Average Error: 41.4 → 0.8
Time: 2.7s
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 code(double x) {
	return ((double) (((double) exp(x)) / ((double) (((double) exp(x)) - 1.0))));
}
↓
double code(double x) {
	return ((double) (((double) sqrt(((double) exp(x)))) / ((double) (((double) (0.041666666666666664 * ((double) pow(x, 3.0)))) + ((double) (((double) (0.0005208333333333333 * ((double) pow(x, 5.0)))) + x))))));
}

Error

Bits error versus x

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Results

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Target

Original41.4
Target41.0
Herbie0.8
\[\frac{1}{1 - e^{-x}}\]

Derivation

  1. Initial program 41.4

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

  3. Applied add-sqr-sqrt41.4

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

  4. Applied associate-/l*41.4

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

  5. Taylor expanded around 0 0.8

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

  6. Final simplification0.8

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

Reproduce

herbie shell --seed 2020126 
(FPCore (x)
  :name "expq2 (section 3.11)"
  :precision binary64

  :herbie-target
  (/ 1.0 (- 1.0 (exp (- x))))

  (/ (exp x) (- (exp x) 1.0)))