Average Error: 0.1 → 0.3
Time: 19.2s
Precision: 64
\[x \cdot \cos y - z \cdot \sin y\]
↓
\[\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{\cos y}\right)\right) \cdot \left({\left(\cos y \cdot \cos y\right)}^{\frac{1}{3}} \cdot x\right) - z \cdot \sin y\]
x \cdot \cos y - z \cdot \sin y
↓
\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{\cos y}\right)\right) \cdot \left({\left(\cos y \cdot \cos y\right)}^{\frac{1}{3}} \cdot x\right) - z \cdot \sin y
double f(double x, double y, double z) {
        double r6184778 = x;
        double r6184779 = y;
        double r6184780 = cos(r6184779);
        double r6184781 = r6184778 * r6184780;
        double r6184782 = z;
        double r6184783 = sin(r6184779);
        double r6184784 = r6184782 * r6184783;
        double r6184785 = r6184781 - r6184784;
        return r6184785;
}
↓
double f(double x, double y, double z) {
        double r6184786 = y;
        double r6184787 = cos(r6184786);
        double r6184788 = cbrt(r6184787);
        double r6184789 = log1p(r6184788);
        double r6184790 = expm1(r6184789);
        double r6184791 = r6184787 * r6184787;
        double r6184792 = 0.3333333333333333;
        double r6184793 = pow(r6184791, r6184792);
        double r6184794 = x;
        double r6184795 = r6184793 * r6184794;
        double r6184796 = r6184790 * r6184795;
        double r6184797 = z;
        double r6184798 = sin(r6184786);
        double r6184799 = r6184797 * r6184798;
        double r6184800 = r6184796 - r6184799;
        return r6184800;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.1

    \[x \cdot \cos y - z \cdot \sin y\]
  2. Using strategy rm

  3. Applied add-cube-cbrt0.4

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

  4. Applied associate-*r*0.4

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

  6. Applied pow1/316.4

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

  7. Applied pow1/316.3

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

  8. Applied pow-prod-down0.2

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

  10. Applied expm1-log1p-u0.3

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

  11. Final simplification0.3

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

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(FPCore (x y z)
  :name "Diagrams.ThreeD.Transform:aboutX from diagrams-lib-1.3.0.3, A"
  (- (* x (cos y)) (* z (sin y))))