sqrtexp (problem 3.4.4)

Average Error: 40.8 → 0.4

Time: 8.6s

Precision: binary64

Cost: 33476

Math

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

↓

\[\begin{array}{l} t_0 := \frac{e^{2 \cdot x} - 1}{e^{x} - 1}\\ \mathbf{if}\;t_0 \leq 1.5:\\ \;\;\;\;\sqrt{t_0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(x + 0.5 \cdot {x}^{2}\right) + \left(2 + 0.16666666666666666 \cdot {x}^{3}\right)}\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ (- (exp (* 2.0 x)) 1.0) (- (exp x) 1.0))))
   (if (<= t_0 1.5)
     (sqrt t_0)
     (sqrt
      (+
       (+ x (* 0.5 (pow x 2.0)))
       (+ 2.0 (* 0.16666666666666666 (pow x 3.0))))))))

C

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

↓

double code(double x) {
	double t_0 = (exp((2.0 * x)) - 1.0) / (exp(x) - 1.0);
	double tmp;
	if (t_0 <= 1.5) {
		tmp = sqrt(t_0);
	} else {
		tmp = sqrt(((x + (0.5 * pow(x, 2.0))) + (2.0 + (0.16666666666666666 * pow(x, 3.0)))));
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (exp((2.0d0 * x)) - 1.0d0) / (exp(x) - 1.0d0)
    if (t_0 <= 1.5d0) then
        tmp = sqrt(t_0)
    else
        tmp = sqrt(((x + (0.5d0 * (x ** 2.0d0))) + (2.0d0 + (0.16666666666666666d0 * (x ** 3.0d0)))))
    end if
    code = tmp
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) {
	double t_0 = (Math.exp((2.0 * x)) - 1.0) / (Math.exp(x) - 1.0);
	double tmp;
	if (t_0 <= 1.5) {
		tmp = Math.sqrt(t_0);
	} else {
		tmp = Math.sqrt(((x + (0.5 * Math.pow(x, 2.0))) + (2.0 + (0.16666666666666666 * Math.pow(x, 3.0)))));
	}
	return tmp;
}

Python

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

↓

def code(x):
	t_0 = (math.exp((2.0 * x)) - 1.0) / (math.exp(x) - 1.0)
	tmp = 0
	if t_0 <= 1.5:
		tmp = math.sqrt(t_0)
	else:
		tmp = math.sqrt(((x + (0.5 * math.pow(x, 2.0))) + (2.0 + (0.16666666666666666 * math.pow(x, 3.0)))))
	return tmp

Julia

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

↓

function code(x)
	t_0 = Float64(Float64(exp(Float64(2.0 * x)) - 1.0) / Float64(exp(x) - 1.0))
	tmp = 0.0
	if (t_0 <= 1.5)
		tmp = sqrt(t_0);
	else
		tmp = sqrt(Float64(Float64(x + Float64(0.5 * (x ^ 2.0))) + Float64(2.0 + Float64(0.16666666666666666 * (x ^ 3.0)))));
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(x)
	t_0 = (exp((2.0 * x)) - 1.0) / (exp(x) - 1.0);
	tmp = 0.0;
	if (t_0 <= 1.5)
		tmp = sqrt(t_0);
	else
		tmp = sqrt(((x + (0.5 * (x ^ 2.0))) + (2.0 + (0.16666666666666666 * (x ^ 3.0)))));
	end
	tmp_2 = tmp;
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_] := Block[{t$95$0 = N[(N[(N[Exp[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] - 1.0), $MachinePrecision] / N[(N[Exp[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 1.5], N[Sqrt[t$95$0], $MachinePrecision], N[Sqrt[N[(N[(x + N[(0.5 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 + N[(0.16666666666666666 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (exp.f64 (*.f64 2 x)) 1) (-.f64 (exp.f64 x) 1)) < 1.5

   1. Initial program 0.1

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

   if 1.5 < (/.f64 (-.f64 (exp.f64 (*.f64 2 x)) 1) (-.f64 (exp.f64 x) 1))

   1. Initial program 61.1

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

   2. Taylor expanded in x around 0 0.6

      \[\leadsto \sqrt{\color{blue}{0.16666666666666666 \cdot {x}^{3} + \left(2 + \left(0.5 \cdot {x}^{2} + x\right)\right)}} \]

   3. Simplified 0.6

      \[\leadsto \sqrt{\color{blue}{\left(x + 0.5 \cdot {x}^{2}\right) + \left(2 + 0.16666666666666666 \cdot {x}^{3}\right)}} \]
      Proof

      [Start] 0.6	\[ \sqrt{0.16666666666666666 \cdot {x}^{3} + \left(2 + \left(0.5 \cdot {x}^{2} + x\right)\right)} \]
      rational_best-simplify-43 [=>] 0.6	\[ \sqrt{\color{blue}{\left(0.5 \cdot {x}^{2} + x\right) + \left(2 + 0.16666666666666666 \cdot {x}^{3}\right)}} \]
      rational_best-simplify-1 [=>] 0.6	\[ \sqrt{\color{blue}{\left(x + 0.5 \cdot {x}^{2}\right)} + \left(2 + 0.16666666666666666 \cdot {x}^{3}\right)} \]

3. Recombined 2 regimes into one program.

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{2 \cdot x} - 1}{e^{x} - 1} \leq 1.5:\\ \;\;\;\;\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(x + 0.5 \cdot {x}^{2}\right) + \left(2 + 0.16666666666666666 \cdot {x}^{3}\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.4
Cost	19908

\[\begin{array}{l} \mathbf{if}\;x \leq -0.000162:\\ \;\;\;\;\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 + \left(0.5 \cdot {x}^{2} + x\right)}\\ \end{array} \]

Alternative 2
Error	18.0
Cost	6464

\[\sqrt{2} \]

Reproduce

herbie shell --seed 2023094 
(FPCore (x)
  :name "sqrtexp (problem 3.4.4)"
  :precision binary64
  (sqrt (/ (- (exp (* 2.0 x)) 1.0) (- (exp x) 1.0))))