Average Error: 57.9 → 0.6
Time: 24.7s
Precision: 64
\[\frac{e^{x} - e^{-x}}{2}\]
↓
\[\frac{\mathsf{fma}\left(\frac{1}{60}, \left({x}^{5}\right), \left(2 \cdot x + \left(\frac{1}{3} \cdot \left(x \cdot x\right)\right) \cdot x\right)\right)}{2}\]
\frac{e^{x} - e^{-x}}{2}
↓
\frac{\mathsf{fma}\left(\frac{1}{60}, \left({x}^{5}\right), \left(2 \cdot x + \left(\frac{1}{3} \cdot \left(x \cdot x\right)\right) \cdot x\right)\right)}{2}
double f(double x) {
        double r9909587 = x;
        double r9909588 = exp(r9909587);
        double r9909589 = -r9909587;
        double r9909590 = exp(r9909589);
        double r9909591 = r9909588 - r9909590;
        double r9909592 = 2.0;
        double r9909593 = r9909591 / r9909592;
        return r9909593;
}
↓
double f(double x) {
        double r9909594 = 0.016666666666666666;
        double r9909595 = x;
        double r9909596 = 5.0;
        double r9909597 = pow(r9909595, r9909596);
        double r9909598 = 2.0;
        double r9909599 = r9909598 * r9909595;
        double r9909600 = 0.3333333333333333;
        double r9909601 = r9909595 * r9909595;
        double r9909602 = r9909600 * r9909601;
        double r9909603 = r9909602 * r9909595;
        double r9909604 = r9909599 + r9909603;
        double r9909605 = fma(r9909594, r9909597, r9909604);
        double r9909606 = r9909605 / r9909598;
        return r9909606;
}

Error

Bits error versus x

Derivation

  1. Initial program 57.9

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

  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

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

  5. Applied fma-udef0.6

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

  6. Applied distribute-rgt-in0.6

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

  7. Final simplification0.6

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

Reproduce

herbie shell --seed 2019125 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic sine"
  (/ (- (exp x) (exp (- x))) 2))