expq2 (section 3.11)

Average Accuracy: 34.6% → 99.3%

Time: 6.5s

Precision: binary64

Cost: 13184

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

Java

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)) / Math.exp(-x);
}

Python

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

↓

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

Julia

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

↓

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

Wolfram

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

↓

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

Target

Original	34.6%
Target	35.2%
Herbie	99.3%

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

Derivation

1. Initial program 34.6%

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

2. Simplified 99.3%

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

   [Start] 34.6	\[ \frac{e^{x}}{e^{x} - 1} \]
   expm1-def [=>] 99.3	\[ \frac{e^{x}}{\color{blue}{\mathsf{expm1}\left(x\right)}} \]

3. Applied egg-rr 99.3%

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

   [Start] 99.3	\[ \frac{e^{x}}{\mathsf{expm1}\left(x\right)} \]
   clear-num [=>] 99.3	\[ \color{blue}{\frac{1}{\frac{\mathsf{expm1}\left(x\right)}{e^{x}}}} \]
   inv-pow [=>] 99.3	\[ \color{blue}{{\left(\frac{\mathsf{expm1}\left(x\right)}{e^{x}}\right)}^{-1}} \]

4. Applied egg-rr 99.3%

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

   [Start] 99.3	\[ {\left(\frac{\mathsf{expm1}\left(x\right)}{e^{x}}\right)}^{-1} \]
   unpow-1 [=>] 99.3	\[ \color{blue}{\frac{1}{\frac{\mathsf{expm1}\left(x\right)}{e^{x}}}} \]
   div-inv [=>] 99.3	\[ \frac{1}{\color{blue}{\mathsf{expm1}\left(x\right) \cdot \frac{1}{e^{x}}}} \]
   associate-/r* [=>] 99.3	\[ \color{blue}{\frac{\frac{1}{\mathsf{expm1}\left(x\right)}}{\frac{1}{e^{x}}}} \]
   rec-exp [=>] 99.3	\[ \frac{\frac{1}{\mathsf{expm1}\left(x\right)}}{\color{blue}{e^{-x}}} \]

5. Final simplification 99.3%

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

Alternatives

Alternative 1
Accuracy	99.3%
Cost	12992

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

Alternative 2
Accuracy	99.1%
Cost	6916

\[\begin{array}{l} \mathbf{if}\;x \leq -0.021:\\ \;\;\;\;\frac{1}{1 - e^{-x}}\\ \mathbf{else}:\\ \;\;\;\;0.5 + \left(\frac{x \cdot \left(\left(x \cdot x\right) \cdot 0.001388888888888889\right)}{-1} + \left(x \cdot 0.08333333333333333 + \frac{1}{x}\right)\right)\\ \end{array} \]

Alternative 3
Accuracy	97.9%
Cost	6592

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

Alternative 4
Accuracy	67.3%
Cost	576

\[0.5 + \left(x \cdot 0.08333333333333333 + \frac{1}{x}\right) \]

Alternative 5
Accuracy	67.3%
Cost	320

\[0.5 + \frac{1}{x} \]

Alternative 6
Accuracy	67.3%
Cost	192

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

Alternative 7
Accuracy	3.3%
Cost	64

\[0.5 \]

Alternative 8
Accuracy	3.9%
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023133 
(FPCore (x)
  :name "expq2 (section 3.11)"
  :precision binary64

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

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