2nthrt (problem 3.4.6)

Average Error: 32.6 → 7.2

Time: 28.8s

Precision: binary64

Cost: 13380

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq 1.32:\\ \;\;\;\;\frac{\log \left(1 + x\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{1}{x}\right)}^{\left(\frac{-1}{n}\right)}}{x \cdot n}\\ \end{array} \]

FPCore

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

↓

(FPCore (x n)
 :precision binary64
 (if (<= x 1.32)
   (/ (- (log (+ 1.0 x)) (log x)) n)
   (/ (pow (/ 1.0 x) (/ -1.0 n)) (* x n))))

C

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

↓

double code(double x, double n) {
	double tmp;
	if (x <= 1.32) {
		tmp = (log((1.0 + x)) - log(x)) / n;
	} else {
		tmp = pow((1.0 / x), (-1.0 / n)) / (x * n);
	}
	return tmp;
}

Fortran

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

↓

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 1.32d0) then
        tmp = (log((1.0d0 + x)) - log(x)) / n
    else
        tmp = ((1.0d0 / x) ** ((-1.0d0) / n)) / (x * n)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	return Math.pow((x + 1.0), (1.0 / n)) - Math.pow(x, (1.0 / n));
}

↓

public static double code(double x, double n) {
	double tmp;
	if (x <= 1.32) {
		tmp = (Math.log((1.0 + x)) - Math.log(x)) / n;
	} else {
		tmp = Math.pow((1.0 / x), (-1.0 / n)) / (x * n);
	}
	return tmp;
}

Python

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

↓

def code(x, n):
	tmp = 0
	if x <= 1.32:
		tmp = (math.log((1.0 + x)) - math.log(x)) / n
	else:
		tmp = math.pow((1.0 / x), (-1.0 / n)) / (x * n)
	return tmp

Julia

function code(x, n)
	return Float64((Float64(x + 1.0) ^ Float64(1.0 / n)) - (x ^ Float64(1.0 / n)))
end

↓

function code(x, n)
	tmp = 0.0
	if (x <= 1.32)
		tmp = Float64(Float64(log(Float64(1.0 + x)) - log(x)) / n);
	else
		tmp = Float64((Float64(1.0 / x) ^ Float64(-1.0 / n)) / Float64(x * n));
	end
	return tmp
end

MATLAB

function tmp = code(x, n)
	tmp = ((x + 1.0) ^ (1.0 / n)) - (x ^ (1.0 / n));
end

↓

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 1.32)
		tmp = (log((1.0 + x)) - log(x)) / n;
	else
		tmp = ((1.0 / x) ^ (-1.0 / n)) / (x * n);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := N[(N[Power[N[(x + 1.0), $MachinePrecision], N[(1.0 / n), $MachinePrecision]], $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

↓

code[x_, n_] := If[LessEqual[x, 1.32], N[(N[(N[Log[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[Power[N[(1.0 / x), $MachinePrecision], N[(-1.0 / n), $MachinePrecision]], $MachinePrecision] / N[(x * n), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.32000000000000006

   1. Initial program 47.0

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

   2. Taylor expanded in n around inf 13.6

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]

   if 1.32000000000000006 < x

   1. Initial program 20.6

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

   2. Taylor expanded in x around inf 1.9

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}} \]

   3. Simplified 1.9

      \[\leadsto \color{blue}{\frac{e^{\frac{\log \left(\frac{1}{x}\right)}{-n}}}{x \cdot n}} \]
      Proof

      [Start] 1.9	\[ \frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x} \]
      rational_best-simplify-10 [=>] 1.9	\[ \frac{e^{\color{blue}{-\frac{\log \left(\frac{1}{x}\right)}{n}}}}{n \cdot x} \]
      rational_best-simplify-9 [=>] 1.9	\[ \frac{e^{\color{blue}{\frac{\frac{\log \left(\frac{1}{x}\right)}{n}}{-1}}}}{n \cdot x} \]
      rational_best-simplify-48 [=>] 1.9	\[ \frac{e^{\color{blue}{\frac{\log \left(\frac{1}{x}\right)}{-1 \cdot n}}}}{n \cdot x} \]
      rational_best-simplify-10 [=>] 1.9	\[ \frac{e^{\frac{\log \left(\frac{1}{x}\right)}{\color{blue}{-n}}}}{n \cdot x} \]
      rational_best-simplify-2 [=>] 1.9	\[ \frac{e^{\frac{\log \left(\frac{1}{x}\right)}{-n}}}{\color{blue}{x \cdot n}} \]

   4. Taylor expanded in x around inf 1.9

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}} \]

   5. Simplified 1.9

      \[\leadsto \color{blue}{\frac{{\left(\frac{1}{x}\right)}^{\left(\frac{-1}{n}\right)}}{x \cdot n}} \]
      Proof

      [Start] 1.9	\[ \frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x} \]
      rational_best-simplify-2 [<=] 1.9	\[ \frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{\color{blue}{x \cdot n}} \]
      rational_best-simplify-10 [=>] 1.9	\[ \frac{e^{\color{blue}{-\frac{\log \left(\frac{1}{x}\right)}{n}}}}{x \cdot n} \]
      rational_best-simplify-10 [<=] 1.9	\[ \frac{e^{\color{blue}{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}}{x \cdot n} \]
      rational_best-simplify-49 [<=] 1.9	\[ \frac{e^{\color{blue}{\frac{\log \left(\frac{1}{x}\right) \cdot -1}{n}}}}{x \cdot n} \]
      rational_best-simplify-2 [=>] 1.9	\[ \frac{e^{\frac{\color{blue}{-1 \cdot \log \left(\frac{1}{x}\right)}}{n}}}{x \cdot n} \]
      rational_best-simplify-49 [=>] 1.9	\[ \frac{e^{\color{blue}{\log \left(\frac{1}{x}\right) \cdot \frac{-1}{n}}}}{x \cdot n} \]
      rational_best-simplify-2 [=>] 1.9	\[ \frac{e^{\color{blue}{\frac{-1}{n} \cdot \log \left(\frac{1}{x}\right)}}}{x \cdot n} \]
      exponential-simplify-11 [<=] 1.9	\[ \frac{e^{\frac{-1}{n} \cdot \log \color{blue}{\left({\left(\frac{1}{x}\right)}^{1}\right)}}}{x \cdot n} \]
      exponential-simplify-29 [=>] 1.9	\[ \frac{e^{\color{blue}{1 \cdot \log \left({\left(\frac{1}{x}\right)}^{\left(\frac{-1}{n}\right)}\right)}}}{x \cdot n} \]
      rational_best-simplify-5 [=>] 1.9	\[ \frac{e^{\color{blue}{\log \left({\left(\frac{1}{x}\right)}^{\left(\frac{-1}{n}\right)}\right)}}}{x \cdot n} \]
      exponential-simplify-7 [=>] 1.9	\[ \frac{\color{blue}{{\left(\frac{1}{x}\right)}^{\left(\frac{-1}{n}\right)}}}{x \cdot n} \]

3. Recombined 2 regimes into one program.

4. Final simplification 7.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.32:\\ \;\;\;\;\frac{\log \left(1 + x\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{1}{x}\right)}^{\left(\frac{-1}{n}\right)}}{x \cdot n}\\ \end{array} \]

Alternatives

Alternative 1
Error	7.4
Cost	7492

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

Alternative 2
Error	7.4
Cost	7172

\[\begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{1}{x}\right)}^{\left(\frac{-1}{n}\right)}}{x \cdot n}\\ \end{array} \]

Alternative 3
Error	15.7
Cost	6852

\[\begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{+125}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 4
Error	15.9
Cost	6788

\[\begin{array}{l} \mathbf{if}\;x \leq 1.35:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 6.1 \cdot 10^{+125}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 5
Error	29.3
Cost	584

\[\begin{array}{l} t_0 := \frac{1}{x \cdot n}\\ \mathbf{if}\;n \leq -21.5:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq 1.56 \cdot 10^{-49}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	28.8
Cost	584

\[\begin{array}{l} t_0 := \frac{\frac{1}{x}}{n}\\ \mathbf{if}\;n \leq -22:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq 2.25 \cdot 10^{-49}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	38.9
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023097 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))