Average Error: 2.0 → 0.2
Time: 8.9s
Precision: binary64
\[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b\]
↓
\[\begin{array}{l} \mathbf{if}\;a \leq -8.47331770737554 \cdot 10^{-45} \lor \neg \left(a \leq 2.676632411961761 \cdot 10^{-75}\right):\\ \;\;\;\;x + \left(y \cdot z + a \cdot \left(t + z \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot t + \left(x + z \cdot \left(y + a \cdot b\right)\right)\\ \end{array}\]
\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 -8.47331770737554 \cdot 10^{-45} \lor \neg \left(a \leq 2.676632411961761 \cdot 10^{-75}\right):\\
\;\;\;\;x + \left(y \cdot z + a \cdot \left(t + z \cdot b\right)\right)\\

\mathbf{else}:\\
\;\;\;\;a \cdot t + \left(x + z \cdot \left(y + a \cdot b\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 -8.47331770737554e-45) (not (<= a 2.676632411961761e-75)))
   (+ x (+ (* y z) (* a (+ t (* z b)))))
   (+ (* a t) (+ x (* z (+ y (* a b)))))))
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 <= -8.47331770737554e-45) || !(a <= 2.676632411961761e-75)) {
		tmp = x + ((y * z) + (a * (t + (z * b))));
	} else {
		tmp = (a * t) + (x + (z * (y + (a * b))));
	}
	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

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original2.0
Target0.4
Herbie0.2
\[\begin{array}{l} \mathbf{if}\;z < -1.1820553527347888 \cdot 10^{+19}:\\ \;\;\;\;z \cdot \left(b \cdot a + y\right) + \left(x + t \cdot a\right)\\ \mathbf{elif}\;z < 4.7589743188364287 \cdot 10^{-122}:\\ \;\;\;\;\left(b \cdot z + t\right) \cdot a + \left(z \cdot y + x\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(b \cdot a + y\right) + \left(x + t \cdot a\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if a < -8.47331770737554031e-45 or 2.676632411961761e-75 < a

    1. Initial program 3.7

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

    2. Simplified0.3

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

    if -8.47331770737554031e-45 < a < 2.676632411961761e-75

    1. Initial program 0.4

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

    2. Taylor expanded around 0 4.6

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

    3. Simplified0.0

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -8.47331770737554 \cdot 10^{-45} \lor \neg \left(a \leq 2.676632411961761 \cdot 10^{-75}\right):\\ \;\;\;\;x + \left(y \cdot z + a \cdot \left(t + z \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot t + \left(x + z \cdot \left(y + a \cdot b\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2021044 
(FPCore (x y z t a b)
  :name "Graphics.Rasterific.CubicBezier:cachedBezierAt from Rasterific-0.6.1"
  :precision binary64

  :herbie-target
  (if (< z -1.1820553527347888e+19) (+ (* 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)))