Average Error: 58.0 → 0.7
Time: 21.5s
Precision: 64
\[\frac{e^{x} - e^{-x}}{2}\]
↓
\[\frac{\mathsf{fma}\left(\left({x}^{5}\right), \frac{1}{60}, \left(\mathsf{fma}\left(x, \left(\frac{1}{3} \cdot x\right), 2\right) \cdot x\right)\right)}{2}\]
\frac{e^{x} - e^{-x}}{2}
↓
\frac{\mathsf{fma}\left(\left({x}^{5}\right), \frac{1}{60}, \left(\mathsf{fma}\left(x, \left(\frac{1}{3} \cdot x\right), 2\right) \cdot x\right)\right)}{2}
double f(double x) {
        double r1700932 = x;
        double r1700933 = exp(r1700932);
        double r1700934 = -r1700932;
        double r1700935 = exp(r1700934);
        double r1700936 = r1700933 - r1700935;
        double r1700937 = 2.0;
        double r1700938 = r1700936 / r1700937;
        return r1700938;
}
↓
double f(double x) {
        double r1700939 = x;
        double r1700940 = 5.0;
        double r1700941 = pow(r1700939, r1700940);
        double r1700942 = 0.016666666666666666;
        double r1700943 = 0.3333333333333333;
        double r1700944 = r1700943 * r1700939;
        double r1700945 = 2.0;
        double r1700946 = fma(r1700939, r1700944, r1700945);
        double r1700947 = r1700946 * r1700939;
        double r1700948 = fma(r1700941, r1700942, r1700947);
        double r1700949 = r1700948 / r1700945;
        return r1700949;
}

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

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

  4. Taylor expanded around 0 0.6

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

  5. Simplified0.7

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

  6. Final simplification0.7

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

Reproduce

herbie shell --seed 2019133 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic sine"
  (/ (- (exp x) (exp (- x))) 2))