left(a, t, x + z \cdot \mathsf{fma}\left(a, b, y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, z, \mathsf{fma}\left(a, \mathsf{fma}\left(z, b, t\right), x\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2021280 
(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)))