Average Error: 58.0 → 0.8
Time: 21.6s
Precision: 64
\[\frac{e^{x} - e^{-x}}{2}\]
↓
\[\frac{\mathsf{fma}\left(\frac{1}{60}, {x}^{5}, 2 \cdot x + \left(\frac{1}{3} \cdot \left(x \cdot x\right)\right) \cdot x\right)}{2}\]
\frac{e^{x} - e^{-x}}{2}
↓
\frac{\mathsf{fma}\left(\frac{1}{60}, {x}^{5}, 2 \cdot x + \left(\frac{1}{3} \cdot \left(x \cdot x\right)\right) \cdot x\right)}{2}
double f(double x) {
        double r1958078 = x;
        double r1958079 = exp(r1958078);
        double r1958080 = -r1958078;
        double r1958081 = exp(r1958080);
        double r1958082 = r1958079 - r1958081;
        double r1958083 = 2.0;
        double r1958084 = r1958082 / r1958083;
        return r1958084;
}
↓
double f(double x) {
        double r1958085 = 0.016666666666666666;
        double r1958086 = x;
        double r1958087 = 5.0;
        double r1958088 = pow(r1958086, r1958087);
        double r1958089 = 2.0;
        double r1958090 = r1958089 * r1958086;
        double r1958091 = 0.3333333333333333;
        double r1958092 = r1958086 * r1958086;
        double r1958093 = r1958091 * r1958092;
        double r1958094 = r1958093 * r1958086;
        double r1958095 = r1958090 + r1958094;
        double r1958096 = fma(r1958085, r1958088, r1958095);
        double r1958097 = r1958096 / r1958089;
        return r1958097;
}

Error

Bits error versus x

Derivation

  1. Initial program 58.0

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

  2. Taylor expanded around 0 0.8

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

  3. Simplified0.8

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

  5. Applied fma-udef0.8

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

  6. Applied distribute-rgt-in0.8

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

  7. Final simplification0.8

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

Reproduce

herbie shell --seed 2019134 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic sine"
  (/ (- (exp x) (exp (- x))) 2))