expq2 (section 3.11)

Average Error: 41.7 → 0.0

Time: 10.0s

Precision: binary64

Cost: 6656

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

Python

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

↓

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

Julia

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

↓

function code(x)
	return Float64(-1.0 / expm1(Float64(-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[(Exp[(-x)] - 1), $MachinePrecision]), $MachinePrecision]

Target

Original	41.7
Target	41.4
Herbie	0.0

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

Derivation

1. Initial program 41.7

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

2. Simplified 0.3

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

3. Applied egg-rr 41.4

   \[\leadsto \color{blue}{\frac{-1}{e^{-x} - 1}} \]

4. Simplified 0.0

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

Alternatives

Alternative 1
Error	1.5
Cost	6592

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

Alternative 2
Error	10.2
Cost	1032

\[\begin{array}{l} \mathbf{if}\;x \leq -6:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;\frac{-1}{x} \ne 0:\\ \;\;\;\;\frac{-\frac{1}{x \cdot x}}{\frac{-1}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 + 0.08333333333333333 \cdot x\right) + \frac{1}{x}\\ \end{array} \]

Alternative 3
Error	20.9
Cost	192

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

Alternative 4
Error	61.5
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023010 
(FPCore (x)
  :name "expq2 (section 3.11)"
  :precision binary64

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

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