Average Error: 40.9 → 1.1
Time: 17.8s
Precision: 64
\[\frac{e^{x}}{e^{x} - 1}\]
↓
\[\frac{e^{x}}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, \frac{1}{6}, \frac{1}{2}\right), x\right)}\]
\frac{e^{x}}{e^{x} - 1}
↓
\frac{e^{x}}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, \frac{1}{6}, \frac{1}{2}\right), x\right)}
double f(double x) {
        double r60946 = x;
        double r60947 = exp(r60946);
        double r60948 = 1.0;
        double r60949 = r60947 - r60948;
        double r60950 = r60947 / r60949;
        return r60950;
}
↓
double f(double x) {
        double r60951 = x;
        double r60952 = exp(r60951);
        double r60953 = 0.16666666666666666;
        double r60954 = 0.5;
        double r60955 = fma(r60951, r60953, r60954);
        double r60956 = r60951 * r60955;
        double r60957 = fma(r60951, r60956, r60951);
        double r60958 = r60952 / r60957;
        return r60958;
}

Error

Bits error versus x

Target

Original40.9
Target40.5
Herbie1.1
\[\frac{1}{1 - e^{-x}}\]

Derivation

  1. Initial program 40.9

    \[\frac{e^{x}}{e^{x} - 1}\]

  2. Taylor expanded around 0 12.1

    \[\leadsto \frac{e^{x}}{\color{blue}{\frac{1}{2} \cdot {x}^{2} + \left(\frac{1}{6} \cdot {x}^{3} + x\right)}}\]

  3. Simplified1.1

    \[\leadsto \frac{e^{x}}{\color{blue}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, \frac{1}{6}, \frac{1}{2}\right), x\right)}}\]

  4. Final simplification1.1

    \[\leadsto \frac{e^{x}}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, \frac{1}{6}, \frac{1}{2}\right), x\right)}\]

Reproduce

herbie shell --seed 2019325 +o rules:numerics
(FPCore (x)
  :name "expq2 (section 3.11)"
  :precision binary64

  :herbie-target
  (/ 1 (- 1 (exp (- x))))

  (/ (exp x) (- (exp x) 1)))