Average Error: 0.2 → 0.2
Time: 26.2s
Precision: 64
\[\cosh x \cdot \frac{\sin y}{y}\]
↓
\[\frac{\left(\frac{1}{2} \cdot \sin y\right) \cdot \left(e^{x} + e^{-x}\right)}{y}\]
\cosh x \cdot \frac{\sin y}{y}
↓
\frac{\left(\frac{1}{2} \cdot \sin y\right) \cdot \left(e^{x} + e^{-x}\right)}{y}
double f(double x, double y) {
        double r283855 = x;
        double r283856 = cosh(r283855);
        double r283857 = y;
        double r283858 = sin(r283857);
        double r283859 = r283858 / r283857;
        double r283860 = r283856 * r283859;
        return r283860;
}
↓
double f(double x, double y) {
        double r283861 = 0.5;
        double r283862 = y;
        double r283863 = sin(r283862);
        double r283864 = r283861 * r283863;
        double r283865 = x;
        double r283866 = exp(r283865);
        double r283867 = -r283865;
        double r283868 = exp(r283867);
        double r283869 = r283866 + r283868;
        double r283870 = r283864 * r283869;
        double r283871 = r283870 / r283862;
        return r283871;
}

Error

Bits error versus x

Bits error versus y

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Results

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Target

Original0.2
Target0.2
Herbie0.2
\[\frac{\cosh x \cdot \sin y}{y}\]

Derivation

  1. Initial program 0.2

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

  3. Applied expm1-log1p-u0.2

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

  4. Taylor expanded around inf 0.2

    \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot \left(\sin y \cdot e^{x}\right) + \frac{1}{2} \cdot \left(\sin y \cdot e^{-x}\right)}{y}}\]

  5. Simplified0.2

    \[\leadsto \color{blue}{\frac{\left(\frac{1}{2} \cdot \sin y\right) \cdot \left(e^{x} + e^{-x}\right)}{y}}\]

  6. Final simplification0.2

    \[\leadsto \frac{\left(\frac{1}{2} \cdot \sin y\right) \cdot \left(e^{x} + e^{-x}\right)}{y}\]

Reproduce

herbie shell --seed 2019303 +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)))