Average Error: 0.0 → 0.6
Time: 5.6s
Precision: 64
\[\sin x \cdot \frac{\sinh y}{y}\]
↓
\[\sin x \cdot \frac{\mathsf{fma}\left(\frac{1}{6}, {y}^{3}, \mathsf{fma}\left(\frac{1}{120}, {y}^{5}, y\right)\right)}{y}\]
\sin x \cdot \frac{\sinh y}{y}
↓
\sin x \cdot \frac{\mathsf{fma}\left(\frac{1}{6}, {y}^{3}, \mathsf{fma}\left(\frac{1}{120}, {y}^{5}, y\right)\right)}{y}
double f(double x, double y) {
        double r691 = x;
        double r692 = sin(r691);
        double r693 = y;
        double r694 = sinh(r693);
        double r695 = r694 / r693;
        double r696 = r692 * r695;
        return r696;
}
↓
double f(double x, double y) {
        double r697 = x;
        double r698 = sin(r697);
        double r699 = 0.16666666666666666;
        double r700 = y;
        double r701 = 3.0;
        double r702 = pow(r700, r701);
        double r703 = 0.008333333333333333;
        double r704 = 5.0;
        double r705 = pow(r700, r704);
        double r706 = fma(r703, r705, r700);
        double r707 = fma(r699, r702, r706);
        double r708 = r707 / r700;
        double r709 = r698 * r708;
        return r709;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 0.0

    \[\sin x \cdot \frac{\sinh y}{y}\]

  2. Taylor expanded around 0 0.6

    \[\leadsto \sin x \cdot \frac{\color{blue}{\frac{1}{6} \cdot {y}^{3} + \left(\frac{1}{120} \cdot {y}^{5} + y\right)}}{y}\]

  3. Simplified0.6

    \[\leadsto \sin x \cdot \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{6}, {y}^{3}, \mathsf{fma}\left(\frac{1}{120}, {y}^{5}, y\right)\right)}}{y}\]

  4. Final simplification0.6

    \[\leadsto \sin x \cdot \frac{\mathsf{fma}\left(\frac{1}{6}, {y}^{3}, \mathsf{fma}\left(\frac{1}{120}, {y}^{5}, y\right)\right)}{y}\]

Reproduce

herbie shell --seed 2020025 +o rules:numerics
(FPCore (x y)
  :name "Linear.Quaternion:$ccos from linear-1.19.1.3"
  :precision binary64
  (* (sin x) (/ (sinh y) y)))