Average Error: 0.1 → 0.3
Time: 22.4s
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 r7725413 = x;
        double r7725414 = y;
        double r7725415 = cos(r7725414);
        double r7725416 = r7725413 * r7725415;
        double r7725417 = z;
        double r7725418 = sin(r7725414);
        double r7725419 = r7725417 * r7725418;
        double r7725420 = r7725416 - r7725419;
        return r7725420;
}
↓
double f(double x, double y, double z) {
        double r7725421 = y;
        double r7725422 = cos(r7725421);
        double r7725423 = cbrt(r7725422);
        double r7725424 = log1p(r7725423);
        double r7725425 = expm1(r7725424);
        double r7725426 = r7725422 * r7725422;
        double r7725427 = 0.3333333333333333;
        double r7725428 = pow(r7725426, r7725427);
        double r7725429 = x;
        double r7725430 = r7725428 * r7725429;
        double r7725431 = r7725425 * r7725430;
        double r7725432 = z;
        double r7725433 = sin(r7725421);
        double r7725434 = r7725432 * r7725433;
        double r7725435 = r7725431 - r7725434;
        return r7725435;
}

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))))