Average Error: 41.5 → 0.0
Time: 2.6s
Precision: binary64
\[\frac{e^{x}}{e^{x} - 1} \]
↓
\[\frac{-1}{\mathsf{expm1}\left(-x\right)} \]
(FPCore (x) :precision binary64 (/ (exp x) (- (exp x) 1.0)))
↓
(FPCore (x) :precision binary64 (/ -1.0 (expm1 (- x))))
double code(double x) {
	return exp(x) / (exp(x) - 1.0);
}
↓
double code(double x) {
	return -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 -1.0 / Math.expm1(-x);
}

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

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

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

\frac{e^{x}}{e^{x} - 1}
↓
\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

Original41.5
Target41.0
Herbie0.0
\[\frac{1}{1 - e^{-x}} \]

Derivation

  1. Initial program 41.5

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

  2. Simplified0.5

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

  3. Applied egg-rr0.5

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

  4. Taylor expanded in x around inf 41.5

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

  5. Simplified0.0

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

  6. Taylor expanded in x around inf 41.0

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

  7. Simplified0.0

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

  8. Final simplification0.0

    \[\leadsto \frac{-1}{\mathsf{expm1}\left(-x\right)} \]

Reproduce

herbie shell --seed 2022150 
(FPCore (x)
  :name "expq2 (section 3.11)"
  :precision binary64

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

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