Average Error: 58.1 → 0.6
Time: 7.8s
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 r30070 = x;
        double r30071 = exp(r30070);
        double r30072 = -r30070;
        double r30073 = exp(r30072);
        double r30074 = r30071 - r30073;
        double r30075 = 2.0;
        double r30076 = r30074 / r30075;
        return r30076;
}
↓
double f(double x) {
        double r30077 = 0.3333333333333333;
        double r30078 = x;
        double r30079 = 3.0;
        double r30080 = pow(r30078, r30079);
        double r30081 = 0.016666666666666666;
        double r30082 = 5.0;
        double r30083 = pow(r30078, r30082);
        double r30084 = 2.0;
        double r30085 = r30084 * r30078;
        double r30086 = fma(r30081, r30083, r30085);
        double r30087 = fma(r30077, r30080, r30086);
        double r30088 = 2.0;
        double r30089 = r30087 / r30088;
        return r30089;
}

Error

Bits error versus x

Derivation

  1. Initial program 58.1

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

  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 2019351 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2))