2nthrt (problem 3.4.6)

Average Error: 32.5 → 6.8

Time: 41.5s

Precision: binary64

Cost: 13636

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq 600000:\\ \;\;\;\;\frac{\log \left(x - -1\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\frac{\frac{-1}{x}}{e^{-\frac{\log x}{n}}}}{n}\\ \end{array} \]

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 600000.0)
   (/ (- (log (- x -1.0)) (log x)) n)
   (- (/ (/ (/ -1.0 x) (exp (- (/ (log x) n)))) 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 <= 600000.0) {
		tmp = (log((x - -1.0)) - log(x)) / n;
	} else {
		tmp = -(((-1.0 / x) / exp(-(log(x) / n))) / 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 <= 600000.0d0) then
        tmp = (log((x - (-1.0d0))) - log(x)) / n
    else
        tmp = -((((-1.0d0) / x) / exp(-(log(x) / n))) / 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 <= 600000.0) {
		tmp = (Math.log((x - -1.0)) - Math.log(x)) / n;
	} else {
		tmp = -(((-1.0 / x) / Math.exp(-(Math.log(x) / n))) / 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 <= 600000.0:
		tmp = (math.log((x - -1.0)) - math.log(x)) / n
	else:
		tmp = -(((-1.0 / x) / math.exp(-(math.log(x) / n))) / 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 <= 600000.0)
		tmp = Float64(Float64(log(Float64(x - -1.0)) - log(x)) / n);
	else
		tmp = Float64(-Float64(Float64(Float64(-1.0 / x) / exp(Float64(-Float64(log(x) / n)))) / 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 <= 600000.0)
		tmp = (log((x - -1.0)) - log(x)) / n;
	else
		tmp = -(((-1.0 / x) / exp(-(log(x) / n))) / 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, 600000.0], N[(N[(N[Log[N[(x - -1.0), $MachinePrecision]], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], (-N[(N[(N[(-1.0 / x), $MachinePrecision] / N[Exp[(-N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision])], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if x < 6e5

   1. Initial program 46.8

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

   2. Taylor expanded in n around inf 14.2

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

   3. Simplified 14.2

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

      [Start] 14.2	\[ \frac{\log \left(1 + x\right) - \log x}{n} \]
      rational.json-simplify-17 [=>] 14.2	\[ \frac{\log \color{blue}{\left(x - -1\right)} - \log x}{n} \]

   if 6e5 < x

   1. Initial program 20.1

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

   2. Taylor expanded in x around inf 1.3

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

   3. Simplified 1.3

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

      [Start] 1.3	\[ \frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x} \]
      rational.json-simplify-2 [=>] 1.3	\[ \frac{e^{\color{blue}{\frac{\log \left(\frac{1}{x}\right)}{n} \cdot -1}}}{n \cdot x} \]
      rational.json-simplify-9 [=>] 1.3	\[ \frac{e^{\color{blue}{-\frac{\log \left(\frac{1}{x}\right)}{n}}}}{n \cdot x} \]
      rational.json-simplify-10 [=>] 1.3	\[ \frac{e^{\color{blue}{\frac{\frac{\log \left(\frac{1}{x}\right)}{n}}{-1}}}}{n \cdot x} \]
      rational.json-simplify-47 [=>] 1.3	\[ \frac{e^{\color{blue}{\frac{\log \left(\frac{1}{x}\right)}{n \cdot -1}}}}{n \cdot x} \]
      rational.json-simplify-9 [=>] 1.3	\[ \frac{e^{\frac{\log \left(\frac{1}{x}\right)}{\color{blue}{-n}}}}{n \cdot x} \]
      rational.json-simplify-2 [=>] 1.3	\[ \frac{e^{\frac{\log \left(\frac{1}{x}\right)}{-n}}}{\color{blue}{x \cdot n}} \]

   4. Applied egg-rr 0.4

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

   5. Taylor expanded in x around 0 0.4

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

   6. Simplified 0.4

      \[\leadsto -\frac{\color{blue}{\frac{\frac{-1}{x}}{e^{-\frac{\log x}{n}}}}}{n} \]
      Proof

      [Start] 0.4	\[ -\frac{\frac{-1}{e^{-1 \cdot \frac{\log x}{n}} \cdot x}}{n} \]
      rational.json-simplify-46 [=>] 0.4	\[ -\frac{\color{blue}{\frac{\frac{-1}{e^{-1 \cdot \frac{\log x}{n}}}}{x}}}{n} \]
      rational.json-simplify-44 [=>] 0.4	\[ -\frac{\color{blue}{\frac{\frac{-1}{x}}{e^{-1 \cdot \frac{\log x}{n}}}}}{n} \]
      rational.json-simplify-2 [=>] 0.4	\[ -\frac{\frac{\frac{-1}{x}}{e^{\color{blue}{\frac{\log x}{n} \cdot -1}}}}{n} \]
      rational.json-simplify-9 [=>] 0.4	\[ -\frac{\frac{\frac{-1}{x}}{e^{\color{blue}{-\frac{\log x}{n}}}}}{n} \]

3. Recombined 2 regimes into one program.

4. Final simplification 6.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 600000:\\ \;\;\;\;\frac{\log \left(x - -1\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\frac{\frac{-1}{x}}{e^{-\frac{\log x}{n}}}}{n}\\ \end{array} \]

Alternatives

Alternative 1
Error	7.5
Cost	13380

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

Alternative 2
Error	7.3
Cost	13380

\[\begin{array}{l} \mathbf{if}\;x \leq 1950000:\\ \;\;\;\;\frac{\log \left(x - -1\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\frac{\log x}{n}}}{x \cdot n}\\ \end{array} \]

Alternative 3
Error	15.8
Cost	7304

\[\begin{array}{l} \mathbf{if}\;x \leq 0.96:\\ \;\;\;\;\left(-\frac{\log x}{n}\right) + \frac{x}{n}\\ \mathbf{elif}\;x \leq 1.92 \cdot 10^{+177}:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{0.5}{{x}^{2}}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{-n \cdot \left(x \cdot x\right)}\\ \end{array} \]

Alternative 4
Error	16.0
Cost	7044

\[\begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\left(-\frac{\log x}{n}\right) + \frac{x}{n}\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{+177}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{-n \cdot \left(x \cdot x\right)}\\ \end{array} \]

Alternative 5
Error	16.0
Cost	6852

\[\begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{+177}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{-n \cdot \left(x \cdot x\right)}\\ \end{array} \]

Alternative 6
Error	16.2
Cost	6788

\[\begin{array}{l} \mathbf{if}\;x \leq 1.25:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{+177}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{-n \cdot \left(x \cdot x\right)}\\ \end{array} \]

Alternative 7
Error	39.3
Cost	840

\[\begin{array}{l} \mathbf{if}\;n \leq -25:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;n \leq -1.02 \cdot 10^{-122}:\\ \;\;\;\;\frac{1}{x \cdot x} \cdot \frac{x}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \end{array} \]

Alternative 8
Error	35.7
Cost	840

\[\begin{array}{l} \mathbf{if}\;n \leq -1.6 \cdot 10^{+21}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;n \leq -6 \cdot 10^{-274}:\\ \;\;\;\;-1 + \left(1 - \frac{\frac{-1}{x}}{n}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \end{array} \]

Alternative 9
Error	34.8
Cost	840

\[\begin{array}{l} \mathbf{if}\;n \leq -31.5:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \mathbf{elif}\;n \leq -2 \cdot 10^{-305}:\\ \;\;\;\;\frac{-x}{-n \cdot \left(x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \end{array} \]

Alternative 10
Error	41.1
Cost	320

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

Alternative 11
Error	40.6
Cost	320

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

Alternative 12
Error	40.6
Cost	320

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

Alternative 13
Error	61.1
Cost	192

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

Reproduce

herbie shell --seed 2023073 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))