Average Error: 0.1 → 0.4
Time: 13.2s
Precision: 64
\[x \cdot \cos y - z \cdot \sin y\]
↓
\[\mathsf{fma}\left(z, -\sin y, \left(x \cdot \left(\sqrt[3]{\cos y} \cdot \sqrt[3]{\cos y}\right)\right) \cdot \sqrt[3]{\cos y}\right)\]
x \cdot \cos y - z \cdot \sin y
↓
\mathsf{fma}\left(z, -\sin y, \left(x \cdot \left(\sqrt[3]{\cos y} \cdot \sqrt[3]{\cos y}\right)\right) \cdot \sqrt[3]{\cos y}\right)
double f(double x, double y, double z) {
        double r211645 = x;
        double r211646 = y;
        double r211647 = cos(r211646);
        double r211648 = r211645 * r211647;
        double r211649 = z;
        double r211650 = sin(r211646);
        double r211651 = r211649 * r211650;
        double r211652 = r211648 - r211651;
        return r211652;
}
↓
double f(double x, double y, double z) {
        double r211653 = z;
        double r211654 = y;
        double r211655 = sin(r211654);
        double r211656 = -r211655;
        double r211657 = x;
        double r211658 = cos(r211654);
        double r211659 = cbrt(r211658);
        double r211660 = r211659 * r211659;
        double r211661 = r211657 * r211660;
        double r211662 = r211661 * r211659;
        double r211663 = fma(r211653, r211656, r211662);
        return r211663;
}

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. Taylor expanded around inf 0.1

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

  3. Simplified0.1

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

  5. Applied add-cube-cbrt0.4

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

  6. Applied associate-*r*0.4

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

  7. Final simplification0.4

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

Reproduce

herbie shell --seed 2020046 +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))))