Average Error: 0.1 → 0.3
Time: 22.4s
Precision: 64
\[x \cdot \cos y - z \cdot \sin y\]
↓
\[\left(x \cdot \sqrt[3]{{\left(\cos y\right)}^{2}}\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{\cos y}\right)\right) - z \cdot \sin y\]
x \cdot \cos y - z \cdot \sin y
↓
\left(x \cdot \sqrt[3]{{\left(\cos y\right)}^{2}}\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{\cos y}\right)\right) - z \cdot \sin y
double f(double x, double y, double z) {
        double r127957 = x;
        double r127958 = y;
        double r127959 = cos(r127958);
        double r127960 = r127957 * r127959;
        double r127961 = z;
        double r127962 = sin(r127958);
        double r127963 = r127961 * r127962;
        double r127964 = r127960 - r127963;
        return r127964;
}
↓
double f(double x, double y, double z) {
        double r127965 = x;
        double r127966 = y;
        double r127967 = cos(r127966);
        double r127968 = 2.0;
        double r127969 = pow(r127967, r127968);
        double r127970 = cbrt(r127969);
        double r127971 = r127965 * r127970;
        double r127972 = cbrt(r127967);
        double r127973 = log1p(r127972);
        double r127974 = expm1(r127973);
        double r127975 = r127971 * r127974;
        double r127976 = z;
        double r127977 = sin(r127966);
        double r127978 = r127976 * r127977;
        double r127979 = r127975 - r127978;
        return r127979;
}

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 cbrt-unprod0.3

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

  7. Simplified0.3

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

  9. Applied expm1-log1p-u0.3

    \[\leadsto \left(x \cdot \sqrt[3]{{\left(\cos y\right)}^{2}}\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{\cos y}\right)\right)} - z \cdot \sin y\]

  10. Final simplification0.3

    \[\leadsto \left(x \cdot \sqrt[3]{{\left(\cos y\right)}^{2}}\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{\cos y}\right)\right) - z \cdot \sin y\]

Reproduce

herbie shell --seed 2019304 +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))))