Average Error: 0.1 → 0.6
Time: 23.4s
Precision: 64
\[x \cdot \cos y - z \cdot \sin y\]
↓
\[\mathsf{fma}\left(x, \cos y, -\left(\sin y \cdot \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right)\right) \cdot \sqrt[3]{z}\right)\]
x \cdot \cos y - z \cdot \sin y
↓
\mathsf{fma}\left(x, \cos y, -\left(\sin y \cdot \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right)\right) \cdot \sqrt[3]{z}\right)
double f(double x, double y, double z) {
        double r183875 = x;
        double r183876 = y;
        double r183877 = cos(r183876);
        double r183878 = r183875 * r183877;
        double r183879 = z;
        double r183880 = sin(r183876);
        double r183881 = r183879 * r183880;
        double r183882 = r183878 - r183881;
        return r183882;
}
↓
double f(double x, double y, double z) {
        double r183883 = x;
        double r183884 = y;
        double r183885 = cos(r183884);
        double r183886 = sin(r183884);
        double r183887 = z;
        double r183888 = cbrt(r183887);
        double r183889 = r183888 * r183888;
        double r183890 = r183886 * r183889;
        double r183891 = r183890 * r183888;
        double r183892 = -r183891;
        double r183893 = fma(r183883, r183885, r183892);
        return r183893;
}

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. Using strategy rm

  3. Applied fma-neg0.1

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

  4. Simplified0.1

    \[\leadsto \mathsf{fma}\left(x, \cos y, \color{blue}{-\sin y \cdot z}\right)\]
  5. Using strategy rm

  6. Applied add-cube-cbrt0.6

    \[\leadsto \mathsf{fma}\left(x, \cos y, -\sin y \cdot \color{blue}{\left(\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}\right)}\right)\]

  7. Applied associate-*r*0.6

    \[\leadsto \mathsf{fma}\left(x, \cos y, -\color{blue}{\left(\sin y \cdot \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right)\right) \cdot \sqrt[3]{z}}\right)\]

  8. Final simplification0.6

    \[\leadsto \mathsf{fma}\left(x, \cos y, -\left(\sin y \cdot \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right)\right) \cdot \sqrt[3]{z}\right)\]

Reproduce

herbie shell --seed 2019303 +o rules:numerics
(FPCore (x y z)
  :name "Diagrams.ThreeD.Transform:aboutX from diagrams-lib-1.3.0.3, A"
  :precision binary64
  (- (* x (cos y)) (* z (sin y))))