expq2 (section 3.11)

Average Error: 41.7 → 0.0

Time: 2.6s

Precision: binary64

Cost: 13056

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

double code(double x) {
	return pow(-expm1(-x), -1.0);
}

Java

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

↓

public static double code(double x) {
	return Math.pow(-Math.expm1(-x), -1.0);
}

Python

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

↓

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

Julia

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

↓

function code(x)
	return Float64(-expm1(Float64(-x))) ^ -1.0
end

Wolfram

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

↓

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

Target

Original	41.7
Target	41.2
Herbie	0.0

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

Derivation

1. Initial program 41.7

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

2. Simplified 0.5

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

   [Start] 41.7	\[ \frac{e^{x}}{e^{x} - 1} \]
   expm1-def [=>] 0.5	\[ \frac{e^{x}}{\color{blue}{\mathsf{expm1}\left(x\right)}} \]

3. Applied egg-rr 0.5

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

4. Taylor expanded in x around inf 41.7

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

5. Simplified 0.0

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

   [Start] 41.7	\[ {\left(\frac{e^{x} - 1}{e^{x}}\right)}^{-1} \]
   expm1-def [=>] 0.5	\[ {\left(\frac{\color{blue}{\mathsf{expm1}\left(x\right)}}{e^{x}}\right)}^{-1} \]
   *-lft-identity [<=] 0.5	\[ {\color{blue}{\left(1 \cdot \frac{\mathsf{expm1}\left(x\right)}{e^{x}}\right)}}^{-1} \]
   metadata-eval [<=] 0.5	\[ {\left(\color{blue}{\frac{-1}{-1}} \cdot \frac{\mathsf{expm1}\left(x\right)}{e^{x}}\right)}^{-1} \]
   times-frac [<=] 0.5	\[ {\color{blue}{\left(\frac{-1 \cdot \mathsf{expm1}\left(x\right)}{-1 \cdot e^{x}}\right)}}^{-1} \]
   neg-mul-1 [<=] 0.5	\[ {\left(\frac{-1 \cdot \mathsf{expm1}\left(x\right)}{\color{blue}{-e^{x}}}\right)}^{-1} \]
   associate-*r/ [<=] 0.5	\[ {\color{blue}{\left(-1 \cdot \frac{\mathsf{expm1}\left(x\right)}{-e^{x}}\right)}}^{-1} \]
   neg-mul-1 [<=] 0.5	\[ {\color{blue}{\left(-\frac{\mathsf{expm1}\left(x\right)}{-e^{x}}\right)}}^{-1} \]
   expm1-def [<=] 41.7	\[ {\left(-\frac{\color{blue}{e^{x} - 1}}{-e^{x}}\right)}^{-1} \]
   div-sub [=>] 62.3	\[ {\left(-\color{blue}{\left(\frac{e^{x}}{-e^{x}} - \frac{1}{-e^{x}}\right)}\right)}^{-1} \]
   neg-mul-1 [=>] 62.3	\[ {\left(-\left(\frac{e^{x}}{-e^{x}} - \frac{1}{\color{blue}{-1 \cdot e^{x}}}\right)\right)}^{-1} \]
   associate-/r* [=>] 62.3	\[ {\left(-\left(\frac{e^{x}}{-e^{x}} - \color{blue}{\frac{\frac{1}{-1}}{e^{x}}}\right)\right)}^{-1} \]
   metadata-eval [=>] 62.3	\[ {\left(-\left(\frac{e^{x}}{-e^{x}} - \frac{\color{blue}{-1}}{e^{x}}\right)\right)}^{-1} \]
   metadata-eval [<=] 62.3	\[ {\left(-\left(\frac{e^{x}}{-e^{x}} - \frac{\color{blue}{-1}}{e^{x}}\right)\right)}^{-1} \]
   distribute-neg-frac [<=] 62.3	\[ {\left(-\left(\frac{e^{x}}{-e^{x}} - \color{blue}{\left(-\frac{1}{e^{x}}\right)}\right)\right)}^{-1} \]
   exp-neg [<=] 62.2	\[ {\left(-\left(\frac{e^{x}}{-e^{x}} - \left(-\color{blue}{e^{-x}}\right)\right)\right)}^{-1} \]
   neg-sub0 [=>] 62.2	\[ {\left(-\left(\frac{e^{x}}{-e^{x}} - \color{blue}{\left(0 - e^{-x}\right)}\right)\right)}^{-1} \]
   associate--r- [=>] 62.2	\[ {\left(-\color{blue}{\left(\left(\frac{e^{x}}{-e^{x}} - 0\right) + e^{-x}\right)}\right)}^{-1} \]

6. Final simplification 0.0

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

Alternatives

Alternative 1
Error	0.5
Cost	12992

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

Alternative 2
Error	21.1
Cost	192

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

Reproduce

herbie shell --seed 2023047 
(FPCore (x)
  :name "expq2 (section 3.11)"
  :precision binary64

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

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