Average Error: 58.0 → 0.7
Time: 13.4s
Precision: 64
\[\frac{e^{x} - e^{-x}}{2}\]
↓
\[\frac{\mathsf{fma}\left(\frac{1}{3}, {x}^{3}, \mathsf{fma}\left(\frac{1}{60}, {x}^{5}, 2 \cdot x\right)\right)}{2}\]
\frac{e^{x} - e^{-x}}{2}
↓
\frac{\mathsf{fma}\left(\frac{1}{3}, {x}^{3}, \mathsf{fma}\left(\frac{1}{60}, {x}^{5}, 2 \cdot x\right)\right)}{2}
double f(double x) {
        double r36564 = x;
        double r36565 = exp(r36564);
        double r36566 = -r36564;
        double r36567 = exp(r36566);
        double r36568 = r36565 - r36567;
        double r36569 = 2.0;
        double r36570 = r36568 / r36569;
        return r36570;
}
↓
double f(double x) {
        double r36571 = 0.3333333333333333;
        double r36572 = x;
        double r36573 = 3.0;
        double r36574 = pow(r36572, r36573);
        double r36575 = 0.016666666666666666;
        double r36576 = 5.0;
        double r36577 = pow(r36572, r36576);
        double r36578 = 2.0;
        double r36579 = r36578 * r36572;
        double r36580 = fma(r36575, r36577, r36579);
        double r36581 = fma(r36571, r36574, r36580);
        double r36582 = 2.0;
        double r36583 = r36581 / r36582;
        return r36583;
}

Error

Bits error versus x

Derivation

  1. Initial program 58.0

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

  2. Taylor expanded around 0 0.7

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

  3. Simplified0.7

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

  4. Final simplification0.7

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

Reproduce

herbie shell --seed 2020047 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2))