Average Error: 39.9 → 1.1
Time: 14.0s
Precision: 64
\[\frac{e^{x}}{e^{x} - 1}\]
↓
\[\frac{e^{x}}{x + \left(\frac{1}{2} + \frac{1}{6} \cdot x\right) \cdot \left(x \cdot x\right)}\]
\frac{e^{x}}{e^{x} - 1}
↓
\frac{e^{x}}{x + \left(\frac{1}{2} + \frac{1}{6} \cdot x\right) \cdot \left(x \cdot x\right)}
double f(double x) {
        double r4088367 = x;
        double r4088368 = exp(r4088367);
        double r4088369 = 1.0;
        double r4088370 = r4088368 - r4088369;
        double r4088371 = r4088368 / r4088370;
        return r4088371;
}
↓
double f(double x) {
        double r4088372 = x;
        double r4088373 = exp(r4088372);
        double r4088374 = 0.5;
        double r4088375 = 0.16666666666666666;
        double r4088376 = r4088375 * r4088372;
        double r4088377 = r4088374 + r4088376;
        double r4088378 = r4088372 * r4088372;
        double r4088379 = r4088377 * r4088378;
        double r4088380 = r4088372 + r4088379;
        double r4088381 = r4088373 / r4088380;
        return r4088381;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original39.9
Target39.5
Herbie1.1
\[\frac{1}{1 - e^{-x}}\]

Derivation

  1. Initial program 39.9

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

  2. Taylor expanded around 0 11.5

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

  3. Simplified1.1

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

  4. Final simplification1.1

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

Reproduce

herbie shell --seed 2019107 
(FPCore (x)
  :name "expq2 (section 3.11)"

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

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