Average Error: 2.2 → 0.6
Time: 4.8s
Precision: 64
\[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.6772471652537959 \cdot 10^{138} \lor \neg \left(b \le 0.18959099764245008\right):\\ \;\;\;\;\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(a, b, y\right), z, \mathsf{fma}\left(a, t, x\right)\right)\\ \end{array}\]
\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b
↓
\begin{array}{l}
\mathbf{if}\;b \le -2.6772471652537959 \cdot 10^{138} \lor \neg \left(b \le 0.18959099764245008\right):\\
\;\;\;\;\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b\\

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

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r632063 = x;
        double r632064 = y;
        double r632065 = z;
        double r632066 = r632064 * r632065;
        double r632067 = r632063 + r632066;
        double r632068 = t;
        double r632069 = a;
        double r632070 = r632068 * r632069;
        double r632071 = r632067 + r632070;
        double r632072 = r632069 * r632065;
        double r632073 = b;
        double r632074 = r632072 * r632073;
        double r632075 = r632071 + r632074;
        return r632075;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r632076 = b;
        double r632077 = -2.677247165253796e+138;
        bool r632078 = r632076 <= r632077;
        double r632079 = 0.18959099764245008;
        bool r632080 = r632076 <= r632079;
        double r632081 = !r632080;
        bool r632082 = r632078 || r632081;
        double r632083 = x;
        double r632084 = y;
        double r632085 = z;
        double r632086 = r632084 * r632085;
        double r632087 = r632083 + r632086;
        double r632088 = t;
        double r632089 = a;
        double r632090 = r632088 * r632089;
        double r632091 = r632087 + r632090;
        double r632092 = r632089 * r632085;
        double r632093 = r632092 * r632076;
        double r632094 = r632091 + r632093;
        double r632095 = fma(r632089, r632076, r632084);
        double r632096 = fma(r632089, r632088, r632083);
        double r632097 = fma(r632095, r632085, r632096);
        double r632098 = r632082 ? r632094 : r632097;
        return r632098;
}

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

Original2.2
Target0.3
Herbie0.6
\[\begin{array}{l} \mathbf{if}\;z \lt -11820553527347888000:\\ \;\;\;\;z \cdot \left(b \cdot a + y\right) + \left(x + t \cdot a\right)\\ \mathbf{elif}\;z \lt 4.75897431883642871 \cdot 10^{-122}:\\ \;\;\;\;\left(b \cdot z + t\right) \cdot a + \left(z \cdot y + x\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(b \cdot a + y\right) + \left(x + t \cdot a\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if b < -2.677247165253796e+138 or 0.18959099764245008 < b

    1. Initial program 0.8

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

    if -2.677247165253796e+138 < b < 0.18959099764245008

    1. Initial program 2.9

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

    2. Simplified0.5

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

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.6772471652537959 \cdot 10^{138} \lor \neg \left(b \le 0.18959099764245008\right):\\ \;\;\;\;\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(a, b, y\right), z, \mathsf{fma}\left(a, t, x\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020036 +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)))