Average Error: 40.9 → 0.9
Time: 19.0s
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 r57654 = x;
        double r57655 = exp(r57654);
        double r57656 = 1.0;
        double r57657 = r57655 - r57656;
        double r57658 = r57655 / r57657;
        return r57658;
}
↓
double f(double x) {
        double r57659 = x;
        double r57660 = exp(r57659);
        double r57661 = sqrt(r57660);
        double r57662 = 0.041666666666666664;
        double r57663 = 3.0;
        double r57664 = pow(r57659, r57663);
        double r57665 = r57662 * r57664;
        double r57666 = 0.0005208333333333333;
        double r57667 = 5.0;
        double r57668 = pow(r57659, r57667);
        double r57669 = r57666 * r57668;
        double r57670 = r57669 + r57659;
        double r57671 = r57665 + r57670;
        double r57672 = r57661 / r57671;
        return r57672;
}

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)))