Average Error: 58.1 → 0.4
Time: 4.5s
Precision: 64
\[\frac{e^{x} - e^{-x}}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 0.0154935820765723566:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{1}{3}, {x}^{3}, \mathsf{fma}\left(\frac{1}{60}, {x}^{5}, 2 \cdot x\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x} + \left(-e^{-x}\right)}{2}\\ \end{array}\]
\frac{e^{x} - e^{-x}}{2}
↓
\begin{array}{l}
\mathbf{if}\;x \le 0.0154935820765723566:\\
\;\;\;\;\frac{\mathsf{fma}\left(\frac{1}{3}, {x}^{3}, \mathsf{fma}\left(\frac{1}{60}, {x}^{5}, 2 \cdot x\right)\right)}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{e^{x} + \left(-e^{-x}\right)}{2}\\

\end{array}
double f(double x) {
        double r68133 = x;
        double r68134 = exp(r68133);
        double r68135 = -r68133;
        double r68136 = exp(r68135);
        double r68137 = r68134 - r68136;
        double r68138 = 2.0;
        double r68139 = r68137 / r68138;
        return r68139;
}
↓
double f(double x) {
        double r68140 = x;
        double r68141 = 0.015493582076572357;
        bool r68142 = r68140 <= r68141;
        double r68143 = 0.3333333333333333;
        double r68144 = 3.0;
        double r68145 = pow(r68140, r68144);
        double r68146 = 0.016666666666666666;
        double r68147 = 5.0;
        double r68148 = pow(r68140, r68147);
        double r68149 = 2.0;
        double r68150 = r68149 * r68140;
        double r68151 = fma(r68146, r68148, r68150);
        double r68152 = fma(r68143, r68145, r68151);
        double r68153 = 2.0;
        double r68154 = r68152 / r68153;
        double r68155 = exp(r68140);
        double r68156 = -r68140;
        double r68157 = exp(r68156);
        double r68158 = -r68157;
        double r68159 = r68155 + r68158;
        double r68160 = r68159 / r68153;
        double r68161 = r68142 ? r68154 : r68160;
        return r68161;
}

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if x < 0.015493582076572357

    1. Initial program 58.5

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

    2. Taylor expanded around 0 0.4

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

    3. Simplified0.4

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

    if 0.015493582076572357 < x

    1. Initial program 0.6

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

    3. Applied add-sqr-sqrt1.1

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

    4. Applied fma-neg1.0

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{e^{x}}, \sqrt{e^{x}}, -e^{-x}\right)}}{2}\]
    5. Using strategy rm

    6. Applied fma-udef1.1

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

    7. Simplified0.6

      \[\leadsto \frac{\color{blue}{e^{x}} + \left(-e^{-x}\right)}{2}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 0.0154935820765723566:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{1}{3}, {x}^{3}, \mathsf{fma}\left(\frac{1}{60}, {x}^{5}, 2 \cdot x\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x} + \left(-e^{-x}\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020035 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2))