Average Error: 40.9 → 0.9
Time: 19.3s
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 r64713 = x;
        double r64714 = exp(r64713);
        double r64715 = 1.0;
        double r64716 = r64714 - r64715;
        double r64717 = r64714 / r64716;
        return r64717;
}
↓
double f(double x) {
        double r64718 = x;
        double r64719 = exp(r64718);
        double r64720 = sqrt(r64719);
        double r64721 = 0.041666666666666664;
        double r64722 = 3.0;
        double r64723 = pow(r64718, r64722);
        double r64724 = r64721 * r64723;
        double r64725 = 0.0005208333333333333;
        double r64726 = 5.0;
        double r64727 = pow(r64718, r64726);
        double r64728 = r64725 * r64727;
        double r64729 = r64728 + r64718;
        double r64730 = r64724 + r64729;
        double r64731 = r64720 / r64730;
        return r64731;
}

Error

Bits error versus x

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Results

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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)))