Average Error: 58.1 → 0.6
Time: 11.3s
Precision: 64
\[\frac{e^{x} - e^{-x}}{2}\]
↓
\[\frac{\mathsf{fma}\left(2, x, \mathsf{fma}\left({x}^{5}, \frac{1}{60}, \frac{1}{3} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right)\right)}{2}\]
\frac{e^{x} - e^{-x}}{2}
↓
\frac{\mathsf{fma}\left(2, x, \mathsf{fma}\left({x}^{5}, \frac{1}{60}, \frac{1}{3} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right)\right)}{2}
double f(double x) {
        double r1177425 = x;
        double r1177426 = exp(r1177425);
        double r1177427 = -r1177425;
        double r1177428 = exp(r1177427);
        double r1177429 = r1177426 - r1177428;
        double r1177430 = 2.0;
        double r1177431 = r1177429 / r1177430;
        return r1177431;
}
↓
double f(double x) {
        double r1177432 = 2.0;
        double r1177433 = x;
        double r1177434 = 5.0;
        double r1177435 = pow(r1177433, r1177434);
        double r1177436 = 0.016666666666666666;
        double r1177437 = 0.3333333333333333;
        double r1177438 = r1177433 * r1177433;
        double r1177439 = r1177438 * r1177433;
        double r1177440 = r1177437 * r1177439;
        double r1177441 = fma(r1177435, r1177436, r1177440);
        double r1177442 = fma(r1177432, r1177433, r1177441);
        double r1177443 = r1177442 / r1177432;
        return r1177443;
}

Error

Bits error versus x

Derivation

  1. Initial program 58.1

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

  3. Applied flip--58.1

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

  4. Taylor expanded around 0 0.6

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

  5. Simplified0.6

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

  6. Final simplification0.6

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

Reproduce

herbie shell --seed 2019153 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic sine"
  (/ (- (exp x) (exp (- x))) 2))