expq2 (section 3.11)

Average Accuracy: 37.1% → 99.3%

Time: 6.9s

Precision: binary64

Cost: 19776

Math

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

↓

\[e^{x} \cdot \frac{e^{x} + 1}{\mathsf{expm1}\left(x + x\right)} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (* (exp x) (/ (+ (exp x) 1.0) (expm1 (+ x x)))))

C

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

↓

double code(double x) {
	return exp(x) * ((exp(x) + 1.0) / expm1((x + 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.exp(x) + 1.0) / Math.expm1((x + x)));
}

Python

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

↓

def code(x):
	return math.exp(x) * ((math.exp(x) + 1.0) / math.expm1((x + x)))

Julia

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

↓

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

Target

Original	37.1%
Target	37.7%
Herbie	99.3%

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

Derivation

1. Initial program 33.0%

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

2. Simplified 99.2%

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

   [Start] 33.0	\[ \frac{e^{x}}{e^{x} - 1} \]
   expm1-def [=>] 99.2	\[ \frac{e^{x}}{\color{blue}{\mathsf{expm1}\left(x\right)}} \]

3. Applied egg-rr 33.0%

   \[\leadsto \frac{e^{x}}{\color{blue}{\frac{{\left(e^{x}\right)}^{2} - 1}{e^{x} + 1}}} \]
   Proof

   [Start] 99.2	\[ \frac{e^{x}}{\mathsf{expm1}\left(x\right)} \]
   expm1-udef [=>] 33.0	\[ \frac{e^{x}}{\color{blue}{e^{x} - 1}} \]
   flip-- [=>] 33.0	\[ \frac{e^{x}}{\color{blue}{\frac{e^{x} \cdot e^{x} - 1 \cdot 1}{e^{x} + 1}}} \]
   pow2 [=>] 33.0	\[ \frac{e^{x}}{\frac{\color{blue}{{\left(e^{x}\right)}^{2}} - 1 \cdot 1}{e^{x} + 1}} \]
   metadata-eval [=>] 33.0	\[ \frac{e^{x}}{\frac{{\left(e^{x}\right)}^{2} - \color{blue}{1}}{e^{x} + 1}} \]

4. Simplified 99.2%

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

   [Start] 33.0	\[ \frac{e^{x}}{\frac{{\left(e^{x}\right)}^{2} - 1}{e^{x} + 1}} \]
   unpow2 [=>] 33.0	\[ \frac{e^{x}}{\frac{\color{blue}{e^{x} \cdot e^{x}} - 1}{e^{x} + 1}} \]
   prod-exp [=>] 33.0	\[ \frac{e^{x}}{\frac{\color{blue}{e^{x + x}} - 1}{e^{x} + 1}} \]
   expm1-def [=>] 99.2	\[ \frac{e^{x}}{\frac{\color{blue}{\mathsf{expm1}\left(x + x\right)}}{e^{x} + 1}} \]
   +-commutative [<=] 99.2	\[ \frac{e^{x}}{\frac{\mathsf{expm1}\left(x + x\right)}{\color{blue}{1 + e^{x}}}} \]

5. Applied egg-rr 99.2%

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

   [Start] 99.2	\[ \frac{e^{x}}{\frac{\mathsf{expm1}\left(x + x\right)}{1 + e^{x}}} \]
   clear-num [=>] 99.2	\[ \color{blue}{\frac{1}{\frac{\frac{\mathsf{expm1}\left(x + x\right)}{1 + e^{x}}}{e^{x}}}} \]
   associate-/r/ [=>] 99.2	\[ \color{blue}{\frac{1}{\frac{\mathsf{expm1}\left(x + x\right)}{1 + e^{x}}} \cdot e^{x}} \]
   clear-num [<=] 99.2	\[ \color{blue}{\frac{1 + e^{x}}{\mathsf{expm1}\left(x + x\right)}} \cdot e^{x} \]
   +-commutative [=>] 99.2	\[ \frac{\color{blue}{e^{x} + 1}}{\mathsf{expm1}\left(x + x\right)} \cdot e^{x} \]

6. Final simplification 99.2%

   \[\leadsto e^{x} \cdot \frac{e^{x} + 1}{\mathsf{expm1}\left(x + x\right)} \]

Alternatives

Alternative 1
Accuracy	99.4%
Cost	12992

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

Alternative 2
Accuracy	98.7%
Cost	7104

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

Alternative 3
Accuracy	97.7%
Cost	6592

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

Alternative 4
Accuracy	66.7%
Cost	576

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

Alternative 5
Accuracy	66.7%
Cost	576

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

Alternative 6
Accuracy	66.7%
Cost	320

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

Alternative 7
Accuracy	66.8%
Cost	192

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

Alternative 8
Accuracy	3.3%
Cost	64

\[0.5 \]

Alternative 9
Accuracy	3.9%
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023159 
(FPCore (x)
  :name "expq2 (section 3.11)"
  :precision binary64

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

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