Average Error: 41.6 → 0.6
Time: 3.1s
Precision: binary64
\[\frac{e^{x}}{e^{x} - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;e^{x} \leq 0.9870522792702024:\\ \;\;\;\;\left(\sqrt[3]{e^{x}} \cdot \frac{\sqrt[3]{e^{x}}}{\sqrt{e^{x}} + \sqrt{1}}\right) \cdot \frac{\sqrt[3]{e^{x}}}{\sqrt{e^{x}} - \sqrt{1}}\\ \mathbf{else}:\\ \;\;\;\;0.5 + \left(x \cdot 0.08333333333333333 + \frac{1}{x}\right)\\ \end{array}\]
\frac{e^{x}}{e^{x} - 1}
↓
\begin{array}{l}
\mathbf{if}\;e^{x} \leq 0.9870522792702024:\\
\;\;\;\;\left(\sqrt[3]{e^{x}} \cdot \frac{\sqrt[3]{e^{x}}}{\sqrt{e^{x}} + \sqrt{1}}\right) \cdot \frac{\sqrt[3]{e^{x}}}{\sqrt{e^{x}} - \sqrt{1}}\\

\mathbf{else}:\\
\;\;\;\;0.5 + \left(x \cdot 0.08333333333333333 + \frac{1}{x}\right)\\

\end{array}
(FPCore (x) :precision binary64 (/ (exp x) (- (exp x) 1.0)))
↓
(FPCore (x)
 :precision binary64
 (if (<= (exp x) 0.9870522792702024)
   (*
    (* (cbrt (exp x)) (/ (cbrt (exp x)) (+ (sqrt (exp x)) (sqrt 1.0))))
    (/ (cbrt (exp x)) (- (sqrt (exp x)) (sqrt 1.0))))
   (+ 0.5 (+ (* x 0.08333333333333333) (/ 1.0 x)))))
double code(double x) {
	return (((double) exp(x)) / ((double) (((double) exp(x)) - 1.0)));
}
↓
double code(double x) {
	double tmp;
	if ((((double) exp(x)) <= 0.9870522792702024)) {
		tmp = ((double) (((double) (((double) cbrt(((double) exp(x)))) * (((double) cbrt(((double) exp(x)))) / ((double) (((double) sqrt(((double) exp(x)))) + ((double) sqrt(1.0))))))) * (((double) cbrt(((double) exp(x)))) / ((double) (((double) sqrt(((double) exp(x)))) - ((double) sqrt(1.0)))))));
	} else {
		tmp = ((double) (0.5 + ((double) (((double) (x * 0.08333333333333333)) + (1.0 / x)))));
	}
	return tmp;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original41.6
Target41.2
Herbie0.6
\[\frac{1}{1 - e^{-x}}\]

Derivation

  1. Split input into 2 regimes

  2. if (exp x) < 0.987052279270202448

    1. Initial program Error: 0.0 bits

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

    3. Applied add-sqr-sqrtError: 0.0 bits

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

    4. Applied add-sqr-sqrtError: 0.0 bits

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

    5. Applied difference-of-squaresError: 0.0 bits

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

    6. Applied add-cube-cbrtError: 0.0 bits

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

    7. Applied times-fracError: 0.0 bits

      \[\leadsto \color{blue}{\frac{\sqrt[3]{e^{x}} \cdot \sqrt[3]{e^{x}}}{\sqrt{e^{x}} + \sqrt{1}} \cdot \frac{\sqrt[3]{e^{x}}}{\sqrt{e^{x}} - \sqrt{1}}}\]

    8. SimplifiedError: 0.0 bits

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

    if 0.987052279270202448 < (exp x)

    1. Initial program Error: 62.1 bits

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

    2. Taylor expanded around 0 Error: 0.9 bits

      \[\leadsto \color{blue}{0.5 + \left(0.08333333333333333 \cdot x + \frac{1}{x}\right)}\]

    3. SimplifiedError: 0.9 bits

      \[\leadsto \color{blue}{0.5 + \left(x \cdot 0.08333333333333333 + \frac{1}{x}\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplificationError: 0.6 bits

    \[\leadsto \begin{array}{l} \mathbf{if}\;e^{x} \leq 0.9870522792702024:\\ \;\;\;\;\left(\sqrt[3]{e^{x}} \cdot \frac{\sqrt[3]{e^{x}}}{\sqrt{e^{x}} + \sqrt{1}}\right) \cdot \frac{\sqrt[3]{e^{x}}}{\sqrt{e^{x}} - \sqrt{1}}\\ \mathbf{else}:\\ \;\;\;\;0.5 + \left(x \cdot 0.08333333333333333 + \frac{1}{x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020204 
(FPCore (x)
  :name "expq2 (section 3.11)"
  :precision binary64

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

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