Average Error: 0.1 → 0.1
Time: 5.3s
Precision: 64
\[\cosh x \cdot \frac{\sin y}{y}\]
↓
\[\frac{\cosh x \cdot \sin y}{y}\]
\cosh x \cdot \frac{\sin y}{y}
↓
\frac{\cosh x \cdot \sin y}{y}
double f(double x, double y) {
        double r289862 = x;
        double r289863 = cosh(r289862);
        double r289864 = y;
        double r289865 = sin(r289864);
        double r289866 = r289865 / r289864;
        double r289867 = r289863 * r289866;
        return r289867;
}
↓
double f(double x, double y) {
        double r289868 = x;
        double r289869 = cosh(r289868);
        double r289870 = y;
        double r289871 = sin(r289870);
        double r289872 = r289869 * r289871;
        double r289873 = r289872 / r289870;
        return r289873;
}

Error

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Bits error versus y

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Results

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Target

Original0.1
Target0.1
Herbie0.1
\[\frac{\cosh x \cdot \sin y}{y}\]

Derivation

  1. Initial program 0.1

    \[\cosh x \cdot \frac{\sin y}{y}\]
  2. Using strategy rm

  3. Applied div-inv0.3

    \[\leadsto \cosh x \cdot \color{blue}{\left(\sin y \cdot \frac{1}{y}\right)}\]

  4. Applied associate-*r*0.3

    \[\leadsto \color{blue}{\left(\cosh x \cdot \sin y\right) \cdot \frac{1}{y}}\]
  5. Using strategy rm

  6. Applied expm1-log1p-u0.5

    \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cosh x \cdot \sin y\right)\right)} \cdot \frac{1}{y}\]
  7. Using strategy rm

  8. Applied associate-*r/0.4

    \[\leadsto \color{blue}{\frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\cosh x \cdot \sin y\right)\right) \cdot 1}{y}}\]

  9. Simplified0.1

    \[\leadsto \frac{\color{blue}{\cosh x \cdot \sin y}}{y}\]

  10. Final simplification0.1

    \[\leadsto \frac{\cosh x \cdot \sin y}{y}\]

Reproduce

herbie shell --seed 2020034 +o rules:numerics
(FPCore (x y)
  :name "Linear.Quaternion:$csinh from linear-1.19.1.3"
  :precision binary64

  :herbie-target
  (/ (* (cosh x) (sin y)) y)

  (* (cosh x) (/ (sin y) y)))