Average Error: 58.3 → 0.6
Time: 12.1s
Precision: 64
\[\frac{e^{x} - e^{-x}}{2}\]
↓
\[\frac{\mathsf{fma}\left({x}^{5}, \frac{1}{60}, x \cdot 2 + x \cdot \left(\frac{1}{3} \cdot \left(x \cdot x\right)\right)\right)}{2}\]
\frac{e^{x} - e^{-x}}{2}
↓
\frac{\mathsf{fma}\left({x}^{5}, \frac{1}{60}, x \cdot 2 + x \cdot \left(\frac{1}{3} \cdot \left(x \cdot x\right)\right)\right)}{2}
double f(double x) {
        double r2371194 = x;
        double r2371195 = exp(r2371194);
        double r2371196 = -r2371194;
        double r2371197 = exp(r2371196);
        double r2371198 = r2371195 - r2371197;
        double r2371199 = 2.0;
        double r2371200 = r2371198 / r2371199;
        return r2371200;
}
↓
double f(double x) {
        double r2371201 = x;
        double r2371202 = 5.0;
        double r2371203 = pow(r2371201, r2371202);
        double r2371204 = 0.016666666666666666;
        double r2371205 = 2.0;
        double r2371206 = r2371201 * r2371205;
        double r2371207 = 0.3333333333333333;
        double r2371208 = r2371201 * r2371201;
        double r2371209 = r2371207 * r2371208;
        double r2371210 = r2371201 * r2371209;
        double r2371211 = r2371206 + r2371210;
        double r2371212 = fma(r2371203, r2371204, r2371211);
        double r2371213 = 2.0;
        double r2371214 = r2371212 / r2371213;
        return r2371214;
}

Error

Bits error versus x

Derivation

  1. Initial program 58.3

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

  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

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

  5. Applied fma-udef0.6

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

  6. Applied distribute-rgt-in0.6

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

  7. Final simplification0.6

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

Reproduce

herbie shell --seed 2019171 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic sine"
  (/ (- (exp x) (exp (- x))) 2.0))