Average Error: 2.0 → 0.5
Time: 3.5s
Precision: 64
\[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -3.31021399312617193 \cdot 10^{133} \lor \neg \left(b \le 7.62927591726979267 \cdot 10^{-75}\right):\\ \;\;\;\;\mathsf{fma}\left(z, y, x\right) + \mathsf{fma}\left(t, a, \left(a \cdot z\right) \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;x + \mathsf{fma}\left(z, y, \mathsf{fma}\left(b, z, t\right) \cdot a\right)\\ \end{array}\]
\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b
↓
\begin{array}{l}
\mathbf{if}\;b \le -3.31021399312617193 \cdot 10^{133} \lor \neg \left(b \le 7.62927591726979267 \cdot 10^{-75}\right):\\
\;\;\;\;\mathsf{fma}\left(z, y, x\right) + \mathsf{fma}\left(t, a, \left(a \cdot z\right) \cdot b\right)\\

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

\end{array}
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 VAR;
	if (((b <= -3.310213993126172e+133) || !(b <= 7.629275917269793e-75))) {
		VAR = (fma(z, y, x) + fma(t, a, ((a * z) * b)));
	} else {
		VAR = (x + fma(z, y, (fma(b, z, t) * a)));
	}
	return VAR;
}

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
Target0.3
Herbie0.5
\[\begin{array}{l} \mathbf{if}\;z \lt -11820553527347888000:\\ \;\;\;\;z \cdot \left(b \cdot a + y\right) + \left(x + t \cdot a\right)\\ \mathbf{elif}\;z \lt 4.75897431883642871 \cdot 10^{-122}:\\ \;\;\;\;\left(b \cdot z + t\right) \cdot a + \left(z \cdot y + x\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(b \cdot a + y\right) + \left(x + t \cdot a\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if b < -3.310213993126172e+133 or 7.629275917269793e-75 < b

    1. Initial program 0.7

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b\]
    2. Using strategy rm

    3. Applied associate-+l+0.7

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

    4. Simplified0.7

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

    5. Taylor expanded around inf 0.7

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

    6. Simplified0.7

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

    if -3.310213993126172e+133 < b < 7.629275917269793e-75

    1. Initial program 2.9

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b\]
    2. Using strategy rm

    3. Applied associate-+l+2.9

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

    4. Simplified2.9

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

    6. Applied associate-+l+2.9

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

    7. Simplified0.4

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -3.31021399312617193 \cdot 10^{133} \lor \neg \left(b \le 7.62927591726979267 \cdot 10^{-75}\right):\\ \;\;\;\;\mathsf{fma}\left(z, y, x\right) + \mathsf{fma}\left(t, a, \left(a \cdot z\right) \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;x + \mathsf{fma}\left(z, y, \mathsf{fma}\left(b, z, t\right) \cdot a\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020075 +o rules:numerics
(FPCore (x y z t a b)
  :name "Graphics.Rasterific.CubicBezier:cachedBezierAt from Rasterific-0.6.1"
  :precision binary64

  :herbie-target
  (if (< z -11820553527347888000) (+ (* 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)))