2nthrt (problem 3.4.6)

Average Error: 32.1 → 7.3

Time: 31.4s

Precision: binary64

Cost: 85700

Math

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

↓

\[\begin{array}{l} t_0 := \log \left(1 + x\right)\\ \mathbf{if}\;x \leq 9200000:\\ \;\;\;\;\left(-\left(\frac{\log x - t_0}{n} + \left({t_0}^{3} - {\log x}^{3}\right) \cdot \frac{-0.16666666666666666}{{n}^{3}}\right)\right) + 0.5 \cdot \left(\frac{{t_0}^{2}}{{n}^{2}} - \frac{{\log x}^{2}}{{n}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\frac{\log \left(\frac{1}{x}\right)}{-n}}}{x \cdot n}\\ \end{array} \]

FPCore

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

↓

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (log (+ 1.0 x))))
   (if (<= x 9200000.0)
     (+
      (-
       (+
        (/ (- (log x) t_0) n)
        (*
         (- (pow t_0 3.0) (pow (log x) 3.0))
         (/ -0.16666666666666666 (pow n 3.0)))))
      (*
       0.5
       (- (/ (pow t_0 2.0) (pow n 2.0)) (/ (pow (log x) 2.0) (pow n 2.0)))))
     (/ (exp (/ (log (/ 1.0 x)) (- 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 t_0 = log((1.0 + x));
	double tmp;
	if (x <= 9200000.0) {
		tmp = -(((log(x) - t_0) / n) + ((pow(t_0, 3.0) - pow(log(x), 3.0)) * (-0.16666666666666666 / pow(n, 3.0)))) + (0.5 * ((pow(t_0, 2.0) / pow(n, 2.0)) - (pow(log(x), 2.0) / pow(n, 2.0))));
	} else {
		tmp = exp((log((1.0 / x)) / -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) :: t_0
    real(8) :: tmp
    t_0 = log((1.0d0 + x))
    if (x <= 9200000.0d0) then
        tmp = -(((log(x) - t_0) / n) + (((t_0 ** 3.0d0) - (log(x) ** 3.0d0)) * ((-0.16666666666666666d0) / (n ** 3.0d0)))) + (0.5d0 * (((t_0 ** 2.0d0) / (n ** 2.0d0)) - ((log(x) ** 2.0d0) / (n ** 2.0d0))))
    else
        tmp = exp((log((1.0d0 / x)) / -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 t_0 = Math.log((1.0 + x));
	double tmp;
	if (x <= 9200000.0) {
		tmp = -(((Math.log(x) - t_0) / n) + ((Math.pow(t_0, 3.0) - Math.pow(Math.log(x), 3.0)) * (-0.16666666666666666 / Math.pow(n, 3.0)))) + (0.5 * ((Math.pow(t_0, 2.0) / Math.pow(n, 2.0)) - (Math.pow(Math.log(x), 2.0) / Math.pow(n, 2.0))));
	} else {
		tmp = Math.exp((Math.log((1.0 / x)) / -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):
	t_0 = math.log((1.0 + x))
	tmp = 0
	if x <= 9200000.0:
		tmp = -(((math.log(x) - t_0) / n) + ((math.pow(t_0, 3.0) - math.pow(math.log(x), 3.0)) * (-0.16666666666666666 / math.pow(n, 3.0)))) + (0.5 * ((math.pow(t_0, 2.0) / math.pow(n, 2.0)) - (math.pow(math.log(x), 2.0) / math.pow(n, 2.0))))
	else:
		tmp = math.exp((math.log((1.0 / x)) / -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)
	t_0 = log(Float64(1.0 + x))
	tmp = 0.0
	if (x <= 9200000.0)
		tmp = Float64(Float64(-Float64(Float64(Float64(log(x) - t_0) / n) + Float64(Float64((t_0 ^ 3.0) - (log(x) ^ 3.0)) * Float64(-0.16666666666666666 / (n ^ 3.0))))) + Float64(0.5 * Float64(Float64((t_0 ^ 2.0) / (n ^ 2.0)) - Float64((log(x) ^ 2.0) / (n ^ 2.0)))));
	else
		tmp = Float64(exp(Float64(log(Float64(1.0 / x)) / Float64(-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)
	t_0 = log((1.0 + x));
	tmp = 0.0;
	if (x <= 9200000.0)
		tmp = -(((log(x) - t_0) / n) + (((t_0 ^ 3.0) - (log(x) ^ 3.0)) * (-0.16666666666666666 / (n ^ 3.0)))) + (0.5 * (((t_0 ^ 2.0) / (n ^ 2.0)) - ((log(x) ^ 2.0) / (n ^ 2.0))));
	else
		tmp = exp((log((1.0 / x)) / -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_] := Block[{t$95$0 = N[Log[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 9200000.0], N[((-N[(N[(N[(N[Log[x], $MachinePrecision] - t$95$0), $MachinePrecision] / n), $MachinePrecision] + N[(N[(N[Power[t$95$0, 3.0], $MachinePrecision] - N[Power[N[Log[x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] * N[(-0.16666666666666666 / N[Power[n, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) + N[(0.5 * N[(N[(N[Power[t$95$0, 2.0], $MachinePrecision] / N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision] - N[(N[Power[N[Log[x], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Exp[N[(N[Log[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] / (-n)), $MachinePrecision]], $MachinePrecision] / N[(x * n), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 9.2e6

   1. Initial program 46.3

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

   2. Taylor expanded in n around -inf 14.3

      \[\leadsto \color{blue}{\left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{{n}^{2}} + \left(-1 \cdot \frac{-0.16666666666666666 \cdot {\log \left(1 + x\right)}^{3} - -0.16666666666666666 \cdot {\log x}^{3}}{{n}^{3}} + -1 \cdot \frac{-1 \cdot \log \left(1 + x\right) - -1 \cdot \log x}{n}\right)\right) - 0.5 \cdot \frac{{\log x}^{2}}{{n}^{2}}} \]

   3. Simplified 14.3

      \[\leadsto \color{blue}{\left(-\left(\frac{\log x - \log \left(1 + x\right)}{n} + \left({\log \left(1 + x\right)}^{3} - {\log x}^{3}\right) \cdot \frac{-0.16666666666666666}{{n}^{3}}\right)\right) + 0.5 \cdot \left(\frac{{\log \left(1 + x\right)}^{2}}{{n}^{2}} - \frac{{\log x}^{2}}{{n}^{2}}\right)} \]
      Proof

      [Start] 14.3	\[ \left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{{n}^{2}} + \left(-1 \cdot \frac{-0.16666666666666666 \cdot {\log \left(1 + x\right)}^{3} - -0.16666666666666666 \cdot {\log x}^{3}}{{n}^{3}} + -1 \cdot \frac{-1 \cdot \log \left(1 + x\right) - -1 \cdot \log x}{n}\right)\right) - 0.5 \cdot \frac{{\log x}^{2}}{{n}^{2}} \]
      rational.json-simplify-48 [=>] 14.3	\[ \color{blue}{\left(-1 \cdot \frac{-0.16666666666666666 \cdot {\log \left(1 + x\right)}^{3} - -0.16666666666666666 \cdot {\log x}^{3}}{{n}^{3}} + -1 \cdot \frac{-1 \cdot \log \left(1 + x\right) - -1 \cdot \log x}{n}\right) + \left(0.5 \cdot \frac{{\log \left(1 + x\right)}^{2}}{{n}^{2}} - 0.5 \cdot \frac{{\log x}^{2}}{{n}^{2}}\right)} \]

   if 9.2e6 < x

   1. Initial program 19.7

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

   2. Taylor expanded in x around inf 1.2

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

   3. Simplified 1.2

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

      [Start] 1.2	\[ \frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x} \]
      rational.json-simplify-2 [=>] 1.2	\[ \frac{e^{\color{blue}{\frac{\log \left(\frac{1}{x}\right)}{n} \cdot -1}}}{n \cdot x} \]
      rational.json-simplify-9 [=>] 1.2	\[ \frac{e^{\color{blue}{-\frac{\log \left(\frac{1}{x}\right)}{n}}}}{n \cdot x} \]
      rational.json-simplify-10 [=>] 1.2	\[ \frac{e^{\color{blue}{\frac{\frac{\log \left(\frac{1}{x}\right)}{n}}{-1}}}}{n \cdot x} \]
      rational.json-simplify-47 [=>] 1.2	\[ \frac{e^{\color{blue}{\frac{\log \left(\frac{1}{x}\right)}{n \cdot -1}}}}{n \cdot x} \]
      rational.json-simplify-9 [=>] 1.2	\[ \frac{e^{\frac{\log \left(\frac{1}{x}\right)}{\color{blue}{-n}}}}{n \cdot x} \]
      rational.json-simplify-2 [=>] 1.2	\[ \frac{e^{\frac{\log \left(\frac{1}{x}\right)}{-n}}}{\color{blue}{x \cdot n}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 7.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 9200000:\\ \;\;\;\;\left(-\left(\frac{\log x - \log \left(1 + x\right)}{n} + \left({\log \left(1 + x\right)}^{3} - {\log x}^{3}\right) \cdot \frac{-0.16666666666666666}{{n}^{3}}\right)\right) + 0.5 \cdot \left(\frac{{\log \left(1 + x\right)}^{2}}{{n}^{2}} - \frac{{\log x}^{2}}{{n}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\frac{\log \left(\frac{1}{x}\right)}{-n}}}{x \cdot n}\\ \end{array} \]

Alternatives

Alternative 1
Error	12.1
Cost	14028

\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -5:\\ \;\;\;\;\frac{0}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -1 \cdot 10^{-123}:\\ \;\;\;\;-\frac{\frac{-1}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-12}:\\ \;\;\;\;\frac{\log \left(1 + x\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{n} + 1\right) - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]

Alternative 2
Error	7.5
Cost	13636

\[\begin{array}{l} \mathbf{if}\;x \leq 9200000:\\ \;\;\;\;\frac{\left(\log \left(1 + x\right) + \left(1 - \log x\right)\right) + -1}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\frac{\log \left(\frac{1}{x}\right)}{-n}}}{x \cdot n}\\ \end{array} \]

Alternative 3
Error	7.5
Cost	13572

\[\begin{array}{l} \mathbf{if}\;x \leq 9200000:\\ \;\;\;\;\frac{\log \left(1 + x\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\frac{\log \left(\frac{1}{x}\right)}{-n}}}{x \cdot n}\\ \end{array} \]

Alternative 4
Error	16.9
Cost	6852

\[\begin{array}{l} \mathbf{if}\;x \leq 1.1:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 2.9 \cdot 10^{+28}:\\ \;\;\;\;\frac{0}{n}\\ \mathbf{elif}\;x \leq 7 \cdot 10^{+187}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{n}\\ \end{array} \]

Alternative 5
Error	17.1
Cost	6788

\[\begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 2.9 \cdot 10^{+28}:\\ \;\;\;\;\frac{0}{n}\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{+189}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{n}\\ \end{array} \]

Alternative 6
Error	28.4
Cost	1032

\[\begin{array}{l} \mathbf{if}\;n \leq -15.5:\\ \;\;\;\;-\frac{\frac{-1}{n}}{x}\\ \mathbf{elif}\;n \leq -4.932652407396749 \cdot 10^{-292}:\\ \;\;\;\;\frac{0}{n}\\ \mathbf{else}:\\ \;\;\;\;-\frac{1}{n \cdot \left(x \cdot -1\right) + n \cdot -0.5}\\ \end{array} \]

Alternative 7
Error	28.4
Cost	840

\[\begin{array}{l} \mathbf{if}\;n \leq -7:\\ \;\;\;\;-\frac{\frac{-1}{n}}{x}\\ \mathbf{elif}\;n \leq -4.932652407396749 \cdot 10^{-292}:\\ \;\;\;\;\frac{0}{n}\\ \mathbf{else}:\\ \;\;\;\;-\frac{1}{n \cdot \left(-0.5 + \left(-x\right)\right)}\\ \end{array} \]

Alternative 8
Error	29.2
Cost	584

\[\begin{array}{l} t_0 := \frac{1}{x \cdot n}\\ \mathbf{if}\;n \leq -18.5:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq 5.5 \cdot 10^{-35}:\\ \;\;\;\;\frac{0}{n}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 9
Error	28.8
Cost	584

\[\begin{array}{l} t_0 := \frac{\frac{1}{x}}{n}\\ \mathbf{if}\;n \leq -10.5:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq 5.5 \cdot 10^{-35}:\\ \;\;\;\;\frac{0}{n}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 10
Error	28.8
Cost	584

\[\begin{array}{l} \mathbf{if}\;n \leq -16:\\ \;\;\;\;-\frac{\frac{-1}{n}}{x}\\ \mathbf{elif}\;n \leq 5.5 \cdot 10^{-35}:\\ \;\;\;\;\frac{0}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \end{array} \]

Alternative 11
Error	41.0
Cost	320

\[\frac{1}{x \cdot n} \]

Alternative 12
Error	61.1
Cost	192

\[\frac{x}{n} \]

Reproduce

herbie shell --seed 2023074 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))