expq2 (section 3.11)

Average Error: 65.38% → 0.72%

Time: 6.1s

Precision: binary64

Cost: 19456

Math

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

↓

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

FPCore

(FPCore (x) :precision binary64 (/ (exp x) (- (exp x) 1.0)))

↓

(FPCore (x) :precision binary64 (pow (/ (expm1 x) (exp x)) -1.0))

C

double code(double x) {
	return exp(x) / (exp(x) - 1.0);
}

↓

double code(double x) {
	return pow((expm1(x) / exp(x)), -1.0);
}

Java

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

↓

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

Python

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

↓

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

Julia

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

↓

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

Wolfram

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

↓

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

Target

Original	65.38%
Target	64.69%
Herbie	0.72%

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

Derivation

1. Initial program 65.38

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

2. Simplified 0.72

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

   [Start] 65.38	\[ \frac{e^{x}}{e^{x} - 1} \]
   expm1-def [=>] 0.72	\[ \frac{e^{x}}{\color{blue}{\mathsf{expm1}\left(x\right)}} \]

3. Applied egg-rr 0.72

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

4. Final simplification 0.72

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

Alternatives

Alternative 1
Error	0.72%
Cost	12992

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

Alternative 2
Error	0.84%
Cost	6916

\[\begin{array}{l} \mathbf{if}\;x \leq -0.00175:\\ \;\;\;\;\frac{1}{1 - e^{-x}}\\ \mathbf{else}:\\ \;\;\;\;0.5 + \left(x \cdot 0.08333333333333333 + \frac{1}{x}\right)\\ \end{array} \]

Alternative 3
Error	32.71%
Cost	192

\[\frac{1}{x} \]

Reproduce

herbie shell --seed 2023089 
(FPCore (x)
  :name "expq2 (section 3.11)"
  :precision binary64

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

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