Average Error: 0.0 → 0.7
Time: 4.4s
Precision: 64
\[\sin x \cdot \frac{\sinh y}{y}\]
↓
\[\sin x \cdot \mathsf{fma}\left(\frac{1}{6}, {y}^{2}, \mathsf{fma}\left(\frac{1}{120}, {y}^{4}, 1\right)\right)\]
\sin x \cdot \frac{\sinh y}{y}
↓
\sin x \cdot \mathsf{fma}\left(\frac{1}{6}, {y}^{2}, \mathsf{fma}\left(\frac{1}{120}, {y}^{4}, 1\right)\right)
double f(double x, double y) {
        double r125105 = x;
        double r125106 = sin(r125105);
        double r125107 = y;
        double r125108 = sinh(r125107);
        double r125109 = r125108 / r125107;
        double r125110 = r125106 * r125109;
        return r125110;
}
↓
double f(double x, double y) {
        double r125111 = x;
        double r125112 = sin(r125111);
        double r125113 = 0.16666666666666666;
        double r125114 = y;
        double r125115 = 2.0;
        double r125116 = pow(r125114, r125115);
        double r125117 = 0.008333333333333333;
        double r125118 = 4.0;
        double r125119 = pow(r125114, r125118);
        double r125120 = 1.0;
        double r125121 = fma(r125117, r125119, r125120);
        double r125122 = fma(r125113, r125116, r125121);
        double r125123 = r125112 * r125122;
        return r125123;
}

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 0.7

    \[\leadsto \sin x \cdot \color{blue}{\left(\frac{1}{6} \cdot {y}^{2} + \left(\frac{1}{120} \cdot {y}^{4} + 1\right)\right)}\]

  3. Simplified0.7

    \[\leadsto \sin x \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{6}, {y}^{2}, \mathsf{fma}\left(\frac{1}{120}, {y}^{4}, 1\right)\right)}\]

  4. Final simplification0.7

    \[\leadsto \sin x \cdot \mathsf{fma}\left(\frac{1}{6}, {y}^{2}, \mathsf{fma}\left(\frac{1}{120}, {y}^{4}, 1\right)\right)\]

Reproduce

herbie shell --seed 2020018 +o rules:numerics
(FPCore (x y)
  :name "Linear.Quaternion:$ccos from linear-1.19.1.3"
  :precision binary64
  (* (sin x) (/ (sinh y) y)))