expq2 (section 3.11)

Average Error: 41.4 → 0.4

Time: 6.7s

Precision: binary64

Cost: 26048

Math

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

↓

\[\frac{\sqrt{e^{x}}}{\frac{\mathsf{expm1}\left(x\right)}{e^{x \cdot 0.5}}} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (/ (sqrt (exp x)) (/ (expm1 x) (exp (* x 0.5)))))

C

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

↓

double code(double x) {
	return sqrt(exp(x)) / (expm1(x) / exp((x * 0.5)));
}

Java

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

↓

public static double code(double x) {
	return Math.sqrt(Math.exp(x)) / (Math.expm1(x) / Math.exp((x * 0.5)));
}

Python

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

↓

def code(x):
	return math.sqrt(math.exp(x)) / (math.expm1(x) / math.exp((x * 0.5)))

Julia

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

↓

function code(x)
	return Float64(sqrt(exp(x)) / Float64(expm1(x) / exp(Float64(x * 0.5))))
end

Wolfram

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

↓

code[x_] := N[(N[Sqrt[N[Exp[x], $MachinePrecision]], $MachinePrecision] / N[(N[(Exp[x] - 1), $MachinePrecision] / N[Exp[N[(x * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	41.4
Target	41.0
Herbie	0.4

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

Derivation

1. Initial program 41.4

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

2. Simplified 0.4

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

   [Start] 41.4	\[ \frac{e^{x}}{e^{x} - 1} \]
   expm1-def [=>] 0.4	\[ \frac{e^{x}}{\color{blue}{\mathsf{expm1}\left(x\right)}} \]

3. Applied egg-rr 0.4

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

4. Applied egg-rr 0.4

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

5. Applied egg-rr 0.4

   \[\leadsto \frac{\sqrt{e^{x}}}{\frac{\mathsf{expm1}\left(x\right)}{\color{blue}{e^{x \cdot 0.5}}}} \]

6. Final simplification 0.4

   \[\leadsto \frac{\sqrt{e^{x}}}{\frac{\mathsf{expm1}\left(x\right)}{e^{x \cdot 0.5}}} \]

Alternatives

Alternative 1
Error	0.4
Cost	19776

\[\begin{array}{l} t_0 := e^{x \cdot 0.5}\\ \frac{t_0}{\frac{\mathsf{expm1}\left(x\right)}{t_0}} \end{array} \]

Alternative 2
Error	0.8
Cost	13124

\[\begin{array}{l} \mathbf{if}\;e^{x} \leq 0.01:\\ \;\;\;\;e^{x} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 + \left(\left(x \cdot 0.08333333333333333 + \frac{1}{x}\right) + \frac{x \cdot \left(\left(x \cdot x\right) \cdot 0.001388888888888889\right)}{-1}\right)\\ \end{array} \]

Alternative 3
Error	0.4
Cost	12992

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

Alternative 4
Error	0.9
Cost	7104

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

Alternative 5
Error	1.1
Cost	6848

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

Alternative 6
Error	1.5
Cost	6592

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

Alternative 7
Error	21.2
Cost	192

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

Alternative 8
Error	61.9
Cost	64

\[-0.5 \]

Alternative 9
Error	61.5
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023055 
(FPCore (x)
  :name "expq2 (section 3.11)"
  :precision binary64

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

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