Average Error: 41.1 → 0.6
Time: 3.5s
Precision: 64
\[\frac{e^{x}}{e^{x} - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.0014783118370881984:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(-1, 1, e^{x + x}\right)}{{\left(e^{x}\right)}^{3} + {1}^{3}}} \cdot \frac{e^{x}}{\mathsf{fma}\left(1, 1 - e^{x}, e^{2 \cdot x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{x}\right) + \frac{1}{2}\\ \end{array}\]
\frac{e^{x}}{e^{x} - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.0014783118370881984:\\
\;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(-1, 1, e^{x + x}\right)}{{\left(e^{x}\right)}^{3} + {1}^{3}}} \cdot \frac{e^{x}}{\mathsf{fma}\left(1, 1 - e^{x}, e^{2 \cdot x}\right)}\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{x}\right) + \frac{1}{2}\\

\end{array}
double code(double x) {
	return (exp(x) / (exp(x) - 1.0));
}
↓
double code(double x) {
	double VAR;
	if ((x <= -0.0014783118370881984)) {
		VAR = ((1.0 / (fma(-1.0, 1.0, exp((x + x))) / (pow(exp(x), 3.0) + pow(1.0, 3.0)))) * (exp(x) / fma(1.0, (1.0 - exp(x)), exp((2.0 * x)))));
	} else {
		VAR = (fma(0.08333333333333333, x, (1.0 / x)) + 0.5);
	}
	return VAR;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original41.1
Target40.7
Herbie0.6
\[\frac{1}{1 - e^{-x}}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -0.0014783118370881984

    1. Initial program 0.0

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

    3. Applied flip--0.0

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

    4. Simplified0.0

      \[\leadsto \frac{e^{x}}{\frac{\color{blue}{\mathsf{fma}\left(-1, 1, e^{x + x}\right)}}{e^{x} + 1}}\]
    5. Using strategy rm

    6. Applied flip3-+0.0

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

    7. Applied associate-/r/0.0

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

    8. Applied *-un-lft-identity0.0

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

    9. Applied times-frac0.0

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

    10. Simplified0.0

      \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(-1, 1, e^{x + x}\right)}{{\left(e^{x}\right)}^{3} + {1}^{3}}} \cdot \color{blue}{\frac{e^{x}}{\mathsf{fma}\left(1, 1 - e^{x}, e^{2 \cdot x}\right)}}\]

    if -0.0014783118370881984 < x

    1. Initial program 61.9

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

    2. Taylor expanded around 0 0.9

      \[\leadsto \color{blue}{\frac{1}{2} + \left(\frac{1}{12} \cdot x + \frac{1}{x}\right)}\]

    3. Simplified0.9

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{x}\right) + \frac{1}{2}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.0014783118370881984:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(-1, 1, e^{x + x}\right)}{{\left(e^{x}\right)}^{3} + {1}^{3}}} \cdot \frac{e^{x}}{\mathsf{fma}\left(1, 1 - e^{x}, e^{2 \cdot x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{x}\right) + \frac{1}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020106 +o rules:numerics
(FPCore (x)
  :name "expq2 (section 3.11)"
  :precision binary64

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

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