sqrtexp (problem 3.4.4)

Average Error: 64.81% → 0.05%

Time: 4.7s

Precision: binary64

Cost: 19328

Math

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

↓

\[\mathsf{hypot}\left(1, \sqrt{e^{x}}\right) \]

FPCore

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

↓

(FPCore (x) :precision binary64 (hypot 1.0 (sqrt (exp x))))

C

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

↓

double code(double x) {
	return hypot(1.0, sqrt(exp(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.hypot(1.0, Math.sqrt(Math.exp(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.hypot(1.0, math.sqrt(math.exp(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 hypot(1.0, sqrt(exp(x)))
end

MATLAB

function tmp = code(x)
	tmp = sqrt(((exp((2.0 * x)) - 1.0) / (exp(x) - 1.0)));
end

↓

function tmp = code(x)
	tmp = hypot(1.0, sqrt(exp(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[1.0 ^ 2 + N[Sqrt[N[Exp[x], $MachinePrecision]], $MachinePrecision] ^ 2], $MachinePrecision]

Derivation

1. Initial program 64.81

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

2. Simplified 0.05

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

   [Start] 64.81	\[ \sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}} \]
   *-commutative [=>] 64.81	\[ \sqrt{\frac{e^{\color{blue}{x \cdot 2}} - 1}{e^{x} - 1}} \]
   exp-lft-sqr [=>] 64.47	\[ \sqrt{\frac{\color{blue}{e^{x} \cdot e^{x}} - 1}{e^{x} - 1}} \]
   difference-of-sqr-1 [=>] 63.95	\[ \sqrt{\frac{\color{blue}{\left(e^{x} + 1\right) \cdot \left(e^{x} - 1\right)}}{e^{x} - 1}} \]
   associate-/l* [=>] 63.95	\[ \sqrt{\color{blue}{\frac{e^{x} + 1}{\frac{e^{x} - 1}{e^{x} - 1}}}} \]
   *-inverses [=>] 0.05	\[ \sqrt{\frac{e^{x} + 1}{\color{blue}{1}}} \]
   /-rgt-identity [=>] 0.05	\[ \sqrt{\color{blue}{e^{x} + 1}} \]
   +-commutative [=>] 0.05	\[ \sqrt{\color{blue}{1 + e^{x}}} \]

3. Applied egg-rr 0.05

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

4. Final simplification 0.05

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

Alternatives

Alternative 1
Error	0.05%
Cost	12992

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

Alternative 2
Error	27.29%
Cost	6464

\[\sqrt{2} \]

Alternative 3
Error	85.73%
Cost	320

\[1 + x \cdot 0.5 \]

Alternative 4
Error	97.08%
Cost	192

\[x \cdot 0.5 \]

Alternative 5
Error	95.71%
Cost	192

\[x \cdot -0.5 \]

Reproduce

herbie shell --seed 2023089 
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  :precision binary64
  (sqrt (/ (- (exp (* 2.0 x)) 1.0) (- (exp x) 1.0))))