Average Error: 0.1 → 0.1
Time: 16.9s
Precision: 64
\[\cosh x \cdot \frac{\sin y}{y}\]
↓
\[\frac{\left(\sin y \cdot \left(e^{x} + e^{-x}\right)\right) \cdot \frac{1}{2}}{y}\]
\cosh x \cdot \frac{\sin y}{y}
↓
\frac{\left(\sin y \cdot \left(e^{x} + e^{-x}\right)\right) \cdot \frac{1}{2}}{y}
double f(double x, double y) {
        double r656811 = x;
        double r656812 = cosh(r656811);
        double r656813 = y;
        double r656814 = sin(r656813);
        double r656815 = r656814 / r656813;
        double r656816 = r656812 * r656815;
        return r656816;
}
↓
double f(double x, double y) {
        double r656817 = y;
        double r656818 = sin(r656817);
        double r656819 = x;
        double r656820 = exp(r656819);
        double r656821 = -r656819;
        double r656822 = exp(r656821);
        double r656823 = r656820 + r656822;
        double r656824 = r656818 * r656823;
        double r656825 = 0.5;
        double r656826 = r656824 * r656825;
        double r656827 = r656826 / r656817;
        return r656827;
}

Error

Bits error versus x

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. Taylor expanded around inf 0.1

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

  3. Simplified0.1

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

  4. Final simplification0.1

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

Reproduce

herbie shell --seed 2020042 +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)))