Average Error: 58.0 → 0.6
Time: 17.7s
Precision: 64
\[\frac{e^{x} - e^{-x}}{2}\]
↓
\[\frac{\mathsf{fma}\left(\frac{1}{60}, {x}^{5}, x \cdot 2\right) + \left(\left(x \cdot x\right) \cdot \frac{1}{3}\right) \cdot x}{2}\]
\frac{e^{x} - e^{-x}}{2}
↓
\frac{\mathsf{fma}\left(\frac{1}{60}, {x}^{5}, x \cdot 2\right) + \left(\left(x \cdot x\right) \cdot \frac{1}{3}\right) \cdot x}{2}
double f(double x) {
        double r1963182 = x;
        double r1963183 = exp(r1963182);
        double r1963184 = -r1963182;
        double r1963185 = exp(r1963184);
        double r1963186 = r1963183 - r1963185;
        double r1963187 = 2.0;
        double r1963188 = r1963186 / r1963187;
        return r1963188;
}
↓
double f(double x) {
        double r1963189 = 0.016666666666666666;
        double r1963190 = x;
        double r1963191 = 5.0;
        double r1963192 = pow(r1963190, r1963191);
        double r1963193 = 2.0;
        double r1963194 = r1963190 * r1963193;
        double r1963195 = fma(r1963189, r1963192, r1963194);
        double r1963196 = r1963190 * r1963190;
        double r1963197 = 0.3333333333333333;
        double r1963198 = r1963196 * r1963197;
        double r1963199 = r1963198 * r1963190;
        double r1963200 = r1963195 + r1963199;
        double r1963201 = r1963200 / r1963193;
        return r1963201;
}

Error

Bits error versus x

Derivation

  1. Initial program 58.0

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

  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. Simplified0.7

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

  5. Applied fma-udef0.7

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

  7. Applied fma-udef0.7

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

  8. Applied distribute-lft-in0.6

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

  9. Applied associate-+l+0.6

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

  10. Simplified0.6

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

  11. Final simplification0.6

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

Reproduce

herbie shell --seed 2019138 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic sine"
  (/ (- (exp x) (exp (- x))) 2))