Average Error: 0.1 → 0.4
Time: 5.8s
Precision: 64
\[x \cdot \cos y - z \cdot \sin y\]
↓
\[\left(x \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{\cos y} \cdot \sqrt[3]{\cos y}\right)\right)\right) \cdot \left(\sqrt[3]{\sqrt[3]{\cos y} \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\sqrt[3]{\cos y}\right)\right)} \cdot \sqrt[3]{\sqrt[3]{\cos y}}\right) - z \cdot \sin y\]
x \cdot \cos y - z \cdot \sin y
↓
\left(x \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{\cos y} \cdot \sqrt[3]{\cos y}\right)\right)\right) \cdot \left(\sqrt[3]{\sqrt[3]{\cos y} \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\sqrt[3]{\cos y}\right)\right)} \cdot \sqrt[3]{\sqrt[3]{\cos y}}\right) - z \cdot \sin y
double f(double x, double y, double z) {
        double r188673 = x;
        double r188674 = y;
        double r188675 = cos(r188674);
        double r188676 = r188673 * r188675;
        double r188677 = z;
        double r188678 = sin(r188674);
        double r188679 = r188677 * r188678;
        double r188680 = r188676 - r188679;
        return r188680;
}
↓
double f(double x, double y, double z) {
        double r188681 = x;
        double r188682 = y;
        double r188683 = cos(r188682);
        double r188684 = cbrt(r188683);
        double r188685 = r188684 * r188684;
        double r188686 = log1p(r188685);
        double r188687 = expm1(r188686);
        double r188688 = r188681 * r188687;
        double r188689 = expm1(r188684);
        double r188690 = log1p(r188689);
        double r188691 = r188684 * r188690;
        double r188692 = cbrt(r188691);
        double r188693 = cbrt(r188684);
        double r188694 = r188692 * r188693;
        double r188695 = r188688 * r188694;
        double r188696 = z;
        double r188697 = sin(r188682);
        double r188698 = r188696 * r188697;
        double r188699 = r188695 - r188698;
        return r188699;
}

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 expm1-log1p-u0.4

    \[\leadsto \left(x \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{\cos y} \cdot \sqrt[3]{\cos y}\right)\right)}\right) \cdot \sqrt[3]{\cos y} - z \cdot \sin y\]
  7. Using strategy rm

  8. Applied add-cube-cbrt0.4

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

  9. Applied cbrt-prod0.4

    \[\leadsto \left(x \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{\cos y} \cdot \sqrt[3]{\cos y}\right)\right)\right) \cdot \color{blue}{\left(\sqrt[3]{\sqrt[3]{\cos y} \cdot \sqrt[3]{\cos y}} \cdot \sqrt[3]{\sqrt[3]{\cos y}}\right)} - z \cdot \sin y\]
  10. Using strategy rm

  11. Applied log1p-expm1-u0.4

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

  12. Final simplification0.4

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

Reproduce

herbie shell --seed 2020021 +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))))