Average Error: 58.3 → 0.6
Time: 4.6s
Precision: 64
\[\frac{e^{x} - e^{-x}}{2}\]
↓
\[\frac{\mathsf{fma}\left(\frac{1}{3}, {x}^{3}, \mathsf{fma}\left(\frac{1}{60}, {x}^{5}, 2 \cdot x\right)\right)}{2}\]
\frac{e^{x} - e^{-x}}{2}
↓
\frac{\mathsf{fma}\left(\frac{1}{3}, {x}^{3}, \mathsf{fma}\left(\frac{1}{60}, {x}^{5}, 2 \cdot x\right)\right)}{2}
double f(double x) {
        double r30942 = x;
        double r30943 = exp(r30942);
        double r30944 = -r30942;
        double r30945 = exp(r30944);
        double r30946 = r30943 - r30945;
        double r30947 = 2.0;
        double r30948 = r30946 / r30947;
        return r30948;
}
↓
double f(double x) {
        double r30949 = 0.3333333333333333;
        double r30950 = x;
        double r30951 = 3.0;
        double r30952 = pow(r30950, r30951);
        double r30953 = 0.016666666666666666;
        double r30954 = 5.0;
        double r30955 = pow(r30950, r30954);
        double r30956 = 2.0;
        double r30957 = r30956 * r30950;
        double r30958 = fma(r30953, r30955, r30957);
        double r30959 = fma(r30949, r30952, r30958);
        double r30960 = 2.0;
        double r30961 = r30959 / r30960;
        return r30961;
}

Error

Bits error versus x

Derivation

  1. Initial program 58.3

    \[\frac{e^{x} - e^{-x}}{2}\]

  2. Taylor expanded around 0 0.6

    \[\leadsto \frac{\color{blue}{\frac{1}{3} \cdot {x}^{3} + \left(\frac{1}{60} \cdot {x}^{5} + 2 \cdot x\right)}}{2}\]

  3. Simplified0.6

    \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{3}, {x}^{3}, \mathsf{fma}\left(\frac{1}{60}, {x}^{5}, 2 \cdot x\right)\right)}}{2}\]

  4. Final simplification0.6

    \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{3}, {x}^{3}, \mathsf{fma}\left(\frac{1}{60}, {x}^{5}, 2 \cdot x\right)\right)}{2}\]

Reproduce

herbie shell --seed 2020033 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2))