Average Error: 0.1 → 0.3
Time: 4.9s
Precision: 64
\[x \cdot \cos y - z \cdot \sin y\]
↓
\[\left(\left(x \cdot {\left({\left(\cos y\right)}^{2}\right)}^{\frac{1}{6}}\right) \cdot {\left({\left(\cos y\right)}^{2}\right)}^{\frac{1}{6}}\right) \cdot \sqrt[3]{\cos y} - z \cdot \sin y\]
x \cdot \cos y - z \cdot \sin y
↓
\left(\left(x \cdot {\left({\left(\cos y\right)}^{2}\right)}^{\frac{1}{6}}\right) \cdot {\left({\left(\cos y\right)}^{2}\right)}^{\frac{1}{6}}\right) \cdot \sqrt[3]{\cos y} - z \cdot \sin y
double f(double x, double y, double z) {
        double r172722 = x;
        double r172723 = y;
        double r172724 = cos(r172723);
        double r172725 = r172722 * r172724;
        double r172726 = z;
        double r172727 = sin(r172723);
        double r172728 = r172726 * r172727;
        double r172729 = r172725 - r172728;
        return r172729;
}
↓
double f(double x, double y, double z) {
        double r172730 = x;
        double r172731 = y;
        double r172732 = cos(r172731);
        double r172733 = 2.0;
        double r172734 = pow(r172732, r172733);
        double r172735 = 0.16666666666666666;
        double r172736 = pow(r172734, r172735);
        double r172737 = r172730 * r172736;
        double r172738 = r172737 * r172736;
        double r172739 = cbrt(r172732);
        double r172740 = r172738 * r172739;
        double r172741 = z;
        double r172742 = sin(r172731);
        double r172743 = r172741 * r172742;
        double r172744 = r172740 - r172743;
        return r172744;
}

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 pow1/316.3

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

  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. Simplified0.2

    \[\leadsto \left(x \cdot {\color{blue}{\left({\left(\cos y\right)}^{2}\right)}}^{\frac{1}{3}}\right) \cdot \sqrt[3]{\cos y} - z \cdot \sin y\]
  10. Using strategy rm

  11. Applied sqr-pow0.3

    \[\leadsto \left(x \cdot \color{blue}{\left({\left({\left(\cos y\right)}^{2}\right)}^{\left(\frac{\frac{1}{3}}{2}\right)} \cdot {\left({\left(\cos y\right)}^{2}\right)}^{\left(\frac{\frac{1}{3}}{2}\right)}\right)}\right) \cdot \sqrt[3]{\cos y} - z \cdot \sin y\]

  12. Applied associate-*r*0.3

    \[\leadsto \color{blue}{\left(\left(x \cdot {\left({\left(\cos y\right)}^{2}\right)}^{\left(\frac{\frac{1}{3}}{2}\right)}\right) \cdot {\left({\left(\cos y\right)}^{2}\right)}^{\left(\frac{\frac{1}{3}}{2}\right)}\right)} \cdot \sqrt[3]{\cos y} - z \cdot \sin y\]

  13. Simplified0.3

    \[\leadsto \left(\color{blue}{\left(x \cdot {\left({\left(\cos y\right)}^{2}\right)}^{\frac{1}{6}}\right)} \cdot {\left({\left(\cos y\right)}^{2}\right)}^{\left(\frac{\frac{1}{3}}{2}\right)}\right) \cdot \sqrt[3]{\cos y} - z \cdot \sin y\]

  14. Final simplification0.3

    \[\leadsto \left(\left(x \cdot {\left({\left(\cos y\right)}^{2}\right)}^{\frac{1}{6}}\right) \cdot {\left({\left(\cos y\right)}^{2}\right)}^{\frac{1}{6}}\right) \cdot \sqrt[3]{\cos y} - z \cdot \sin y\]

Reproduce

herbie shell --seed 2019353 +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))))