Average Error: 39.3 → 1.0
Time: 13.8s
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 r4247099 = x;
        double r4247100 = exp(r4247099);
        double r4247101 = 1.0;
        double r4247102 = r4247100 - r4247101;
        double r4247103 = r4247100 / r4247102;
        return r4247103;
}
↓
double f(double x) {
        double r4247104 = x;
        double r4247105 = exp(r4247104);
        double r4247106 = 0.5;
        double r4247107 = 0.16666666666666666;
        double r4247108 = r4247107 * r4247104;
        double r4247109 = r4247106 + r4247108;
        double r4247110 = r4247104 * r4247104;
        double r4247111 = r4247109 * r4247110;
        double r4247112 = r4247104 + r4247111;
        double r4247113 = r4247105 / r4247112;
        return r4247113;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original39.3
Target38.9
Herbie1.0
\[\frac{1}{1 - e^{-x}}\]

Derivation

  1. Initial program 39.3

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

  2. Taylor expanded around 0 11.8

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

  3. Simplified1.0

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

  4. Final simplification1.0

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

Reproduce

herbie shell --seed 2019165 
(FPCore (x)
  :name "expq2 (section 3.11)"

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

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