expq2 (section 3.11)

Average Error: 41.4 → 0.0

Time: 39.8s

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	41.4
Target	40.9
Herbie	0.0

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

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

3. Taylor expanded in x around inf 41.4

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

4. Simplified 0.0

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

Alternatives

Alternative 1
Error	1.7
Cost	6592

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

Alternative 2
Error	18.0
Cost	1732

\[\begin{array}{l} t_0 := x \cdot \left(-2 + x\right) - -4\\ \mathbf{if}\;x + 2 \ne 0:\\ \;\;\;\;\frac{t_0}{\frac{\left(x + x\right) \cdot t_0}{x + 2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 - x}{-2 \cdot x}\\ \end{array} \]

Alternative 3
Error	21.3
Cost	192

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

Alternative 4
Error	61.9
Cost	64

\[0.5 \]

Reproduce

herbie shell --seed 2023033 
(FPCore (x)
  :name "expq2 (section 3.11)"
  :precision binary64

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

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