?

Average Error: 41.3 → 1.1
Time: 7.0s
Precision: binary64
Cost: 13312

?

\[\frac{e^{x}}{e^{x} - 1} \]
\[\frac{e^{x}}{0.5 \cdot {x}^{2} + x} \]
(FPCore (x) :precision binary64 (/ (exp x) (- (exp x) 1.0)))
(FPCore (x) :precision binary64 (/ (exp x) (+ (* 0.5 (pow x 2.0)) x)))
double code(double x) {
	return exp(x) / (exp(x) - 1.0);
}
double code(double x) {
	return exp(x) / ((0.5 * pow(x, 2.0)) + x);
}
real(8) function code(x)
    real(8), intent (in) :: x
    code = exp(x) / (exp(x) - 1.0d0)
end function
real(8) function code(x)
    real(8), intent (in) :: x
    code = exp(x) / ((0.5d0 * (x ** 2.0d0)) + x)
end function
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) / ((0.5 * Math.pow(x, 2.0)) + x);
}
def code(x):
	return math.exp(x) / (math.exp(x) - 1.0)
def code(x):
	return math.exp(x) / ((0.5 * math.pow(x, 2.0)) + x)
function code(x)
	return Float64(exp(x) / Float64(exp(x) - 1.0))
end
function code(x)
	return Float64(exp(x) / Float64(Float64(0.5 * (x ^ 2.0)) + x))
end
function tmp = code(x)
	tmp = exp(x) / (exp(x) - 1.0);
end
function tmp = code(x)
	tmp = exp(x) / ((0.5 * (x ^ 2.0)) + 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[(N[(0.5 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]
\frac{e^{x}}{e^{x} - 1}
\frac{e^{x}}{0.5 \cdot {x}^{2} + x}

Error?

Try it out?

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original41.3
Target41.0
Herbie1.1
\[\frac{1}{1 - e^{-x}} \]

Derivation?

  1. Initial program 41.3

    \[\frac{e^{x}}{e^{x} - 1} \]
  2. Taylor expanded in x around 0 1.1

    \[\leadsto \frac{e^{x}}{\color{blue}{0.5 \cdot {x}^{2} + x}} \]
  3. Final simplification1.1

    \[\leadsto \frac{e^{x}}{0.5 \cdot {x}^{2} + x} \]

Alternatives

Alternative 1
Error1.5
Cost6720
\[\frac{1}{x} \cdot e^{x} \]
Alternative 2
Error1.5
Cost6592
\[\frac{e^{x}}{x} \]
Alternative 3
Error61.8
Cost192
\[x \cdot 0.25 \]
Alternative 4
Error21.3
Cost192
\[\frac{1}{x} \]

Error

Reproduce?

herbie shell --seed 2023075 
(FPCore (x)
  :name "expq2 (section 3.11)"
  :precision binary64

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

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