Average Error: 0.2 → 0.2
Time: 13.3s
Precision: 64
\[\cosh x \cdot \frac{\sin y}{y}\]
↓
\[\frac{\cosh x}{\frac{y}{\sin y}}\]
\cosh x \cdot \frac{\sin y}{y}
↓
\frac{\cosh x}{\frac{y}{\sin y}}
double f(double x, double y) {
        double r549515 = x;
        double r549516 = cosh(r549515);
        double r549517 = y;
        double r549518 = sin(r549517);
        double r549519 = r549518 / r549517;
        double r549520 = r549516 * r549519;
        return r549520;
}
↓
double f(double x, double y) {
        double r549521 = x;
        double r549522 = cosh(r549521);
        double r549523 = y;
        double r549524 = sin(r549523);
        double r549525 = r549523 / r549524;
        double r549526 = r549522 / r549525;
        return r549526;
}

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. Taylor expanded around inf 0.2

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

  3. Simplified0.2

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

  4. Final simplification0.2

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

Reproduce

herbie shell --seed 2020047 +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)))