Average Error: 0.1 → 0.1
Time: 18.7s
Precision: 64
\[x \cdot \cos y + z \cdot \sin y\]
↓
\[\mathsf{fma}\left(z, \sin y, x \cdot \cos y\right)\]
x \cdot \cos y + z \cdot \sin y
↓
\mathsf{fma}\left(z, \sin y, x \cdot \cos y\right)
double f(double x, double y, double z) {
        double r132160 = x;
        double r132161 = y;
        double r132162 = cos(r132161);
        double r132163 = r132160 * r132162;
        double r132164 = z;
        double r132165 = sin(r132161);
        double r132166 = r132164 * r132165;
        double r132167 = r132163 + r132166;
        return r132167;
}
↓
double f(double x, double y, double z) {
        double r132168 = z;
        double r132169 = y;
        double r132170 = sin(r132169);
        double r132171 = x;
        double r132172 = cos(r132169);
        double r132173 = r132171 * r132172;
        double r132174 = fma(r132168, r132170, r132173);
        return r132174;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

  1. Initial program 0.1

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

  2. Simplified0.1

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

  3. Taylor expanded around inf 0.1

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

  4. Simplified0.1

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

  5. Final simplification0.1

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

Reproduce

herbie shell --seed 2019212 +o rules:numerics
(FPCore (x y z)
  :name "Diagrams.ThreeD.Transform:aboutY from diagrams-lib-1.3.0.3"
  :precision binary64
  (+ (* x (cos y)) (* z (sin y))))