expq2 (section 3.11)

Average Error: 41.4 → 0.4

Time: 10.2s

Precision: binary64

Cost: 12992

Math

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

↓

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

FPCore

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

↓

(FPCore (x) :precision binary64 (/ (exp x) (expm1 x)))

C

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

↓

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

Python

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

↓

def code(x):
	return 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(exp(x) / expm1(x))
end

Wolfram

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

↓

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

Target

Original	41.4
Target	41.1
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. Final simplification 0.4

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

Alternatives

Alternative 1
Error	4.7
Cost	10888

\[\begin{array}{l} t_0 := x \cdot 0.08333333333333333 + \frac{1}{x}\\ t_1 := x \cdot -0.08333333333333333 + \frac{-1}{x}\\ \mathbf{if}\;x \leq -5 \cdot 10^{+103}:\\ \;\;\;\;\frac{-24}{{x}^{4}}\\ \mathbf{elif}\;x \leq -1 \cdot 10^{+47}:\\ \;\;\;\;\frac{0.125 + {t_0}^{3}}{0.0625 - \frac{\left(\left(-0.5 - t_1\right) \cdot \left(x \cdot t_1\right)\right) \cdot \left(4.8225308641975306 \cdot 10^{-5} \cdot \left(x \cdot x\right) + -0.001736111111111111\right)}{-0.041666666666666664 + x \cdot -0.006944444444444444}} \cdot \left(0.25 + t_0 \cdot \left(0.5 + t_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 + 0.5\\ \end{array} \]

Alternative 2
Error	5.6
Cost	6788

\[\begin{array}{l} \mathbf{if}\;x \leq -4:\\ \;\;\;\;\frac{-24}{{x}^{4}}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 0.08333333333333333 + \frac{1}{x}\right) + 0.5\\ \end{array} \]

Alternative 3
Error	21.3
Cost	576

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

Alternative 4
Error	21.3
Cost	320

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

Alternative 5
Error	21.2
Cost	192

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

Alternative 6
Error	61.9
Cost	64

\[0.5 \]

Reproduce

herbie shell --seed 2023011 
(FPCore (x)
  :name "expq2 (section 3.11)"
  :precision binary64

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

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