expq2 (section 3.11)

Percentage Accurate: 37.5% → 100.0%

Time: 5.6s

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	37.5%
Target	38.1%
Herbie	100.0%

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

Derivation

1. Initial program 34.8%

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

2. Simplified 98.8%

   \[\leadsto \color{blue}{\frac{e^{x}}{\mathsf{expm1}\left(x\right)}} \]
   Step-by-step derivation

   [Start] 34.8	\[ \frac{e^{x}}{e^{x} - 1} \]
   expm1-def [=>] 98.8	\[ \frac{e^{x}}{\color{blue}{\mathsf{expm1}\left(x\right)}} \]

3. Applied egg-rr 98.8%

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

   [Start] 98.8	\[ \frac{e^{x}}{\mathsf{expm1}\left(x\right)} \]
   clear-num [=>] 98.8	\[ \color{blue}{\frac{1}{\frac{\mathsf{expm1}\left(x\right)}{e^{x}}}} \]
   associate-/r/ [=>] 98.8	\[ \color{blue}{\frac{1}{\mathsf{expm1}\left(x\right)} \cdot e^{x}} \]

4. Applied egg-rr 98.8%

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

   [Start] 98.8	\[ \frac{1}{\mathsf{expm1}\left(x\right)} \cdot e^{x} \]
   associate-*l/ [=>] 98.8	\[ \color{blue}{\frac{1 \cdot e^{x}}{\mathsf{expm1}\left(x\right)}} \]
   associate-/l* [=>] 98.8	\[ \color{blue}{\frac{1}{\frac{\mathsf{expm1}\left(x\right)}{e^{x}}}} \]

5. Taylor expanded in x around inf 34.8%

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

6. Simplified 100.0%

   \[\leadsto \frac{1}{\color{blue}{-\mathsf{expm1}\left(-x\right)}} \]
   Step-by-step derivation

   [Start] 34.8	\[ \frac{1}{\frac{e^{x} - 1}{e^{x}}} \]
   expm1-def [=>] 98.8	\[ \frac{1}{\frac{\color{blue}{\mathsf{expm1}\left(x\right)}}{e^{x}}} \]
   *-rgt-identity [<=] 98.8	\[ \frac{1}{\frac{\color{blue}{\mathsf{expm1}\left(x\right) \cdot 1}}{e^{x}}} \]
   associate-*r/ [<=] 98.8	\[ \frac{1}{\color{blue}{\mathsf{expm1}\left(x\right) \cdot \frac{1}{e^{x}}}} \]
   exp-neg [<=] 98.8	\[ \frac{1}{\mathsf{expm1}\left(x\right) \cdot \color{blue}{e^{-x}}} \]
   remove-double-neg [<=] 98.8	\[ \frac{1}{\mathsf{expm1}\left(x\right) \cdot \color{blue}{\left(-\left(-e^{-x}\right)\right)}} \]
   distribute-rgt-neg-in [<=] 98.8	\[ \frac{1}{\color{blue}{-\mathsf{expm1}\left(x\right) \cdot \left(-e^{-x}\right)}} \]
   distribute-lft-neg-in [=>] 98.8	\[ \frac{1}{\color{blue}{\left(-\mathsf{expm1}\left(x\right)\right) \cdot \left(-e^{-x}\right)}} \]
   distribute-rgt-neg-out [=>] 98.8	\[ \frac{1}{\color{blue}{-\left(-\mathsf{expm1}\left(x\right)\right) \cdot e^{-x}}} \]
   distribute-lft-neg-in [<=] 98.8	\[ \frac{1}{-\color{blue}{\left(-\mathsf{expm1}\left(x\right) \cdot e^{-x}\right)}} \]
   exp-neg [=>] 98.8	\[ \frac{1}{-\left(-\mathsf{expm1}\left(x\right) \cdot \color{blue}{\frac{1}{e^{x}}}\right)} \]
   associate-*r/ [=>] 98.8	\[ \frac{1}{-\left(-\color{blue}{\frac{\mathsf{expm1}\left(x\right) \cdot 1}{e^{x}}}\right)} \]
   *-rgt-identity [=>] 98.8	\[ \frac{1}{-\left(-\frac{\color{blue}{\mathsf{expm1}\left(x\right)}}{e^{x}}\right)} \]
   expm1-def [<=] 34.8	\[ \frac{1}{-\left(-\frac{\color{blue}{e^{x} - 1}}{e^{x}}\right)} \]
   div-sub [=>] 6.3	\[ \frac{1}{-\left(-\color{blue}{\left(\frac{e^{x}}{e^{x}} - \frac{1}{e^{x}}\right)}\right)} \]
   exp-neg [<=] 6.3	\[ \frac{1}{-\left(-\left(\frac{e^{x}}{e^{x}} - \color{blue}{e^{-x}}\right)\right)} \]
   sub-neg [=>] 6.3	\[ \frac{1}{-\left(-\color{blue}{\left(\frac{e^{x}}{e^{x}} + \left(-e^{-x}\right)\right)}\right)} \]
   *-inverses [=>] 36.0	\[ \frac{1}{-\left(-\left(\color{blue}{1} + \left(-e^{-x}\right)\right)\right)} \]
   +-commutative [=>] 36.0	\[ \frac{1}{-\left(-\color{blue}{\left(\left(-e^{-x}\right) + 1\right)}\right)} \]
   distribute-neg-in [=>] 35.9	\[ \frac{1}{-\color{blue}{\left(\left(-\left(-e^{-x}\right)\right) + \left(-1\right)\right)}} \]
   remove-double-neg [=>] 35.9	\[ \frac{1}{-\left(\color{blue}{e^{-x}} + \left(-1\right)\right)} \]

7. Taylor expanded in x around inf 35.9%

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

8. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{-1}{\mathsf{expm1}\left(-x\right)}} \]
   Step-by-step derivation

   [Start] 35.9	\[ \frac{-1}{e^{-x} - 1} \]
   expm1-def [=>] 100.0	\[ \frac{-1}{\color{blue}{\mathsf{expm1}\left(-x\right)}} \]

9. Final simplification 100.0%

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

Alternatives

Alternative 1
Accuracy	98.7%
Cost	13124

\[\begin{array}{l} \mathbf{if}\;e^{x} \leq 2 \cdot 10^{-5}:\\ \;\;\;\;e^{x} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 + \left(x \cdot 0.08333333333333333 + \frac{1}{x}\right)\\ \end{array} \]

Alternative 2
Accuracy	98.7%
Cost	708

\[\begin{array}{l} \mathbf{if}\;x \leq -350:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;0.5 + \left(x \cdot 0.08333333333333333 + \frac{1}{x}\right)\\ \end{array} \]

Alternative 3
Accuracy	98.3%
Cost	452

\[\begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;0.5 + \frac{1}{x}\\ \end{array} \]

Alternative 4
Accuracy	97.5%
Cost	324

\[\begin{array}{l} \mathbf{if}\;x \leq -360:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]

Alternative 5
Accuracy	3.3%
Cost	64

\[0.5 \]

Alternative 6
Accuracy	34.4%
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023160 
(FPCore (x)
  :name "expq2 (section 3.11)"
  :precision binary64

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

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