sqrtexp (problem 3.4.4)

Average Accuracy: 35.7% → 99.9%

Time: 3.0s

Precision: binary64

Cost: 12992

Math

\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}} \]

↓

\[\sqrt{2 + \mathsf{expm1}\left(x\right)} \]

FPCore

(FPCore (x)
 :precision binary64
 (sqrt (/ (- (exp (* 2.0 x)) 1.0) (- (exp x) 1.0))))

↓

(FPCore (x) :precision binary64 (sqrt (+ 2.0 (expm1 x))))

C

double code(double x) {
	return sqrt(((exp((2.0 * x)) - 1.0) / (exp(x) - 1.0)));
}

↓

double code(double x) {
	return sqrt((2.0 + expm1(x)));
}

Java

public static double code(double x) {
	return Math.sqrt(((Math.exp((2.0 * x)) - 1.0) / (Math.exp(x) - 1.0)));
}

↓

public static double code(double x) {
	return Math.sqrt((2.0 + Math.expm1(x)));
}

Python

def code(x):
	return math.sqrt(((math.exp((2.0 * x)) - 1.0) / (math.exp(x) - 1.0)))

↓

def code(x):
	return math.sqrt((2.0 + math.expm1(x)))

Julia

function code(x)
	return sqrt(Float64(Float64(exp(Float64(2.0 * x)) - 1.0) / Float64(exp(x) - 1.0)))
end

↓

function code(x)
	return sqrt(Float64(2.0 + expm1(x)))
end

Wolfram

code[x_] := N[Sqrt[N[(N[(N[Exp[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] - 1.0), $MachinePrecision] / N[(N[Exp[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

↓

code[x_] := N[Sqrt[N[(2.0 + N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 35.7%

   \[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}} \]

2. Simplified 99.9%

   \[\leadsto \color{blue}{\sqrt{1 + e^{x}}} \]
   Proof

   [Start] 35.7	\[ \sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}} \]
   *-commutative [=>] 35.7	\[ \sqrt{\frac{e^{\color{blue}{x \cdot 2}} - 1}{e^{x} - 1}} \]
   exp-lft-sqr [=>] 36.1	\[ \sqrt{\frac{\color{blue}{e^{x} \cdot e^{x}} - 1}{e^{x} - 1}} \]
   difference-of-sqr-1 [=>] 36.7	\[ \sqrt{\frac{\color{blue}{\left(e^{x} + 1\right) \cdot \left(e^{x} - 1\right)}}{e^{x} - 1}} \]
   associate-/l* [=>] 36.7	\[ \sqrt{\color{blue}{\frac{e^{x} + 1}{\frac{e^{x} - 1}{e^{x} - 1}}}} \]
   *-inverses [=>] 99.9	\[ \sqrt{\frac{e^{x} + 1}{\color{blue}{1}}} \]
   /-rgt-identity [=>] 99.9	\[ \sqrt{\color{blue}{e^{x} + 1}} \]
   +-commutative [=>] 99.9	\[ \sqrt{\color{blue}{1 + e^{x}}} \]

3. Applied egg-rr 99.9%

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

   [Start] 99.9	\[ \sqrt{1 + e^{x}} \]
   expm1-log1p-u [=>] 98.9	\[ \sqrt{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 + e^{x}\right)\right)}} \]
   expm1-udef [=>] 98.9	\[ \sqrt{\color{blue}{e^{\mathsf{log1p}\left(1 + e^{x}\right)} - 1}} \]
   log1p-udef [=>] 99.9	\[ \sqrt{e^{\color{blue}{\log \left(1 + \left(1 + e^{x}\right)\right)}} - 1} \]
   add-exp-log [<=] 99.9	\[ \sqrt{\color{blue}{\left(1 + \left(1 + e^{x}\right)\right)} - 1} \]

4. Simplified 99.9%

   \[\leadsto \sqrt{\color{blue}{2 + \mathsf{expm1}\left(x\right)}} \]
   Proof

   [Start] 99.9	\[ \sqrt{\left(1 + \left(1 + e^{x}\right)\right) - 1} \]
   associate-+r+ [=>] 99.9	\[ \sqrt{\color{blue}{\left(\left(1 + 1\right) + e^{x}\right)} - 1} \]
   metadata-eval [=>] 99.9	\[ \sqrt{\left(\color{blue}{2} + e^{x}\right) - 1} \]
   associate--l+ [=>] 99.9	\[ \sqrt{\color{blue}{2 + \left(e^{x} - 1\right)}} \]
   expm1-def [=>] 99.9	\[ \sqrt{2 + \color{blue}{\mathsf{expm1}\left(x\right)}} \]

5. Final simplification 99.9%

   \[\leadsto \sqrt{2 + \mathsf{expm1}\left(x\right)} \]

Alternatives

Alternative 1
Accuracy	99.9%
Cost	12992

\[\sqrt{1 + e^{x}} \]

Alternative 2
Accuracy	72.3%
Cost	6464

\[\sqrt{2} \]

Reproduce

herbie shell --seed 2023141 
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  :precision binary64
  (sqrt (/ (- (exp (* 2.0 x)) 1.0) (- (exp x) 1.0))))