sqrtexp (problem 3.4.4)

Average Error: 41.0 → 0.0

Time: 4.4s

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 41.0

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

2. Simplified 0.0

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

   [Start] 41.0	\[ \sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}} \]
   *-commutative [=>] 41.0	\[ \sqrt{\frac{e^{\color{blue}{x \cdot 2}} - 1}{e^{x} - 1}} \]
   exp-lft-sqr [=>] 40.8	\[ \sqrt{\frac{\color{blue}{e^{x} \cdot e^{x}} - 1}{e^{x} - 1}} \]
   difference-of-sqr-1 [=>] 40.4	\[ \sqrt{\frac{\color{blue}{\left(e^{x} + 1\right) \cdot \left(e^{x} - 1\right)}}{e^{x} - 1}} \]
   associate-/l* [=>] 40.4	\[ \sqrt{\color{blue}{\frac{e^{x} + 1}{\frac{e^{x} - 1}{e^{x} - 1}}}} \]
   *-inverses [=>] 0.0	\[ \sqrt{\frac{e^{x} + 1}{\color{blue}{1}}} \]
   /-rgt-identity [=>] 0.0	\[ \sqrt{\color{blue}{e^{x} + 1}} \]
   +-commutative [=>] 0.0	\[ \sqrt{\color{blue}{1 + e^{x}}} \]

3. Applied egg-rr 0.7

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

4. Simplified 0.0

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

   [Start] 0.7	\[ \left(1 + \mathsf{hypot}\left(1, \sqrt{e^{x}}\right)\right) - 1 \]
   +-commutative [=>] 0.7	\[ \color{blue}{\left(\mathsf{hypot}\left(1, \sqrt{e^{x}}\right) + 1\right)} - 1 \]
   associate--l+ [=>] 0.0	\[ \color{blue}{\mathsf{hypot}\left(1, \sqrt{e^{x}}\right) + \left(1 - 1\right)} \]
   metadata-eval [=>] 0.0	\[ \mathsf{hypot}\left(1, \sqrt{e^{x}}\right) + \color{blue}{0} \]
   +-rgt-identity [=>] 0.0	\[ \color{blue}{\mathsf{hypot}\left(1, \sqrt{e^{x}}\right)} \]
   hypot-1-def [<=] 0.0	\[ \color{blue}{\sqrt{1 + \sqrt{e^{x}} \cdot \sqrt{e^{x}}}} \]
   rem-square-sqrt [=>] 0.0	\[ \sqrt{1 + \color{blue}{e^{x}}} \]
   +-commutative [=>] 0.0	\[ \sqrt{\color{blue}{e^{x} + 1}} \]
   metadata-eval [<=] 0.0	\[ \sqrt{e^{x} + \color{blue}{\left(2 - 1\right)}} \]
   associate--l+ [<=] 0.0	\[ \sqrt{\color{blue}{\left(e^{x} + 2\right) - 1}} \]
   +-commutative [<=] 0.0	\[ \sqrt{\color{blue}{\left(2 + e^{x}\right)} - 1} \]
   associate--l+ [=>] 0.0	\[ \sqrt{\color{blue}{2 + \left(e^{x} - 1\right)}} \]
   expm1-def [=>] 0.0	\[ \sqrt{2 + \color{blue}{\mathsf{expm1}\left(x\right)}} \]

5. Final simplification 0.0

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

Alternatives

Alternative 1
Error	0.0
Cost	12992

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

Alternative 2
Error	17.8
Cost	6464

\[\sqrt{2} \]

Reproduce

herbie shell --seed 2023017 
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  :precision binary64
  (sqrt (/ (- (exp (* 2.0 x)) 1.0) (- (exp x) 1.0))))