2nthrt (problem 3.4.6)

Average Accuracy: 49.4% → 89.4%

Time: 23.8s

Precision: binary64

Cost: 7044

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq 8200:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{x}}{n}\\ \end{array} \]

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 8200.0) (/ (log (/ (+ x 1.0) x)) n) (/ (/ (pow 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 <= 8200.0) {
		tmp = log(((x + 1.0) / x)) / n;
	} else {
		tmp = (pow(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 <= 8200.0d0) then
        tmp = log(((x + 1.0d0) / x)) / n
    else
        tmp = ((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 <= 8200.0) {
		tmp = Math.log(((x + 1.0) / x)) / n;
	} else {
		tmp = (Math.pow(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 <= 8200.0:
		tmp = math.log(((x + 1.0) / x)) / n
	else:
		tmp = (math.pow(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 <= 8200.0)
		tmp = Float64(log(Float64(Float64(x + 1.0) / x)) / n);
	else
		tmp = Float64(Float64((x ^ Float64(1.0 / n)) / 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 <= 8200.0)
		tmp = log(((x + 1.0) / x)) / n;
	else
		tmp = ((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, 8200.0], N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(N[(N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 8200

   1. Initial program 26.9%

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

   2. Taylor expanded in n around inf 77.7%

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

   3. Simplified 77.7%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]
      Proof

      [Start] 77.7	\[ \frac{\log \left(1 + x\right) - \log x}{n} \]
      log1p-def [=>] 77.7	\[ \frac{\color{blue}{\mathsf{log1p}\left(x\right)} - \log x}{n} \]

   4. Applied egg-rr 77.7%

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

      [Start] 77.7	\[ \frac{\mathsf{log1p}\left(x\right) - \log x}{n} \]
      add-log-exp [=>] 77.7	\[ \frac{\color{blue}{\log \left(e^{\mathsf{log1p}\left(x\right) - \log x}\right)}}{n} \]
      exp-diff [=>] 77.7	\[ \frac{\log \color{blue}{\left(\frac{e^{\mathsf{log1p}\left(x\right)}}{e^{\log x}}\right)}}{n} \]
      log1p-udef [=>] 77.7	\[ \frac{\log \left(\frac{e^{\color{blue}{\log \left(1 + x\right)}}}{e^{\log x}}\right)}{n} \]
      add-exp-log [<=] 77.7	\[ \frac{\log \left(\frac{\color{blue}{1 + x}}{e^{\log x}}\right)}{n} \]
      +-commutative [=>] 77.7	\[ \frac{\log \left(\frac{\color{blue}{x + 1}}{e^{\log x}}\right)}{n} \]
      add-exp-log [<=] 77.7	\[ \frac{\log \left(\frac{x + 1}{\color{blue}{x}}\right)}{n} \]

   if 8200 < x

   1. Initial program 68.1%

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

   2. Taylor expanded in x around inf 97.6%

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

   3. Simplified 97.6%

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

      [Start] 97.6	\[ \frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x} \]
      mul-1-neg [=>] 97.6	\[ \frac{e^{\color{blue}{-\frac{\log \left(\frac{1}{x}\right)}{n}}}}{n \cdot x} \]
      log-rec [=>] 97.6	\[ \frac{e^{-\frac{\color{blue}{-\log x}}{n}}}{n \cdot x} \]
      mul-1-neg [<=] 97.6	\[ \frac{e^{-\frac{\color{blue}{-1 \cdot \log x}}{n}}}{n \cdot x} \]
      distribute-neg-frac [=>] 97.6	\[ \frac{e^{\color{blue}{\frac{--1 \cdot \log x}{n}}}}{n \cdot x} \]
      mul-1-neg [=>] 97.6	\[ \frac{e^{\frac{-\color{blue}{\left(-\log x\right)}}{n}}}{n \cdot x} \]
      remove-double-neg [=>] 97.6	\[ \frac{e^{\frac{\color{blue}{\log x}}{n}}}{n \cdot x} \]
      *-commutative [=>] 97.6	\[ \frac{e^{\frac{\log x}{n}}}{\color{blue}{x \cdot n}} \]

   4. Applied egg-rr 97.1%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}}\right)}^{3}} \]
      Proof

      [Start] 97.6	\[ \frac{e^{\frac{\log x}{n}}}{x \cdot n} \]
      add-cube-cbrt [=>] 97.1	\[ \color{blue}{\left(\sqrt[3]{\frac{e^{\frac{\log x}{n}}}{x \cdot n}} \cdot \sqrt[3]{\frac{e^{\frac{\log x}{n}}}{x \cdot n}}\right) \cdot \sqrt[3]{\frac{e^{\frac{\log x}{n}}}{x \cdot n}}} \]
      pow3 [=>] 97.1	\[ \color{blue}{{\left(\sqrt[3]{\frac{e^{\frac{\log x}{n}}}{x \cdot n}}\right)}^{3}} \]
      div-inv [=>] 97.1	\[ {\left(\sqrt[3]{\frac{e^{\color{blue}{\log x \cdot \frac{1}{n}}}}{x \cdot n}}\right)}^{3} \]
      exp-to-pow [=>] 97.1	\[ {\left(\sqrt[3]{\frac{\color{blue}{{x}^{\left(\frac{1}{n}\right)}}}{x \cdot n}}\right)}^{3} \]

   5. Applied egg-rr 99.1%

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

      [Start] 97.1	\[ {\left(\sqrt[3]{\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}}\right)}^{3} \]
      rem-cube-cbrt [=>] 97.6	\[ \color{blue}{\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}} \]
      associate-/r* [=>] 99.1	\[ \color{blue}{\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{x}}{n}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 8200:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{x}}{n}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	89.4%
Cost	7044

\[\begin{array}{l} \mathbf{if}\;x \leq 8200:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \end{array} \]

Alternative 2
Accuracy	76.3%
Cost	6980

\[\begin{array}{l} \mathbf{if}\;x \leq 800:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{+144}:\\ \;\;\;\;\frac{\frac{1}{\frac{\frac{0.041666666666666664}{x} + -0.08333333333333333}{x} + \left(x + 0.5\right)}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{n} \cdot 0\\ \end{array} \]

Alternative 3
Accuracy	75.9%
Cost	6852

\[\begin{array}{l} \mathbf{if}\;x \leq 0.52:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{+144}:\\ \;\;\;\;\frac{\frac{1}{\frac{\frac{0.041666666666666664}{x} + -0.08333333333333333}{x} + \left(x + 0.5\right)}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{n} \cdot 0\\ \end{array} \]

Alternative 4
Accuracy	73.5%
Cost	6788

\[\begin{array}{l} \mathbf{if}\;x \leq 0.35:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+191}:\\ \;\;\;\;\frac{\frac{1}{\frac{\frac{0.041666666666666664}{x} + -0.08333333333333333}{x} + \left(x + 0.5\right)}}{n}\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(1 + \frac{1}{x \cdot n}\right)\\ \end{array} \]

Alternative 5
Accuracy	75.6%
Cost	6788

\[\begin{array}{l} \mathbf{if}\;x \leq 0.35:\\ \;\;\;\;\frac{-\log x}{n}\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{+144}:\\ \;\;\;\;\frac{\frac{1}{\frac{\frac{0.041666666666666664}{x} + -0.08333333333333333}{x} + \left(x + 0.5\right)}}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{n} \cdot 0\\ \end{array} \]

Alternative 6
Accuracy	49.4%
Cost	836

\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -1000000000000:\\ \;\;\;\;-1 + \left(1 + \frac{1}{x \cdot n}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x + 0.5}}{n}\\ \end{array} \]

Alternative 7
Accuracy	42.1%
Cost	448

\[\frac{\frac{1}{x + 0.5}}{n} \]

Alternative 8
Accuracy	37.3%
Cost	320

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

Alternative 9
Accuracy	38.1%
Cost	320

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

Alternative 10
Accuracy	38.1%
Cost	320

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

Alternative 11
Accuracy	4.6%
Cost	192

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

Reproduce

herbie shell --seed 2023135 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))