sqrtexp (problem 3.4.4)

Percentage Accurate: 3.3% → 100.0%

Time: 12.0s

Precision: binary64

Cost: 13184

• 24.2% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

  1. x = -4.9463031944417784e+207

Math

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

↓

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

FPCore

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

↓

(FPCore (x) :precision binary64 (pow (/ 1.0 (+ 1.0 (exp x))) -0.5))

C

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

↓

double code(double x) {
	return pow((1.0 / (1.0 + exp(x))), -0.5);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = sqrt(((exp((2.0d0 * x)) - 1.0d0) / (exp(x) - 1.0d0)))
end function

↓

real(8) function code(x)
    real(8), intent (in) :: x
    code = (1.0d0 / (1.0d0 + exp(x))) ** (-0.5d0)
end function

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.pow((1.0 / (1.0 + Math.exp(x))), -0.5);
}

Python

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

↓

def code(x):
	return math.pow((1.0 / (1.0 + math.exp(x))), -0.5)

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 Float64(1.0 / Float64(1.0 + exp(x))) ^ -0.5
end

MATLAB

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

↓

function tmp = code(x)
	tmp = (1.0 / (1.0 + exp(x))) ^ -0.5;
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[Power[N[(1.0 / N[(1.0 + N[Exp[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]

Derivation

1. Initial program 3.9%

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

2. Simplified 100.0%

   \[\leadsto \color{blue}{\sqrt{1 + e^{x}}} \]
   Step-by-step derivation

   [Start] 3.9%	\[ \sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}} \]
   *-commutative [=>] 3.9%	\[ \sqrt{\frac{e^{\color{blue}{x \cdot 2}} - 1}{e^{x} - 1}} \]
   exp-lft-sqr [=>] 4.4%	\[ \sqrt{\frac{\color{blue}{e^{x} \cdot e^{x}} - 1}{e^{x} - 1}} \]
   difference-of-sqr-1 [=>] 4.7%	\[ \sqrt{\frac{\color{blue}{\left(e^{x} + 1\right) \cdot \left(e^{x} - 1\right)}}{e^{x} - 1}} \]
   associate-/l* [=>] 4.7%	\[ \sqrt{\color{blue}{\frac{e^{x} + 1}{\frac{e^{x} - 1}{e^{x} - 1}}}} \]
   *-inverses [=>] 100.0%	\[ \sqrt{\frac{e^{x} + 1}{\color{blue}{1}}} \]
   /-rgt-identity [=>] 100.0%	\[ \sqrt{\color{blue}{e^{x} + 1}} \]
   +-commutative [=>] 100.0%	\[ \sqrt{\color{blue}{1 + e^{x}}} \]

3. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{{\left(\frac{1}{1 + e^{x}}\right)}^{-0.5}} \]
   Step-by-step derivation

   [Start] 100.0%	\[ \sqrt{1 + e^{x}} \]
   pow1/2 [=>] 100.0%	\[ \color{blue}{{\left(1 + e^{x}\right)}^{0.5}} \]
   +-commutative [=>] 100.0%	\[ {\color{blue}{\left(e^{x} + 1\right)}}^{0.5} \]
   flip-+ [=>] 4.4%	\[ {\color{blue}{\left(\frac{e^{x} \cdot e^{x} - 1 \cdot 1}{e^{x} - 1}\right)}}^{0.5} \]
   metadata-eval [=>] 4.4%	\[ {\left(\frac{e^{x} \cdot e^{x} - \color{blue}{1}}{e^{x} - 1}\right)}^{0.5} \]
   prod-exp [=>] 3.9%	\[ {\left(\frac{\color{blue}{e^{x + x}} - 1}{e^{x} - 1}\right)}^{0.5} \]
   expm1-udef [<=] 5.4%	\[ {\left(\frac{\color{blue}{\mathsf{expm1}\left(x + x\right)}}{e^{x} - 1}\right)}^{0.5} \]
   expm1-udef [<=] 68.3%	\[ {\left(\frac{\mathsf{expm1}\left(x + x\right)}{\color{blue}{\mathsf{expm1}\left(x\right)}}\right)}^{0.5} \]
   clear-num [=>] 68.3%	\[ {\color{blue}{\left(\frac{1}{\frac{\mathsf{expm1}\left(x\right)}{\mathsf{expm1}\left(x + x\right)}}\right)}}^{0.5} \]
   inv-pow [=>] 68.3%	\[ {\color{blue}{\left({\left(\frac{\mathsf{expm1}\left(x\right)}{\mathsf{expm1}\left(x + x\right)}\right)}^{-1}\right)}}^{0.5} \]
   metadata-eval [<=] 68.3%	\[ {\left({\left(\frac{\mathsf{expm1}\left(x\right)}{\mathsf{expm1}\left(x + x\right)}\right)}^{\color{blue}{\left(-1\right)}}\right)}^{0.5} \]
   pow-pow [=>] 68.4%	\[ \color{blue}{{\left(\frac{\mathsf{expm1}\left(x\right)}{\mathsf{expm1}\left(x + x\right)}\right)}^{\left(\left(-1\right) \cdot 0.5\right)}} \]
   clear-num [=>] 68.4%	\[ {\color{blue}{\left(\frac{1}{\frac{\mathsf{expm1}\left(x + x\right)}{\mathsf{expm1}\left(x\right)}}\right)}}^{\left(\left(-1\right) \cdot 0.5\right)} \]
   expm1-udef [=>] 5.5%	\[ {\left(\frac{1}{\frac{\color{blue}{e^{x + x} - 1}}{\mathsf{expm1}\left(x\right)}}\right)}^{\left(\left(-1\right) \cdot 0.5\right)} \]
   prod-exp [<=] 5.4%	\[ {\left(\frac{1}{\frac{\color{blue}{e^{x} \cdot e^{x}} - 1}{\mathsf{expm1}\left(x\right)}}\right)}^{\left(\left(-1\right) \cdot 0.5\right)} \]
   metadata-eval [<=] 5.4%	\[ {\left(\frac{1}{\frac{e^{x} \cdot e^{x} - \color{blue}{1 \cdot 1}}{\mathsf{expm1}\left(x\right)}}\right)}^{\left(\left(-1\right) \cdot 0.5\right)} \]
   expm1-udef [=>] 4.4%	\[ {\left(\frac{1}{\frac{e^{x} \cdot e^{x} - 1 \cdot 1}{\color{blue}{e^{x} - 1}}}\right)}^{\left(\left(-1\right) \cdot 0.5\right)} \]
   flip-+ [<=] 100.0%	\[ {\left(\frac{1}{\color{blue}{e^{x} + 1}}\right)}^{\left(\left(-1\right) \cdot 0.5\right)} \]
   +-commutative [<=] 100.0%	\[ {\left(\frac{1}{\color{blue}{1 + e^{x}}}\right)}^{\left(\left(-1\right) \cdot 0.5\right)} \]

4. Final simplification 100.0%

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

Alternatives

Alternative 1
Accuracy	100.0%
Cost	13184

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

Alternative 2
Accuracy	100.0%
Cost	12992

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

Alternative 3
Accuracy	91.0%
Cost	7812

\[\begin{array}{l} \mathbf{if}\;x \leq 1.34 \cdot 10^{+154}:\\ \;\;\;\;{\left(\frac{2 - x}{\frac{16 - \left(x \cdot x\right) \cdot \left(x \cdot x\right)}{4 + x \cdot x}}\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x \cdot \left(x \cdot 0.125\right)}\\ \end{array} \]

Alternative 4
Accuracy	83.0%
Cost	7040

\[{\left(\frac{2 - x}{4 - x \cdot x}\right)}^{-0.5} \]

Alternative 5
Accuracy	83.3%
Cost	6916

\[\begin{array}{l} \mathbf{if}\;x \leq 1.96:\\ \;\;\;\;{\left(0.5 + x \cdot -0.25\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x \cdot \left(x \cdot 0.125\right)}\\ \end{array} \]

Alternative 6
Accuracy	83.0%
Cost	6852

\[\begin{array}{l} \mathbf{if}\;x \leq 9.5:\\ \;\;\;\;\sqrt{x + 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x \cdot \left(x \cdot 0.125\right)}\\ \end{array} \]

Alternative 7
Accuracy	67.8%
Cost	6592

\[\sqrt{x + 2} \]

Alternative 8
Accuracy	66.9%
Cost	6464

\[\sqrt{2} \]

Reproduce

herbie shell --seed 2023272 
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  :precision binary64
  (sqrt (/ (- (exp (* 2.0 x)) 1.0) (- (exp x) 1.0))))