Average Error: 57.9 → 0.6
Time: 31.6s
Precision: 64
\[\frac{e^{x} - e^{-x}}{2}\]
↓
\[\frac{\mathsf{fma}\left({x}^{5}, \frac{1}{60}, x \cdot 2 + \left(\frac{1}{3} \cdot \left(x \cdot x\right)\right) \cdot x\right)}{2}\]
\frac{e^{x} - e^{-x}}{2}
↓
\frac{\mathsf{fma}\left({x}^{5}, \frac{1}{60}, x \cdot 2 + \left(\frac{1}{3} \cdot \left(x \cdot x\right)\right) \cdot x\right)}{2}
double f(double x) {
        double r2616146 = x;
        double r2616147 = exp(r2616146);
        double r2616148 = -r2616146;
        double r2616149 = exp(r2616148);
        double r2616150 = r2616147 - r2616149;
        double r2616151 = 2.0;
        double r2616152 = r2616150 / r2616151;
        return r2616152;
}
↓
double f(double x) {
        double r2616153 = x;
        double r2616154 = 5.0;
        double r2616155 = pow(r2616153, r2616154);
        double r2616156 = 0.016666666666666666;
        double r2616157 = 2.0;
        double r2616158 = r2616153 * r2616157;
        double r2616159 = 0.3333333333333333;
        double r2616160 = r2616153 * r2616153;
        double r2616161 = r2616159 * r2616160;
        double r2616162 = r2616161 * r2616153;
        double r2616163 = r2616158 + r2616162;
        double r2616164 = fma(r2616155, r2616156, r2616163);
        double r2616165 = 2.0;
        double r2616166 = r2616164 / r2616165;
        return r2616166;
}

Error

Bits error versus x

Derivation

  1. Initial program 57.9

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

  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. Simplified0.7

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

  5. Applied fma-udef0.7

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

  6. Applied distribute-lft-in0.6

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

  7. Final simplification0.6

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

Reproduce

herbie shell --seed 2019200 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic sine"
  (/ (- (exp x) (exp (- x))) 2.0))