Average Error: 41.1 → 0.5
Time: 2.8s
Precision: binary64
\[\frac{e^{x}}{e^{x} - 1} \]
↓
\[e^{x} \cdot \frac{1}{\mathsf{expm1}\left(x\right)} \]
(FPCore (x) :precision binary64 (/ (exp x) (- (exp x) 1.0)))
↓
(FPCore (x) :precision binary64 (* (exp x) (/ 1.0 (expm1 x))))
double code(double x) {
	return exp(x) / (exp(x) - 1.0);
}
↓
double code(double x) {
	return exp(x) * (1.0 / 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) * (1.0 / Math.expm1(x));
}

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

function code(x)
	return Float64(exp(x) / Float64(exp(x) - 1.0))
end
↓
function code(x)
	return Float64(exp(x) * Float64(1.0 / 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[(1.0 / N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

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

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original41.1
Target40.7
Herbie0.5
\[\frac{1}{1 - e^{-x}} \]

Derivation

  1. Initial program 41.1

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

  2. Simplified0.5

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

  3. Applied div-inv_binary640.5

    \[\leadsto \color{blue}{e^{x} \cdot \frac{1}{\mathsf{expm1}\left(x\right)}} \]

  4. Final simplification0.5

    \[\leadsto e^{x} \cdot \frac{1}{\mathsf{expm1}\left(x\right)} \]

Reproduce

herbie shell --seed 2022131 
(FPCore (x)
  :name "expq2 (section 3.11)"
  :precision binary64

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

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