Average Error: 41.4 → 0.5
Time: 6.7s
Precision: binary64
Cost: 12992
\[\frac{e^{x}}{e^{x} - 1} \]
↓
\[\frac{e^{x}}{\mathsf{expm1}\left(x\right)} \]
(FPCore (x) :precision binary64 (/ (exp x) (- (exp x) 1.0)))
↓
(FPCore (x) :precision binary64 (/ (exp x) (expm1 x)))
double code(double x) {
	return exp(x) / (exp(x) - 1.0);
}
↓
double code(double x) {
	return exp(x) / expm1(x);
}

public static double code(double x) {
	return Math.exp(x) / (Math.exp(x) - 1.0);
}
↓
public static double code(double x) {
	return Math.exp(x) / Math.expm1(x);
}

def code(x):
	return math.exp(x) / (math.exp(x) - 1.0)
↓
def code(x):
	return math.exp(x) / math.expm1(x)

function code(x)
	return Float64(exp(x) / Float64(exp(x) - 1.0))
end
↓
function code(x)
	return Float64(exp(x) / expm1(x))
end

code[x_] := N[(N[Exp[x], $MachinePrecision] / N[(N[Exp[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]
↓
code[x_] := N[(N[Exp[x], $MachinePrecision] / N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision]

\frac{e^{x}}{e^{x} - 1}
↓
\frac{e^{x}}{\mathsf{expm1}\left(x\right)}

Error

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original41.4
Target40.9
Herbie0.5
\[\frac{1}{1 - e^{-x}} \]

Derivation

  1. Initial program 41.4

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

  2. Simplified0.5

    \[\leadsto \color{blue}{\frac{e^{x}}{\mathsf{expm1}\left(x\right)}} \]
    Proof

    [Start]41.4

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

    expm1-def [=>]0.5

    \[ \frac{e^{x}}{\color{blue}{\mathsf{expm1}\left(x\right)}} \]

  3. Final simplification0.5

    \[\leadsto \frac{e^{x}}{\mathsf{expm1}\left(x\right)} \]

Alternatives

Alternative 1
Error0.9
Cost13124
\[\begin{array}{l} \mathbf{if}\;e^{x} \leq 0:\\ \;\;\;\;e^{x} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 0.08333333333333333 + \frac{1}{x}\right) + 0.5\\ \end{array} \]
Alternative 2
Error0.9
Cost7104
\[e^{x} \cdot \left(\left(x \cdot 0.08333333333333333 + \frac{1}{x}\right) + -0.5\right) \]
Alternative 3
Error1.2
Cost6848
\[e^{x} \cdot \left(\frac{1}{x} + -0.5\right) \]
Alternative 4
Error10.7
Cost836
\[\begin{array}{l} \mathbf{if}\;x \leq -4:\\ \;\;\;\;\frac{\frac{-1}{x \cdot x}}{0.5 + \frac{-1}{x}}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 0.08333333333333333 + \frac{1}{x}\right) + 0.5\\ \end{array} \]
Alternative 5
Error11.3
Cost832
\[\frac{1}{x} \cdot \frac{\frac{1}{\frac{1}{x} + -0.5}}{x} \]
Alternative 6
Error10.7
Cost708
\[\begin{array}{l} \mathbf{if}\;x \leq -4.5:\\ \;\;\;\;\frac{-2}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 0.08333333333333333 + \frac{1}{x}\right) + 0.5\\ \end{array} \]
Alternative 7
Error10.9
Cost452
\[\begin{array}{l} \mathbf{if}\;x \leq -1.8:\\ \;\;\;\;\frac{-2}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} + 0.5\\ \end{array} \]
Alternative 8
Error21.2
Cost320
\[\frac{1}{x} + 0.5 \]
Alternative 9
Error21.3
Cost192
\[\frac{1}{x} \]
Alternative 10
Error61.8
Cost64
\[0.5 \]

Error

Reproduce

herbie shell --seed 2022356 
(FPCore (x)
  :name "expq2 (section 3.11)"
  :precision binary64

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

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