Average Error: 58.0 → 0.6
Time: 14.9s
Precision: 64
\[\frac{e^{x} - e^{-x}}{2}\]
↓
\[\frac{\mathsf{fma}\left(2, x, \mathsf{fma}\left({x}^{5}, \frac{1}{60}, \frac{1}{3} \cdot {x}^{3}\right)\right)}{2}\]
\frac{e^{x} - e^{-x}}{2}
↓
\frac{\mathsf{fma}\left(2, x, \mathsf{fma}\left({x}^{5}, \frac{1}{60}, \frac{1}{3} \cdot {x}^{3}\right)\right)}{2}
double f(double x) {
        double r52166 = x;
        double r52167 = exp(r52166);
        double r52168 = -r52166;
        double r52169 = exp(r52168);
        double r52170 = r52167 - r52169;
        double r52171 = 2.0;
        double r52172 = r52170 / r52171;
        return r52172;
}
↓
double f(double x) {
        double r52173 = 2.0;
        double r52174 = x;
        double r52175 = 5.0;
        double r52176 = pow(r52174, r52175);
        double r52177 = 0.016666666666666666;
        double r52178 = 0.3333333333333333;
        double r52179 = 3.0;
        double r52180 = pow(r52174, r52179);
        double r52181 = r52178 * r52180;
        double r52182 = fma(r52176, r52177, r52181);
        double r52183 = fma(r52173, r52174, r52182);
        double r52184 = 2.0;
        double r52185 = r52183 / r52184;
        return r52185;
}

Error

Bits error versus x

Derivation

  1. Initial program 58.0

    \[\frac{e^{x} - e^{-x}}{2}\]
  2. Using strategy rm

  3. Applied flip--58.1

    \[\leadsto \frac{\color{blue}{\frac{e^{x} \cdot e^{x} - e^{-x} \cdot e^{-x}}{e^{x} + e^{-x}}}}{2}\]

  4. Simplified58.0

    \[\leadsto \frac{\frac{\color{blue}{{\left(e^{x}\right)}^{2} - e^{x \cdot -2}}}{e^{x} + e^{-x}}}{2}\]

  5. Taylor expanded around 0 0.6

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

  6. Simplified0.6

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

  7. Final simplification0.6

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

Reproduce

herbie shell --seed 2019195 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic sine"
  (/ (- (exp x) (exp (- x))) 2.0))