Average Error: 58.2 → 0.6
Time: 11.8s
Precision: 64
\[\frac{e^{x} - e^{-x}}{2}\]
↓
\[\frac{\mathsf{fma}\left(\frac{1}{60}, \left({x}^{5}\right), \left(2 \cdot x + \left(\left(x \cdot \frac{1}{3}\right) \cdot x\right) \cdot x\right)\right)}{2}\]
\frac{e^{x} - e^{-x}}{2}
↓
\frac{\mathsf{fma}\left(\frac{1}{60}, \left({x}^{5}\right), \left(2 \cdot x + \left(\left(x \cdot \frac{1}{3}\right) \cdot x\right) \cdot x\right)\right)}{2}
double f(double x) {
        double r1068146 = x;
        double r1068147 = exp(r1068146);
        double r1068148 = -r1068146;
        double r1068149 = exp(r1068148);
        double r1068150 = r1068147 - r1068149;
        double r1068151 = 2.0;
        double r1068152 = r1068150 / r1068151;
        return r1068152;
}
↓
double f(double x) {
        double r1068153 = 0.016666666666666666;
        double r1068154 = x;
        double r1068155 = 5.0;
        double r1068156 = pow(r1068154, r1068155);
        double r1068157 = 2.0;
        double r1068158 = r1068157 * r1068154;
        double r1068159 = 0.3333333333333333;
        double r1068160 = r1068154 * r1068159;
        double r1068161 = r1068160 * r1068154;
        double r1068162 = r1068161 * r1068154;
        double r1068163 = r1068158 + r1068162;
        double r1068164 = fma(r1068153, r1068156, r1068163);
        double r1068165 = r1068164 / r1068157;
        return r1068165;
}

Error

Bits error versus x

Derivation

  1. Initial program 58.2

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

  2. Taylor expanded around 0 0.6

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

  3. Simplified0.6

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

  5. Applied distribute-rgt-in0.6

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

  6. Final simplification0.6

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

Reproduce

herbie shell --seed 2019128 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic sine"
  (/ (- (exp x) (exp (- x))) 2))