Average Error: 58.0 → 0.6
Time: 17.8s
Precision: 64
\[\frac{e^{x} - e^{-x}}{2}\]
↓
\[\frac{\mathsf{fma}\left({x}^{5}, \frac{1}{60}, x \cdot 2 + x \cdot \left(\frac{1}{3} \cdot \left(x \cdot x\right)\right)\right)}{2}\]
\frac{e^{x} - e^{-x}}{2}
↓
\frac{\mathsf{fma}\left({x}^{5}, \frac{1}{60}, x \cdot 2 + x \cdot \left(\frac{1}{3} \cdot \left(x \cdot x\right)\right)\right)}{2}
double f(double x) {
        double r1976297 = x;
        double r1976298 = exp(r1976297);
        double r1976299 = -r1976297;
        double r1976300 = exp(r1976299);
        double r1976301 = r1976298 - r1976300;
        double r1976302 = 2.0;
        double r1976303 = r1976301 / r1976302;
        return r1976303;
}
↓
double f(double x) {
        double r1976304 = x;
        double r1976305 = 5.0;
        double r1976306 = pow(r1976304, r1976305);
        double r1976307 = 0.016666666666666666;
        double r1976308 = 2.0;
        double r1976309 = r1976304 * r1976308;
        double r1976310 = 0.3333333333333333;
        double r1976311 = r1976304 * r1976304;
        double r1976312 = r1976310 * r1976311;
        double r1976313 = r1976304 * r1976312;
        double r1976314 = r1976309 + r1976313;
        double r1976315 = fma(r1976306, r1976307, r1976314);
        double r1976316 = 2.0;
        double r1976317 = r1976315 / r1976316;
        return r1976317;
}

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. Simplified0.6

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

  5. Applied fma-udef0.6

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

  6. Applied distribute-lft-in0.6

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

  7. Final simplification0.6

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

Reproduce

herbie shell --seed 2019170 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic sine"
  (/ (- (exp x) (exp (- x))) 2.0))