expq2 (section 3.11)

Average Accuracy: 35.6% → 99.4%

Time: 8.1s

Precision: binary64

Cost: 13120

Math

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

↓

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

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(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	35.6%
Target	36.2%
Herbie	99.4%

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

Derivation

1. Initial program 35.6%

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

2. Simplified 99.4%

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

   [Start] 35.6	\[ \frac{e^{x}}{e^{x} - 1} \]
   expm1-def [=>] 99.4	\[ \frac{e^{x}}{\color{blue}{\mathsf{expm1}\left(x\right)}} \]

3. Applied egg-rr 99.4%

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

4. Final simplification 99.4%

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

Alternatives

Alternative 1
Accuracy	98.7%
Cost	13124

\[\begin{array}{l} \mathbf{if}\;e^{x} \leq 5 \cdot 10^{-53}:\\ \;\;\;\;e^{x} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\left(1.5 + \left(\frac{1}{x} + x \cdot 0.08333333333333333\right)\right) + -1\\ \end{array} \]

Alternative 2
Accuracy	99.4%
Cost	12992

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

Alternative 3
Accuracy	98.3%
Cost	6848

\[e^{x} \cdot \left(\frac{1}{x} + -0.5\right) \]

Alternative 4
Accuracy	66.5%
Cost	192

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

Alternative 5
Accuracy	3.3%
Cost	64

\[0.5 \]

Reproduce

herbie shell --seed 2023129 
(FPCore (x)
  :name "expq2 (section 3.11)"
  :precision binary64

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

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