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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -1.2938895012169612 \cdot 10^{-14}:\\ \;\;\;\;\mathsf{fma}\left(t, a, \mathsf{fma}\left(z, \mathsf{fma}\left(a, b, y\right), x\right)\right)\\ \mathbf{elif}\;z \le 4.1149667559960053 \cdot 10^{-44}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(z, b, t\right) + \left(z \cdot y + x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, a, \mathsf{fma}\left(z, \mathsf{fma}\left(a, b, y\right), x\right)\right)\\ \end{array}\]

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

↓

\begin{array}{l}
\mathbf{if}\;z \le -1.2938895012169612 \cdot 10^{-14}:\\
\;\;\;\;\mathsf{fma}\left(t, a, \mathsf{fma}\left(z, \mathsf{fma}\left(a, b, y\right), x\right)\right)\\

\mathbf{elif}\;z \le 4.1149667559960053 \cdot 10^{-44}:\\
\;\;\;\;a \cdot \mathsf{fma}\left(z, b, t\right) + \left(z \cdot y + x\right)\\

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

\end{array}

double f(double x, double y, double z, double t, double a, double b) {
        double r29403207 = x;
        double r29403208 = y;
        double r29403209 = z;
        double r29403210 = r29403208 * r29403209;
        double r29403211 = r29403207 + r29403210;
        double r29403212 = t;
        double r29403213 = a;
        double r29403214 = r29403212 * r29403213;
        double r29403215 = r29403211 + r29403214;
        double r29403216 = r29403213 * r29403209;
        double r29403217 = b;
        double r29403218 = r29403216 * r29403217;
        double r29403219 = r29403215 + r29403218;
        return r29403219;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r29403220 = z;
        double r29403221 = -1.2938895012169612e-14;
        bool r29403222 = r29403220 <= r29403221;
        double r29403223 = t;
        double r29403224 = a;
        double r29403225 = b;
        double r29403226 = y;
        double r29403227 = fma(r29403224, r29403225, r29403226);
        double r29403228 = x;
        double r29403229 = fma(r29403220, r29403227, r29403228);
        double r29403230 = fma(r29403223, r29403224, r29403229);
        double r29403231 = 4.1149667559960053e-44;
        bool r29403232 = r29403220 <= r29403231;
        double r29403233 = fma(r29403220, r29403225, r29403223);
        double r29403234 = r29403224 * r29403233;
        double r29403235 = r29403220 * r29403226;
        double r29403236 = r29403235 + r29403228;
        double r29403237 = r29403234 + r29403236;
        double r29403238 = r29403232 ? r29403237 : r29403230;
        double r29403239 = r29403222 ? r29403230 : r29403238;
        return r29403239;
}

Original	2.0
Target	0.4
Herbie	0.1

Derivation

Split input into 2 regimes
if z < -1.2938895012169612e-14 or 4.1149667559960053e-44 < z
1. Initial program 4.0
  \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b\]
2. Simplified0.2
  \[\leadsto \color{blue}{\mathsf{fma}\left(t, a, \mathsf{fma}\left(z, \mathsf{fma}\left(a, b, y\right), x\right)\right)}\]
if -1.2938895012169612e-14 < z < 4.1149667559960053e-44
1. Initial program 0.5
  \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b\]
2. Using strategy rm
3. Applied associate-+l+0.5
  \[\leadsto \color{blue}{\left(x + y \cdot z\right) + \left(t \cdot a + \left(a \cdot z\right) \cdot b\right)}\]
4. Simplified0.0
  \[\leadsto \left(x + y \cdot z\right) + \color{blue}{\mathsf{fma}\left(z, b, t\right) \cdot a}\]
Recombined 2 regimes into one program.
Final simplification0.1
\[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1.2938895012169612 \cdot 10^{-14}:\\ \;\;\;\;\mathsf{fma}\left(t, a, \mathsf{fma}\left(z, \mathsf{fma}\left(a, b, y\right), x\right)\right)\\ \mathbf{elif}\;z \le 4.1149667559960053 \cdot 10^{-44}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(z, b, t\right) + \left(z \cdot y + x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, a, \mathsf{fma}\left(z, \mathsf{fma}\left(a, b, y\right), x\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019165 +o rules:numerics
(FPCore (x y z t a b)
  :name "Graphics.Rasterific.CubicBezier:cachedBezierAt from Rasterific-0.6.1"

  :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)))

Error

Target

Derivation

`if z < -1.2938895012169612e-14 or 4.1149667559960053e-44 < z`

`if -1.2938895012169612e-14 < z < 4.1149667559960053e-44`

Reproduce