Average Error: 40.9 → 0.9
Time: 19.9s
Precision: 64
\[\frac{e^{x}}{e^{x} - 1}\]
↓
\[\frac{\sqrt{e^{x}}}{\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{1920} \cdot {x}^{5} + x\right)}\]
\frac{e^{x}}{e^{x} - 1}
↓
\frac{\sqrt{e^{x}}}{\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{1920} \cdot {x}^{5} + x\right)}
double f(double x) {
        double r61146 = x;
        double r61147 = exp(r61146);
        double r61148 = 1.0;
        double r61149 = r61147 - r61148;
        double r61150 = r61147 / r61149;
        return r61150;
}
↓
double f(double x) {
        double r61151 = x;
        double r61152 = exp(r61151);
        double r61153 = sqrt(r61152);
        double r61154 = 0.041666666666666664;
        double r61155 = 3.0;
        double r61156 = pow(r61151, r61155);
        double r61157 = r61154 * r61156;
        double r61158 = 0.0005208333333333333;
        double r61159 = 5.0;
        double r61160 = pow(r61151, r61159);
        double r61161 = r61158 * r61160;
        double r61162 = r61161 + r61151;
        double r61163 = r61157 + r61162;
        double r61164 = r61153 / r61163;
        return r61164;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original40.9
Target40.5
Herbie0.9
\[\frac{1}{1 - e^{-x}}\]

Derivation

  1. Initial program 40.9

    \[\frac{e^{x}}{e^{x} - 1}\]
  2. Using strategy rm

  3. Applied add-sqr-sqrt40.9

    \[\leadsto \frac{\color{blue}{\sqrt{e^{x}} \cdot \sqrt{e^{x}}}}{e^{x} - 1}\]

  4. Applied associate-/l*40.9

    \[\leadsto \color{blue}{\frac{\sqrt{e^{x}}}{\frac{e^{x} - 1}{\sqrt{e^{x}}}}}\]

  5. Taylor expanded around 0 0.9

    \[\leadsto \frac{\sqrt{e^{x}}}{\color{blue}{\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{1920} \cdot {x}^{5} + x\right)}}\]

  6. Final simplification0.9

    \[\leadsto \frac{\sqrt{e^{x}}}{\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{1920} \cdot {x}^{5} + x\right)}\]

Reproduce

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

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

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