expq2 (section 3.11)

Average Error: 64.31% → 0.64%

Time: 6.3s

Precision: binary64

Cost: 13184

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

Python

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

↓

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

Julia

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

↓

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

Wolfram

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

↓

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

Target

Original	64.31%
Target	63.7%
Herbie	0.64%

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

Derivation

1. Initial program 64.31

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

2. Simplified 0.63

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

   [Start] 64.31	\[ \frac{e^{x}}{e^{x} - 1} \]
   expm1-def [=>] 0.63	\[ \frac{e^{x}}{\color{blue}{\mathsf{expm1}\left(x\right)}} \]

3. Applied egg-rr 0.69

   \[\leadsto \color{blue}{\frac{{\left(\sqrt[3]{e^{x}}\right)}^{2}}{1} \cdot \frac{\sqrt[3]{e^{x}}}{\mathsf{expm1}\left(x\right)}} \]

4. Applied egg-rr 67.29

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

5. Simplified 0.64

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

   [Start] 67.29	\[ \frac{\frac{1}{\sqrt{\mathsf{expm1}\left(x\right)}}}{\frac{1}{\frac{e^{x}}{\sqrt{\mathsf{expm1}\left(x\right)}}}} \]
   associate-/l/ [=>] 67.24	\[ \color{blue}{\frac{1}{\frac{1}{\frac{e^{x}}{\sqrt{\mathsf{expm1}\left(x\right)}}} \cdot \sqrt{\mathsf{expm1}\left(x\right)}}} \]
   associate-/r/ [=>] 67.22	\[ \frac{1}{\color{blue}{\left(\frac{1}{e^{x}} \cdot \sqrt{\mathsf{expm1}\left(x\right)}\right)} \cdot \sqrt{\mathsf{expm1}\left(x\right)}} \]
   associate-*l* [=>] 67.21	\[ \frac{1}{\color{blue}{\frac{1}{e^{x}} \cdot \left(\sqrt{\mathsf{expm1}\left(x\right)} \cdot \sqrt{\mathsf{expm1}\left(x\right)}\right)}} \]
   rec-exp [=>] 67.21	\[ \frac{1}{\color{blue}{e^{-x}} \cdot \left(\sqrt{\mathsf{expm1}\left(x\right)} \cdot \sqrt{\mathsf{expm1}\left(x\right)}\right)} \]
   rem-square-sqrt [=>] 0.64	\[ \frac{1}{e^{-x} \cdot \color{blue}{\mathsf{expm1}\left(x\right)}} \]

6. Final simplification 0.64

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

Alternatives

Alternative 1
Error	0.63%
Cost	12992

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

Alternative 2
Error	1.33%
Cost	7104

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

Alternative 3
Error	0.88%
Cost	6916

\[\begin{array}{l} \mathbf{if}\;x \leq -0.017:\\ \;\;\;\;\frac{1}{1 - e^{-x}}\\ \mathbf{else}:\\ \;\;\;\;0.5 + \left(\left(x \cdot 0.08333333333333333 + \frac{1}{x}\right) + \frac{x \cdot \left(\left(x \cdot x\right) \cdot 0.001388888888888889\right)}{-1}\right)\\ \end{array} \]

Alternative 4
Error	2.44%
Cost	6720

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

Alternative 5
Error	2.44%
Cost	6592

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

Alternative 6
Error	33.48%
Cost	576

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

Alternative 7
Error	33.56%
Cost	320

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

Alternative 8
Error	33.57%
Cost	192

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

Reproduce

herbie shell --seed 2023102 
(FPCore (x)
  :name "expq2 (section 3.11)"
  :precision binary64

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

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