Average Error: 0.1 → 0.3
Time: 23.9s
Precision: 64
\[x \cdot \sin y + z \cdot \cos y\]
↓
\[\mathsf{fma}\left(x, \sin y, \mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{\cos y}\right)\right) \cdot \left(z \cdot {\left(\cos y \cdot \cos y\right)}^{\frac{1}{3}}\right)\right)\]
x \cdot \sin y + z \cdot \cos y
↓
\mathsf{fma}\left(x, \sin y, \mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{\cos y}\right)\right) \cdot \left(z \cdot {\left(\cos y \cdot \cos y\right)}^{\frac{1}{3}}\right)\right)
double f(double x, double y, double z) {
        double r10886042 = x;
        double r10886043 = y;
        double r10886044 = sin(r10886043);
        double r10886045 = r10886042 * r10886044;
        double r10886046 = z;
        double r10886047 = cos(r10886043);
        double r10886048 = r10886046 * r10886047;
        double r10886049 = r10886045 + r10886048;
        return r10886049;
}
↓
double f(double x, double y, double z) {
        double r10886050 = x;
        double r10886051 = y;
        double r10886052 = sin(r10886051);
        double r10886053 = cos(r10886051);
        double r10886054 = cbrt(r10886053);
        double r10886055 = log1p(r10886054);
        double r10886056 = expm1(r10886055);
        double r10886057 = z;
        double r10886058 = r10886053 * r10886053;
        double r10886059 = 0.3333333333333333;
        double r10886060 = pow(r10886058, r10886059);
        double r10886061 = r10886057 * r10886060;
        double r10886062 = r10886056 * r10886061;
        double r10886063 = fma(r10886050, r10886052, r10886062);
        return r10886063;
}

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. Using strategy rm

  6. Applied add-cube-cbrt0.4

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

  7. Applied associate-*r*0.4

    \[\leadsto \mathsf{fma}\left(x, \sin y, \color{blue}{\left(z \cdot \left(\sqrt[3]{\cos y} \cdot \sqrt[3]{\cos y}\right)\right) \cdot \sqrt[3]{\cos y}}\right)\]
  8. Using strategy rm

  9. Applied pow1/316.1

    \[\leadsto \mathsf{fma}\left(x, \sin y, \left(z \cdot \left(\sqrt[3]{\cos y} \cdot \color{blue}{{\left(\cos y\right)}^{\frac{1}{3}}}\right)\right) \cdot \sqrt[3]{\cos y}\right)\]

  10. Applied pow1/316.0

    \[\leadsto \mathsf{fma}\left(x, \sin y, \left(z \cdot \left(\color{blue}{{\left(\cos y\right)}^{\frac{1}{3}}} \cdot {\left(\cos y\right)}^{\frac{1}{3}}\right)\right) \cdot \sqrt[3]{\cos y}\right)\]

  11. Applied pow-prod-down0.2

    \[\leadsto \mathsf{fma}\left(x, \sin y, \left(z \cdot \color{blue}{{\left(\cos y \cdot \cos y\right)}^{\frac{1}{3}}}\right) \cdot \sqrt[3]{\cos y}\right)\]
  12. Using strategy rm

  13. Applied expm1-log1p-u0.3

    \[\leadsto \mathsf{fma}\left(x, \sin y, \left(z \cdot {\left(\cos y \cdot \cos y\right)}^{\frac{1}{3}}\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{\cos y}\right)\right)}\right)\]

  14. Final simplification0.3

    \[\leadsto \mathsf{fma}\left(x, \sin y, \mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{\cos y}\right)\right) \cdot \left(z \cdot {\left(\cos y \cdot \cos y\right)}^{\frac{1}{3}}\right)\right)\]

Reproduce

herbie shell --seed 2019170 +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))))