Average Error: 58.1 → 0.5
Time: 18.0s
Precision: 64
\[\frac{e^{x} - e^{-x}}{2}\]
↓
\[\frac{\mathsf{fma}\left(2, x, \left(\mathsf{fma}\left(\left({x}^{5}\right), \frac{1}{60}, \left(\frac{1}{3} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right)\right)\right)\right)}{2}\]
\frac{e^{x} - e^{-x}}{2}
↓
\frac{\mathsf{fma}\left(2, x, \left(\mathsf{fma}\left(\left({x}^{5}\right), \frac{1}{60}, \left(\frac{1}{3} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right)\right)\right)\right)}{2}
double f(double x) {
        double r2166561 = x;
        double r2166562 = exp(r2166561);
        double r2166563 = -r2166561;
        double r2166564 = exp(r2166563);
        double r2166565 = r2166562 - r2166564;
        double r2166566 = 2.0;
        double r2166567 = r2166565 / r2166566;
        return r2166567;
}
↓
double f(double x) {
        double r2166568 = 2.0;
        double r2166569 = x;
        double r2166570 = 5.0;
        double r2166571 = pow(r2166569, r2166570);
        double r2166572 = 0.016666666666666666;
        double r2166573 = 0.3333333333333333;
        double r2166574 = r2166569 * r2166569;
        double r2166575 = r2166574 * r2166569;
        double r2166576 = r2166573 * r2166575;
        double r2166577 = fma(r2166571, r2166572, r2166576);
        double r2166578 = fma(r2166568, r2166569, r2166577);
        double r2166579 = r2166578 / r2166568;
        return r2166579;
}

Error

Bits error versus x

Derivation

  1. Initial program 58.1

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

  3. Applied flip--58.2

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

  4. Taylor expanded around 0 0.5

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

  5. Simplified0.5

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

  6. Final simplification0.5

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

Reproduce

herbie shell --seed 2019132 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic sine"
  (/ (- (exp x) (exp (- x))) 2))