Average Error: 0.1 → 0.1
Time: 19.7s
Precision: 64
\[x \cdot \sin y + z \cdot \cos y\]
↓
\[\mathsf{fma}\left(x, \sin y, z \cdot \cos y\right)\]
x \cdot \sin y + z \cdot \cos y
↓
\mathsf{fma}\left(x, \sin y, z \cdot \cos y\right)
double f(double x, double y, double z) {
        double r6179064 = x;
        double r6179065 = y;
        double r6179066 = sin(r6179065);
        double r6179067 = r6179064 * r6179066;
        double r6179068 = z;
        double r6179069 = cos(r6179065);
        double r6179070 = r6179068 * r6179069;
        double r6179071 = r6179067 + r6179070;
        return r6179071;
}
↓
double f(double x, double y, double z) {
        double r6179072 = x;
        double r6179073 = y;
        double r6179074 = sin(r6179073);
        double r6179075 = z;
        double r6179076 = cos(r6179073);
        double r6179077 = r6179075 * r6179076;
        double r6179078 = fma(r6179072, r6179074, r6179077);
        return r6179078;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

  1. Initial program 0.1

    \[x \cdot \sin y + z \cdot \cos y\]

  2. Simplified0.1

    \[\leadsto \color{blue}{\mathsf{fma}\left(\cos y, z, x \cdot \sin y\right)}\]

  3. Taylor expanded around inf 0.1

    \[\leadsto \color{blue}{z \cdot \cos y + x \cdot \sin y}\]

  4. Simplified0.1

    \[\leadsto \color{blue}{\mathsf{fma}\left(x, \sin y, z \cdot \cos y\right)}\]

  5. Final simplification0.1

    \[\leadsto \mathsf{fma}\left(x, \sin y, z \cdot \cos y\right)\]

Reproduce

herbie shell --seed 2019171 +o rules:numerics
(FPCore (x y z)
  :name "Diagrams.ThreeD.Transform:aboutX from diagrams-lib-1.3.0.3, B"
  (+ (* x (sin y)) (* z (cos y))))