Average Error: 0.0 → 0.6
Time: 13.0s
Precision: 64
\[\cos x \cdot \frac{\sinh y}{y}\]
↓
\[\cos x \cdot \frac{\mathsf{fma}\left(\frac{1}{6}, {y}^{3}, \mathsf{fma}\left(\frac{1}{120}, {y}^{5}, y\right)\right)}{y}\]
\cos x \cdot \frac{\sinh y}{y}
↓
\cos 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 r78140 = x;
        double r78141 = cos(r78140);
        double r78142 = y;
        double r78143 = sinh(r78142);
        double r78144 = r78143 / r78142;
        double r78145 = r78141 * r78144;
        return r78145;
}
↓
double f(double x, double y) {
        double r78146 = x;
        double r78147 = cos(r78146);
        double r78148 = 0.16666666666666666;
        double r78149 = y;
        double r78150 = 3.0;
        double r78151 = pow(r78149, r78150);
        double r78152 = 0.008333333333333333;
        double r78153 = 5.0;
        double r78154 = pow(r78149, r78153);
        double r78155 = fma(r78152, r78154, r78149);
        double r78156 = fma(r78148, r78151, r78155);
        double r78157 = r78156 / r78149;
        double r78158 = r78147 * r78157;
        return r78158;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 0.0

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

  2. Taylor expanded around 0 0.6

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

  3. Simplified0.6

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

  4. Final simplification0.6

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

Reproduce

herbie shell --seed 2019212 +o rules:numerics
(FPCore (x y)
  :name "Linear.Quaternion:$csin from linear-1.19.1.3"
  :precision binary64
  (* (cos x) (/ (sinh y) y)))