Average Error: 2.3 → 0.4
Time: 11.1s
Precision: binary64
\[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b\]
↓
\[\begin{array}{l} \mathbf{if}\;z \leq -1.2211996473497586 \cdot 10^{-54} \lor \neg \left(z \leq 6.70947441644599 \cdot 10^{+110}\right):\\ \;\;\;\;\left(x + z \cdot y\right) + \left(t \cdot a + z \cdot \left(a \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(z \cdot b\right) + \left(x + \left(z \cdot y + t \cdot a\right)\right)\\ \end{array}\]
\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b
↓
\begin{array}{l}
\mathbf{if}\;z \leq -1.2211996473497586 \cdot 10^{-54} \lor \neg \left(z \leq 6.70947441644599 \cdot 10^{+110}\right):\\
\;\;\;\;\left(x + z \cdot y\right) + \left(t \cdot a + z \cdot \left(a \cdot b\right)\right)\\

\mathbf{else}:\\
\;\;\;\;a \cdot \left(z \cdot b\right) + \left(x + \left(z \cdot y + t \cdot a\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 (<= z -1.2211996473497586e-54) (not (<= z 6.70947441644599e+110)))
   (+ (+ x (* z y)) (+ (* t a) (* z (* a b))))
   (+ (* a (* z b)) (+ x (+ (* z y) (* t 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 <= -1.2211996473497586e-54) || !(z <= 6.70947441644599e+110)) {
		tmp = (x + (z * y)) + ((t * a) + (z * (a * b)));
	} else {
		tmp = (a * (z * b)) + (x + ((z * y) + (t * 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

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original2.3
Target0.3
Herbie0.4
\[\begin{array}{l} \mathbf{if}\;z < -1.1820553527347888 \cdot 10^{+19}:\\ \;\;\;\;z \cdot \left(b \cdot a + y\right) + \left(x + t \cdot a\right)\\ \mathbf{elif}\;z < 4.7589743188364287 \cdot 10^{-122}:\\ \;\;\;\;\left(b \cdot z + t\right) \cdot a + \left(z \cdot y + x\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(b \cdot a + y\right) + \left(x + t \cdot a\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -1.22119964734975857e-54 or 6.7094744164459896e110 < z

    1. Initial program 5.5

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

    2. Simplified6.5

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

    3. Taylor expanded around 0 6.5

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

    4. Simplified0.3

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

    if -1.22119964734975857e-54 < z < 6.7094744164459896e110

    1. Initial program 0.7

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

    2. Simplified0.5

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

    3. Taylor expanded around 0 0.5

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.2211996473497586 \cdot 10^{-54} \lor \neg \left(z \leq 6.70947441644599 \cdot 10^{+110}\right):\\ \;\;\;\;\left(x + z \cdot y\right) + \left(t \cdot a + z \cdot \left(a \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(z \cdot b\right) + \left(x + \left(z \cdot y + t \cdot a\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2021176 
(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)))