2nthrt (problem 3.4.6)

Percentage Accurate: 53.0% → 91.3%

Time: 1.3min

Alternatives: 15

Speedup: 1.8×

• could not determine a ground truth

  1. x = -2
  2. n = 4.94066e-324

Specification

Math

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

FPCore

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

C

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Initial Program: 53.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Alternative 1: 91.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\frac{x}{n} - \mathsf{expm1}\left(\frac{\log x}{n}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 1.0)
   (- (/ x n) (expm1 (/ (log x) n)))
   (/ (pow x (/ 1.0 n)) (* x n))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 1.0) {
		tmp = (x / n) - expm1((log(x) / n));
	} else {
		tmp = pow(x, (1.0 / n)) / (x * n);
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 1.0) {
		tmp = (x / n) - Math.expm1((Math.log(x) / n));
	} else {
		tmp = Math.pow(x, (1.0 / n)) / (x * n);
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 1.0:
		tmp = (x / n) - math.expm1((math.log(x) / n))
	else:
		tmp = math.pow(x, (1.0 / n)) / (x * n)
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 1.0)
		tmp = Float64(Float64(x / n) - expm1(Float64(log(x) / n)));
	else
		tmp = Float64((x ^ Float64(1.0 / n)) / Float64(x * n));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[x, 1.0], N[(N[(x / n), $MachinePrecision] - N[(Exp[N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision], N[(N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision] / N[(x * n), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 49.1%

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

   3. Taylor expanded in x around 0 49.0%

      \[\leadsto \color{blue}{\left(1 + \frac{x}{n}\right) - e^{\frac{\log x}{n}}} \]
   4. Step-by-step derivation

      1. *-rgt-identity 49.0%

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

      2. associate-/l* 49.0%

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

      3. exp-to-pow 49.0%

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

   5. Simplified 49.0%

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

   6. Taylor expanded in x around 0 49.0%

      \[\leadsto \color{blue}{\left(1 + \frac{x}{n}\right) - e^{\frac{\log x}{n}}} \]
   7. Step-by-step derivation

      1. associate--l+ 49.0%

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

      2. *-rgt-identity 49.0%

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

      3. associate-*r/ 49.0%

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

      4. exp-to-pow 49.0%

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

      5. +-commutative 49.0%

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

      6. associate-+l- 49.3%

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

      7. exp-to-pow 49.3%

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

      8. associate-*r/ 49.3%

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

      9. *-rgt-identity 49.3%

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

      10. expm1-define 93.6%

          \[\leadsto \frac{x}{n} - \color{blue}{\mathsf{expm1}\left(\frac{\log x}{n}\right)} \]

   8. Simplified 93.6%

      \[\leadsto \color{blue}{\frac{x}{n} - \mathsf{expm1}\left(\frac{\log x}{n}\right)} \]

   if 1 < x

   1. Initial program 67.4%

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

   3. Taylor expanded in x around inf 96.0%

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}} \]
   4. Step-by-step derivation

      1. log-rec 96.0%

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

      2. mul-1-neg 96.0%

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

      3. associate-*r/ 96.0%

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

      4. neg-mul-1 96.0%

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

      5. mul-1-neg 96.0%

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

      6. remove-double-neg 96.0%

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

      7. *-rgt-identity 96.0%

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

      8. associate-/l* 96.0%

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

      9. exp-to-pow 96.0%

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

      10. *-commutative 96.0%

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

   5. Simplified 96.0%

      \[\leadsto \color{blue}{\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 78.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{if}\;\frac{1}{n} \leq -40000:\\ \;\;\;\;\frac{t\_0}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq -2 \cdot 10^{-75}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-90}:\\ \;\;\;\;\frac{\frac{1 + \frac{-0.5 + \frac{0.3333333333333333}{x}}{x}}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-6}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{x}{n} + 2\right) + -1\right) - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))) (t_1 (/ (log (/ (+ x 1.0) x)) n)))
   (if (<= (/ 1.0 n) -40000.0)
     (/ t_0 (* x n))
     (if (<= (/ 1.0 n) -2e-75)
       t_1
       (if (<= (/ 1.0 n) -5e-90)
         (/ (/ (+ 1.0 (/ (+ -0.5 (/ 0.3333333333333333 x)) x)) x) n)
         (if (<= (/ 1.0 n) 2e-6) t_1 (- (+ (+ (/ x n) 2.0) -1.0) t_0)))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = log(((x + 1.0) / x)) / n;
	double tmp;
	if ((1.0 / n) <= -40000.0) {
		tmp = t_0 / (x * n);
	} else if ((1.0 / n) <= -2e-75) {
		tmp = t_1;
	} else if ((1.0 / n) <= -5e-90) {
		tmp = ((1.0 + ((-0.5 + (0.3333333333333333 / x)) / x)) / x) / n;
	} else if ((1.0 / n) <= 2e-6) {
		tmp = t_1;
	} else {
		tmp = (((x / n) + 2.0) + -1.0) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    t_1 = log(((x + 1.0d0) / x)) / n
    if ((1.0d0 / n) <= (-40000.0d0)) then
        tmp = t_0 / (x * n)
    else if ((1.0d0 / n) <= (-2d-75)) then
        tmp = t_1
    else if ((1.0d0 / n) <= (-5d-90)) then
        tmp = ((1.0d0 + (((-0.5d0) + (0.3333333333333333d0 / x)) / x)) / x) / n
    else if ((1.0d0 / n) <= 2d-6) then
        tmp = t_1
    else
        tmp = (((x / n) + 2.0d0) + (-1.0d0)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double t_1 = Math.log(((x + 1.0) / x)) / n;
	double tmp;
	if ((1.0 / n) <= -40000.0) {
		tmp = t_0 / (x * n);
	} else if ((1.0 / n) <= -2e-75) {
		tmp = t_1;
	} else if ((1.0 / n) <= -5e-90) {
		tmp = ((1.0 + ((-0.5 + (0.3333333333333333 / x)) / x)) / x) / n;
	} else if ((1.0 / n) <= 2e-6) {
		tmp = t_1;
	} else {
		tmp = (((x / n) + 2.0) + -1.0) - t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	t_1 = math.log(((x + 1.0) / x)) / n
	tmp = 0
	if (1.0 / n) <= -40000.0:
		tmp = t_0 / (x * n)
	elif (1.0 / n) <= -2e-75:
		tmp = t_1
	elif (1.0 / n) <= -5e-90:
		tmp = ((1.0 + ((-0.5 + (0.3333333333333333 / x)) / x)) / x) / n
	elif (1.0 / n) <= 2e-6:
		tmp = t_1
	else:
		tmp = (((x / n) + 2.0) + -1.0) - t_0
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = Float64(log(Float64(Float64(x + 1.0) / x)) / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -40000.0)
		tmp = Float64(t_0 / Float64(x * n));
	elseif (Float64(1.0 / n) <= -2e-75)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= -5e-90)
		tmp = Float64(Float64(Float64(1.0 + Float64(Float64(-0.5 + Float64(0.3333333333333333 / x)) / x)) / x) / n);
	elseif (Float64(1.0 / n) <= 2e-6)
		tmp = t_1;
	else
		tmp = Float64(Float64(Float64(Float64(x / n) + 2.0) + -1.0) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	t_1 = log(((x + 1.0) / x)) / n;
	tmp = 0.0;
	if ((1.0 / n) <= -40000.0)
		tmp = t_0 / (x * n);
	elseif ((1.0 / n) <= -2e-75)
		tmp = t_1;
	elseif ((1.0 / n) <= -5e-90)
		tmp = ((1.0 + ((-0.5 + (0.3333333333333333 / x)) / x)) / x) / n;
	elseif ((1.0 / n) <= 2e-6)
		tmp = t_1;
	else
		tmp = (((x / n) + 2.0) + -1.0) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -40000.0], N[(t$95$0 / N[(x * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-75], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-90], N[(N[(N[(1.0 + N[(N[(-0.5 + N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-6], t$95$1, N[(N[(N[(N[(x / n), $MachinePrecision] + 2.0), $MachinePrecision] + -1.0), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -4e4

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 100.0%

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}} \]
   4. Step-by-step derivation

      1. log-rec 100.0%

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

      2. mul-1-neg 100.0%

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

      3. associate-*r/ 100.0%

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

      4. neg-mul-1 100.0%

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

      5. mul-1-neg 100.0%

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

      6. remove-double-neg 100.0%

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

      7. *-rgt-identity 100.0%

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

      8. associate-/l* 100.0%

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

      9. exp-to-pow 100.0%

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

      10. *-commutative 100.0%

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

   5. Simplified 100.0%

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

   if -4e4 < (/.f64 #s(literal 1 binary64) n) < -1.9999999999999999e-75 or -5.00000000000000019e-90 < (/.f64 #s(literal 1 binary64) n) < 1.99999999999999991e-6

   1. Initial program 31.2%

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

   3. Taylor expanded in n around inf 81.3%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   4. Step-by-step derivation

      1. log1p-define 81.3%

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

   5. Simplified 81.3%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]
   6. Step-by-step derivation

      1. log1p-undefine 81.3%

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

      2. diff-log 81.4%

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

   7. Applied egg-rr 81.4%

      \[\leadsto \frac{\color{blue}{\log \left(\frac{1 + x}{x}\right)}}{n} \]
   8. Step-by-step derivation

      1. +-commutative 81.4%

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

   9. Simplified 81.4%

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

   if -1.9999999999999999e-75 < (/.f64 #s(literal 1 binary64) n) < -5.00000000000000019e-90

   1. Initial program 4.6%

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

   3. Taylor expanded in n around inf 25.2%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   4. Step-by-step derivation

      1. log1p-define 25.2%

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

   5. Simplified 25.2%

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

   6. Taylor expanded in x around -inf 89.1%

      \[\leadsto \frac{\color{blue}{-1 \cdot \frac{-1 \cdot \frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{x} - 1}{x}}}{n} \]

   7. Taylor expanded in n around 0 88.7%

      \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{x} - 1}{n \cdot x}} \]

   9. Simplified 89.1%

      \[\leadsto \color{blue}{\frac{\frac{1 + \frac{-0.5 + \frac{0.3333333333333333}{x}}{x}}{x}}{n}} \]

   if 1.99999999999999991e-6 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 67.2%

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

   3. Taylor expanded in x around 0 66.7%

      \[\leadsto \color{blue}{\left(1 + \frac{x}{n}\right) - e^{\frac{\log x}{n}}} \]
   4. Step-by-step derivation

      1. *-rgt-identity 66.7%

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

      2. associate-/l* 66.7%

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

      3. exp-to-pow 66.7%

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

   5. Simplified 66.7%

      \[\leadsto \color{blue}{\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}} \]
   6. Step-by-step derivation

      1. expm1-log1p-u 66.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 + \frac{x}{n}\right)\right)} - {x}^{\left(\frac{1}{n}\right)} \]

      2. expm1-undefine 66.7%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(1 + \frac{x}{n}\right)} - 1\right)} - {x}^{\left(\frac{1}{n}\right)} \]

   7. Applied egg-rr 66.7%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(1 + \frac{x}{n}\right)} - 1\right)} - {x}^{\left(\frac{1}{n}\right)} \]
   8. Step-by-step derivation

      1. sub-neg 66.7%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(1 + \frac{x}{n}\right)} + \left(-1\right)\right)} - {x}^{\left(\frac{1}{n}\right)} \]

      2. log1p-undefine 66.7%

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

      3. rem-exp-log 66.7%

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

      4. associate-+r+ 66.8%

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

      5. metadata-eval 66.8%

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

      6. metadata-eval 66.8%

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

   9. Simplified 66.8%

      \[\leadsto \color{blue}{\left(\left(2 + \frac{x}{n}\right) + -1\right)} - {x}^{\left(\frac{1}{n}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -40000:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq -2 \cdot 10^{-75}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-90}:\\ \;\;\;\;\frac{\frac{1 + \frac{-0.5 + \frac{0.3333333333333333}{x}}{x}}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{x}{n} + 2\right) + -1\right) - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{if}\;\frac{1}{n} \leq -40000:\\ \;\;\;\;\frac{t\_0}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq -2 \cdot 10^{-75}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-90}:\\ \;\;\;\;\frac{\frac{1 + \frac{-0.5 + \frac{0.3333333333333333}{x}}{x}}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-6}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))) (t_1 (/ (log (/ (+ x 1.0) x)) n)))
   (if (<= (/ 1.0 n) -40000.0)
     (/ t_0 (* x n))
     (if (<= (/ 1.0 n) -2e-75)
       t_1
       (if (<= (/ 1.0 n) -5e-90)
         (/ (/ (+ 1.0 (/ (+ -0.5 (/ 0.3333333333333333 x)) x)) x) n)
         (if (<= (/ 1.0 n) 2e-6) t_1 (- (+ 1.0 (/ x n)) t_0)))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = log(((x + 1.0) / x)) / n;
	double tmp;
	if ((1.0 / n) <= -40000.0) {
		tmp = t_0 / (x * n);
	} else if ((1.0 / n) <= -2e-75) {
		tmp = t_1;
	} else if ((1.0 / n) <= -5e-90) {
		tmp = ((1.0 + ((-0.5 + (0.3333333333333333 / x)) / x)) / x) / n;
	} else if ((1.0 / n) <= 2e-6) {
		tmp = t_1;
	} else {
		tmp = (1.0 + (x / n)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    t_1 = log(((x + 1.0d0) / x)) / n
    if ((1.0d0 / n) <= (-40000.0d0)) then
        tmp = t_0 / (x * n)
    else if ((1.0d0 / n) <= (-2d-75)) then
        tmp = t_1
    else if ((1.0d0 / n) <= (-5d-90)) then
        tmp = ((1.0d0 + (((-0.5d0) + (0.3333333333333333d0 / x)) / x)) / x) / n
    else if ((1.0d0 / n) <= 2d-6) then
        tmp = t_1
    else
        tmp = (1.0d0 + (x / n)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double t_1 = Math.log(((x + 1.0) / x)) / n;
	double tmp;
	if ((1.0 / n) <= -40000.0) {
		tmp = t_0 / (x * n);
	} else if ((1.0 / n) <= -2e-75) {
		tmp = t_1;
	} else if ((1.0 / n) <= -5e-90) {
		tmp = ((1.0 + ((-0.5 + (0.3333333333333333 / x)) / x)) / x) / n;
	} else if ((1.0 / n) <= 2e-6) {
		tmp = t_1;
	} else {
		tmp = (1.0 + (x / n)) - t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	t_1 = math.log(((x + 1.0) / x)) / n
	tmp = 0
	if (1.0 / n) <= -40000.0:
		tmp = t_0 / (x * n)
	elif (1.0 / n) <= -2e-75:
		tmp = t_1
	elif (1.0 / n) <= -5e-90:
		tmp = ((1.0 + ((-0.5 + (0.3333333333333333 / x)) / x)) / x) / n
	elif (1.0 / n) <= 2e-6:
		tmp = t_1
	else:
		tmp = (1.0 + (x / n)) - t_0
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = Float64(log(Float64(Float64(x + 1.0) / x)) / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -40000.0)
		tmp = Float64(t_0 / Float64(x * n));
	elseif (Float64(1.0 / n) <= -2e-75)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= -5e-90)
		tmp = Float64(Float64(Float64(1.0 + Float64(Float64(-0.5 + Float64(0.3333333333333333 / x)) / x)) / x) / n);
	elseif (Float64(1.0 / n) <= 2e-6)
		tmp = t_1;
	else
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	t_1 = log(((x + 1.0) / x)) / n;
	tmp = 0.0;
	if ((1.0 / n) <= -40000.0)
		tmp = t_0 / (x * n);
	elseif ((1.0 / n) <= -2e-75)
		tmp = t_1;
	elseif ((1.0 / n) <= -5e-90)
		tmp = ((1.0 + ((-0.5 + (0.3333333333333333 / x)) / x)) / x) / n;
	elseif ((1.0 / n) <= 2e-6)
		tmp = t_1;
	else
		tmp = (1.0 + (x / n)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -40000.0], N[(t$95$0 / N[(x * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-75], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-90], N[(N[(N[(1.0 + N[(N[(-0.5 + N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-6], t$95$1, N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -4e4

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 100.0%

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}} \]
   4. Step-by-step derivation

      1. log-rec 100.0%

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

      2. mul-1-neg 100.0%

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

      3. associate-*r/ 100.0%

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

      4. neg-mul-1 100.0%

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

      5. mul-1-neg 100.0%

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

      6. remove-double-neg 100.0%

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

      7. *-rgt-identity 100.0%

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

      8. associate-/l* 100.0%

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

      9. exp-to-pow 100.0%

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

      10. *-commutative 100.0%

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

   5. Simplified 100.0%

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

   if -4e4 < (/.f64 #s(literal 1 binary64) n) < -1.9999999999999999e-75 or -5.00000000000000019e-90 < (/.f64 #s(literal 1 binary64) n) < 1.99999999999999991e-6

   1. Initial program 31.2%

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

   3. Taylor expanded in n around inf 81.3%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   4. Step-by-step derivation

      1. log1p-define 81.3%

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

   5. Simplified 81.3%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]
   6. Step-by-step derivation

      1. log1p-undefine 81.3%

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

      2. diff-log 81.4%

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

   7. Applied egg-rr 81.4%

      \[\leadsto \frac{\color{blue}{\log \left(\frac{1 + x}{x}\right)}}{n} \]
   8. Step-by-step derivation

      1. +-commutative 81.4%

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

   9. Simplified 81.4%

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

   if -1.9999999999999999e-75 < (/.f64 #s(literal 1 binary64) n) < -5.00000000000000019e-90

   1. Initial program 4.6%

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

   3. Taylor expanded in n around inf 25.2%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   4. Step-by-step derivation

      1. log1p-define 25.2%

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

   5. Simplified 25.2%

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

   6. Taylor expanded in x around -inf 89.1%

      \[\leadsto \frac{\color{blue}{-1 \cdot \frac{-1 \cdot \frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{x} - 1}{x}}}{n} \]

   7. Taylor expanded in n around 0 88.7%

      \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{x} - 1}{n \cdot x}} \]

   9. Simplified 89.1%

      \[\leadsto \color{blue}{\frac{\frac{1 + \frac{-0.5 + \frac{0.3333333333333333}{x}}{x}}{x}}{n}} \]

   if 1.99999999999999991e-6 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 67.2%

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

   3. Taylor expanded in x around 0 66.7%

      \[\leadsto \color{blue}{\left(1 + \frac{x}{n}\right) - e^{\frac{\log x}{n}}} \]
   4. Step-by-step derivation

      1. *-rgt-identity 66.7%

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

      2. associate-/l* 66.7%

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

      3. exp-to-pow 66.7%

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

   5. Simplified 66.7%

      \[\leadsto \color{blue}{\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 4: 77.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{if}\;\frac{1}{n} \leq -40000:\\ \;\;\;\;\frac{t\_0}{x \cdot n}\\ \mathbf{elif}\;\frac{1}{n} \leq -2 \cdot 10^{-75}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-90}:\\ \;\;\;\;\frac{\frac{1 + \frac{-0.5 + \frac{0.3333333333333333}{x}}{x}}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-6}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;1 - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))) (t_1 (/ (log (/ (+ x 1.0) x)) n)))
   (if (<= (/ 1.0 n) -40000.0)
     (/ t_0 (* x n))
     (if (<= (/ 1.0 n) -2e-75)
       t_1
       (if (<= (/ 1.0 n) -5e-90)
         (/ (/ (+ 1.0 (/ (+ -0.5 (/ 0.3333333333333333 x)) x)) x) n)
         (if (<= (/ 1.0 n) 2e-6) t_1 (- 1.0 t_0)))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double t_1 = log(((x + 1.0) / x)) / n;
	double tmp;
	if ((1.0 / n) <= -40000.0) {
		tmp = t_0 / (x * n);
	} else if ((1.0 / n) <= -2e-75) {
		tmp = t_1;
	} else if ((1.0 / n) <= -5e-90) {
		tmp = ((1.0 + ((-0.5 + (0.3333333333333333 / x)) / x)) / x) / n;
	} else if ((1.0 / n) <= 2e-6) {
		tmp = t_1;
	} else {
		tmp = 1.0 - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    t_1 = log(((x + 1.0d0) / x)) / n
    if ((1.0d0 / n) <= (-40000.0d0)) then
        tmp = t_0 / (x * n)
    else if ((1.0d0 / n) <= (-2d-75)) then
        tmp = t_1
    else if ((1.0d0 / n) <= (-5d-90)) then
        tmp = ((1.0d0 + (((-0.5d0) + (0.3333333333333333d0 / x)) / x)) / x) / n
    else if ((1.0d0 / n) <= 2d-6) then
        tmp = t_1
    else
        tmp = 1.0d0 - t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double t_1 = Math.log(((x + 1.0) / x)) / n;
	double tmp;
	if ((1.0 / n) <= -40000.0) {
		tmp = t_0 / (x * n);
	} else if ((1.0 / n) <= -2e-75) {
		tmp = t_1;
	} else if ((1.0 / n) <= -5e-90) {
		tmp = ((1.0 + ((-0.5 + (0.3333333333333333 / x)) / x)) / x) / n;
	} else if ((1.0 / n) <= 2e-6) {
		tmp = t_1;
	} else {
		tmp = 1.0 - t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	t_1 = math.log(((x + 1.0) / x)) / n
	tmp = 0
	if (1.0 / n) <= -40000.0:
		tmp = t_0 / (x * n)
	elif (1.0 / n) <= -2e-75:
		tmp = t_1
	elif (1.0 / n) <= -5e-90:
		tmp = ((1.0 + ((-0.5 + (0.3333333333333333 / x)) / x)) / x) / n
	elif (1.0 / n) <= 2e-6:
		tmp = t_1
	else:
		tmp = 1.0 - t_0
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	t_1 = Float64(log(Float64(Float64(x + 1.0) / x)) / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -40000.0)
		tmp = Float64(t_0 / Float64(x * n));
	elseif (Float64(1.0 / n) <= -2e-75)
		tmp = t_1;
	elseif (Float64(1.0 / n) <= -5e-90)
		tmp = Float64(Float64(Float64(1.0 + Float64(Float64(-0.5 + Float64(0.3333333333333333 / x)) / x)) / x) / n);
	elseif (Float64(1.0 / n) <= 2e-6)
		tmp = t_1;
	else
		tmp = Float64(1.0 - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	t_1 = log(((x + 1.0) / x)) / n;
	tmp = 0.0;
	if ((1.0 / n) <= -40000.0)
		tmp = t_0 / (x * n);
	elseif ((1.0 / n) <= -2e-75)
		tmp = t_1;
	elseif ((1.0 / n) <= -5e-90)
		tmp = ((1.0 + ((-0.5 + (0.3333333333333333 / x)) / x)) / x) / n;
	elseif ((1.0 / n) <= 2e-6)
		tmp = t_1;
	else
		tmp = 1.0 - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -40000.0], N[(t$95$0 / N[(x * n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-75], t$95$1, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-90], N[(N[(N[(1.0 + N[(N[(-0.5 + N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-6], t$95$1, N[(1.0 - t$95$0), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -4e4

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 100.0%

      \[\leadsto \color{blue}{\frac{e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}}{n \cdot x}} \]
   4. Step-by-step derivation

      1. log-rec 100.0%

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

      2. mul-1-neg 100.0%

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

      3. associate-*r/ 100.0%

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

      4. neg-mul-1 100.0%

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

      5. mul-1-neg 100.0%

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

      6. remove-double-neg 100.0%

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

      7. *-rgt-identity 100.0%

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

      8. associate-/l* 100.0%

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

      9. exp-to-pow 100.0%

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

      10. *-commutative 100.0%

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

   5. Simplified 100.0%

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

   if -4e4 < (/.f64 #s(literal 1 binary64) n) < -1.9999999999999999e-75 or -5.00000000000000019e-90 < (/.f64 #s(literal 1 binary64) n) < 1.99999999999999991e-6

   1. Initial program 31.2%

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

   3. Taylor expanded in n around inf 81.3%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   4. Step-by-step derivation

      1. log1p-define 81.3%

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

   5. Simplified 81.3%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]
   6. Step-by-step derivation

      1. log1p-undefine 81.3%

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

      2. diff-log 81.4%

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

   7. Applied egg-rr 81.4%

      \[\leadsto \frac{\color{blue}{\log \left(\frac{1 + x}{x}\right)}}{n} \]
   8. Step-by-step derivation

      1. +-commutative 81.4%

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

   9. Simplified 81.4%

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

   if -1.9999999999999999e-75 < (/.f64 #s(literal 1 binary64) n) < -5.00000000000000019e-90

   1. Initial program 4.6%

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

   3. Taylor expanded in n around inf 25.2%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   4. Step-by-step derivation

      1. log1p-define 25.2%

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

   5. Simplified 25.2%

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

   6. Taylor expanded in x around -inf 89.1%

      \[\leadsto \frac{\color{blue}{-1 \cdot \frac{-1 \cdot \frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{x} - 1}{x}}}{n} \]

   7. Taylor expanded in n around 0 88.7%

      \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{x} - 1}{n \cdot x}} \]

   9. Simplified 89.1%

      \[\leadsto \color{blue}{\frac{\frac{1 + \frac{-0.5 + \frac{0.3333333333333333}{x}}{x}}{x}}{n}} \]

   if 1.99999999999999991e-6 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 67.2%

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

   3. Taylor expanded in x around 0 63.9%

      \[\leadsto \color{blue}{1 - e^{\frac{\log x}{n}}} \]
   4. Step-by-step derivation

      1. *-rgt-identity 63.9%

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

      2. associate-/l* 63.8%

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

      3. exp-to-pow 63.9%

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

   5. Simplified 63.9%

      \[\leadsto \color{blue}{1 - {x}^{\left(\frac{1}{n}\right)}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 5: 70.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\log \left(\frac{x + 1}{x}\right)}{n}\\ \mathbf{if}\;\frac{1}{n} \leq -40000:\\ \;\;\;\;{x}^{-3} \cdot \frac{0.3333333333333333}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq -2 \cdot 10^{-75}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\frac{1}{n} \leq -5 \cdot 10^{-90}:\\ \;\;\;\;\frac{\frac{1 + \frac{-0.5 + \frac{0.3333333333333333}{x}}{x}}{x}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{-6}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (log (/ (+ x 1.0) x)) n)))
   (if (<= (/ 1.0 n) -40000.0)
     (* (pow x -3.0) (/ 0.3333333333333333 n))
     (if (<= (/ 1.0 n) -2e-75)
       t_0
       (if (<= (/ 1.0 n) -5e-90)
         (/ (/ (+ 1.0 (/ (+ -0.5 (/ 0.3333333333333333 x)) x)) x) n)
         (if (<= (/ 1.0 n) 2e-6) t_0 (- 1.0 (pow x (/ 1.0 n)))))))))

C

double code(double x, double n) {
	double t_0 = log(((x + 1.0) / x)) / n;
	double tmp;
	if ((1.0 / n) <= -40000.0) {
		tmp = pow(x, -3.0) * (0.3333333333333333 / n);
	} else if ((1.0 / n) <= -2e-75) {
		tmp = t_0;
	} else if ((1.0 / n) <= -5e-90) {
		tmp = ((1.0 + ((-0.5 + (0.3333333333333333 / x)) / x)) / x) / n;
	} else if ((1.0 / n) <= 2e-6) {
		tmp = t_0;
	} else {
		tmp = 1.0 - pow(x, (1.0 / n));
	}
	return tmp;
}

Fortran

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(((x + 1.0d0) / x)) / n
    if ((1.0d0 / n) <= (-40000.0d0)) then
        tmp = (x ** (-3.0d0)) * (0.3333333333333333d0 / n)
    else if ((1.0d0 / n) <= (-2d-75)) then
        tmp = t_0
    else if ((1.0d0 / n) <= (-5d-90)) then
        tmp = ((1.0d0 + (((-0.5d0) + (0.3333333333333333d0 / x)) / x)) / x) / n
    else if ((1.0d0 / n) <= 2d-6) then
        tmp = t_0
    else
        tmp = 1.0d0 - (x ** (1.0d0 / n))
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = Math.log(((x + 1.0) / x)) / n;
	double tmp;
	if ((1.0 / n) <= -40000.0) {
		tmp = Math.pow(x, -3.0) * (0.3333333333333333 / n);
	} else if ((1.0 / n) <= -2e-75) {
		tmp = t_0;
	} else if ((1.0 / n) <= -5e-90) {
		tmp = ((1.0 + ((-0.5 + (0.3333333333333333 / x)) / x)) / x) / n;
	} else if ((1.0 / n) <= 2e-6) {
		tmp = t_0;
	} else {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.log(((x + 1.0) / x)) / n
	tmp = 0
	if (1.0 / n) <= -40000.0:
		tmp = math.pow(x, -3.0) * (0.3333333333333333 / n)
	elif (1.0 / n) <= -2e-75:
		tmp = t_0
	elif (1.0 / n) <= -5e-90:
		tmp = ((1.0 + ((-0.5 + (0.3333333333333333 / x)) / x)) / x) / n
	elif (1.0 / n) <= 2e-6:
		tmp = t_0
	else:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	return tmp

Julia

function code(x, n)
	t_0 = Float64(log(Float64(Float64(x + 1.0) / x)) / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -40000.0)
		tmp = Float64((x ^ -3.0) * Float64(0.3333333333333333 / n));
	elseif (Float64(1.0 / n) <= -2e-75)
		tmp = t_0;
	elseif (Float64(1.0 / n) <= -5e-90)
		tmp = Float64(Float64(Float64(1.0 + Float64(Float64(-0.5 + Float64(0.3333333333333333 / x)) / x)) / x) / n);
	elseif (Float64(1.0 / n) <= 2e-6)
		tmp = t_0;
	else
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = log(((x + 1.0) / x)) / n;
	tmp = 0.0;
	if ((1.0 / n) <= -40000.0)
		tmp = (x ^ -3.0) * (0.3333333333333333 / n);
	elseif ((1.0 / n) <= -2e-75)
		tmp = t_0;
	elseif ((1.0 / n) <= -5e-90)
		tmp = ((1.0 + ((-0.5 + (0.3333333333333333 / x)) / x)) / x) / n;
	elseif ((1.0 / n) <= 2e-6)
		tmp = t_0;
	else
		tmp = 1.0 - (x ^ (1.0 / n));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[Log[N[(N[(x + 1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -40000.0], N[(N[Power[x, -3.0], $MachinePrecision] * N[(0.3333333333333333 / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], -2e-75], t$95$0, If[LessEqual[N[(1.0 / n), $MachinePrecision], -5e-90], N[(N[(N[(1.0 + N[(N[(-0.5 + N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e-6], t$95$0, N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -4e4

   1. Initial program 100.0%

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

   3. Taylor expanded in n around inf 42.4%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   4. Step-by-step derivation

      1. log1p-define 42.4%

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

   5. Simplified 42.4%

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

   6. Taylor expanded in x around -inf 55.5%

      \[\leadsto \frac{\color{blue}{-1 \cdot \frac{-1 \cdot \frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{x} - 1}{x}}}{n} \]

   7. Taylor expanded in x around 0 79.1%

      \[\leadsto \color{blue}{\frac{0.3333333333333333}{n \cdot {x}^{3}}} \]
   8. Step-by-step derivation

      1. associate-/r* 79.1%

         \[\leadsto \color{blue}{\frac{\frac{0.3333333333333333}{n}}{{x}^{3}}} \]

   9. Simplified 79.1%

      \[\leadsto \color{blue}{\frac{\frac{0.3333333333333333}{n}}{{x}^{3}}} \]
   10. Step-by-step derivation

       1. clear-num 79.1%

          \[\leadsto \color{blue}{\frac{1}{\frac{{x}^{3}}{\frac{0.3333333333333333}{n}}}} \]

       2. associate-/r/ 79.1%

          \[\leadsto \color{blue}{\frac{1}{{x}^{3}} \cdot \frac{0.3333333333333333}{n}} \]

       3. pow-flip 79.1%

          \[\leadsto \color{blue}{{x}^{\left(-3\right)}} \cdot \frac{0.3333333333333333}{n} \]

       4. metadata-eval 79.1%

          \[\leadsto {x}^{\color{blue}{-3}} \cdot \frac{0.3333333333333333}{n} \]

   11. Applied egg-rr 79.1%

       \[\leadsto \color{blue}{{x}^{-3} \cdot \frac{0.3333333333333333}{n}} \]

   if -4e4 < (/.f64 #s(literal 1 binary64) n) < -1.9999999999999999e-75 or -5.00000000000000019e-90 < (/.f64 #s(literal 1 binary64) n) < 1.99999999999999991e-6

   1. Initial program 31.2%

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

   3. Taylor expanded in n around inf 81.3%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   4. Step-by-step derivation

      1. log1p-define 81.3%

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

   5. Simplified 81.3%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(x\right) - \log x}{n}} \]
   6. Step-by-step derivation

      1. log1p-undefine 81.3%

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

      2. diff-log 81.4%

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

   7. Applied egg-rr 81.4%

      \[\leadsto \frac{\color{blue}{\log \left(\frac{1 + x}{x}\right)}}{n} \]
   8. Step-by-step derivation

      1. +-commutative 81.4%

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

   9. Simplified 81.4%

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

   if -1.9999999999999999e-75 < (/.f64 #s(literal 1 binary64) n) < -5.00000000000000019e-90

   1. Initial program 4.6%

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

   3. Taylor expanded in n around inf 25.2%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   4. Step-by-step derivation

      1. log1p-define 25.2%

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

   5. Simplified 25.2%

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

   6. Taylor expanded in x around -inf 89.1%

      \[\leadsto \frac{\color{blue}{-1 \cdot \frac{-1 \cdot \frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{x} - 1}{x}}}{n} \]

   7. Taylor expanded in n around 0 88.7%

      \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{x} - 1}{n \cdot x}} \]

   9. Simplified 89.1%

      \[\leadsto \color{blue}{\frac{\frac{1 + \frac{-0.5 + \frac{0.3333333333333333}{x}}{x}}{x}}{n}} \]

   if 1.99999999999999991e-6 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 67.2%

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

   3. Taylor expanded in x around 0 63.9%

      \[\leadsto \color{blue}{1 - e^{\frac{\log x}{n}}} \]
   4. Step-by-step derivation

      1. *-rgt-identity 63.9%

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

      2. associate-/l* 63.8%

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

      3. exp-to-pow 63.9%

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

   5. Simplified 63.9%

      \[\leadsto \color{blue}{1 - {x}^{\left(\frac{1}{n}\right)}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 6: 58.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - {x}^{\left(\frac{1}{n}\right)}\\ t_1 := \frac{\log x}{-n}\\ \mathbf{if}\;x \leq 4.7 \cdot 10^{-287}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-224}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{-189}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 4.9 \cdot 10^{-36}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{+127}:\\ \;\;\;\;\frac{\frac{1}{x} + \frac{\frac{-0.5 + \frac{0.3333333333333333}{x}}{x}}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-3} \cdot \frac{0.3333333333333333}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (- 1.0 (pow x (/ 1.0 n)))) (t_1 (/ (log x) (- n))))
   (if (<= x 4.7e-287)
     t_0
     (if (<= x 2e-224)
       t_1
       (if (<= x 2.7e-189)
         t_0
         (if (<= x 4.9e-36)
           t_1
           (if (<= x 3.1e+127)
             (/ (+ (/ 1.0 x) (/ (/ (+ -0.5 (/ 0.3333333333333333 x)) x) x)) n)
             (* (pow x -3.0) (/ 0.3333333333333333 n)))))))))

C

double code(double x, double n) {
	double t_0 = 1.0 - pow(x, (1.0 / n));
	double t_1 = log(x) / -n;
	double tmp;
	if (x <= 4.7e-287) {
		tmp = t_0;
	} else if (x <= 2e-224) {
		tmp = t_1;
	} else if (x <= 2.7e-189) {
		tmp = t_0;
	} else if (x <= 4.9e-36) {
		tmp = t_1;
	} else if (x <= 3.1e+127) {
		tmp = ((1.0 / x) + (((-0.5 + (0.3333333333333333 / x)) / x) / x)) / n;
	} else {
		tmp = pow(x, -3.0) * (0.3333333333333333 / n);
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 - (x ** (1.0d0 / n))
    t_1 = log(x) / -n
    if (x <= 4.7d-287) then
        tmp = t_0
    else if (x <= 2d-224) then
        tmp = t_1
    else if (x <= 2.7d-189) then
        tmp = t_0
    else if (x <= 4.9d-36) then
        tmp = t_1
    else if (x <= 3.1d+127) then
        tmp = ((1.0d0 / x) + ((((-0.5d0) + (0.3333333333333333d0 / x)) / x) / x)) / n
    else
        tmp = (x ** (-3.0d0)) * (0.3333333333333333d0 / n)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = 1.0 - Math.pow(x, (1.0 / n));
	double t_1 = Math.log(x) / -n;
	double tmp;
	if (x <= 4.7e-287) {
		tmp = t_0;
	} else if (x <= 2e-224) {
		tmp = t_1;
	} else if (x <= 2.7e-189) {
		tmp = t_0;
	} else if (x <= 4.9e-36) {
		tmp = t_1;
	} else if (x <= 3.1e+127) {
		tmp = ((1.0 / x) + (((-0.5 + (0.3333333333333333 / x)) / x) / x)) / n;
	} else {
		tmp = Math.pow(x, -3.0) * (0.3333333333333333 / n);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = 1.0 - math.pow(x, (1.0 / n))
	t_1 = math.log(x) / -n
	tmp = 0
	if x <= 4.7e-287:
		tmp = t_0
	elif x <= 2e-224:
		tmp = t_1
	elif x <= 2.7e-189:
		tmp = t_0
	elif x <= 4.9e-36:
		tmp = t_1
	elif x <= 3.1e+127:
		tmp = ((1.0 / x) + (((-0.5 + (0.3333333333333333 / x)) / x) / x)) / n
	else:
		tmp = math.pow(x, -3.0) * (0.3333333333333333 / n)
	return tmp

Julia

function code(x, n)
	t_0 = Float64(1.0 - (x ^ Float64(1.0 / n)))
	t_1 = Float64(log(x) / Float64(-n))
	tmp = 0.0
	if (x <= 4.7e-287)
		tmp = t_0;
	elseif (x <= 2e-224)
		tmp = t_1;
	elseif (x <= 2.7e-189)
		tmp = t_0;
	elseif (x <= 4.9e-36)
		tmp = t_1;
	elseif (x <= 3.1e+127)
		tmp = Float64(Float64(Float64(1.0 / x) + Float64(Float64(Float64(-0.5 + Float64(0.3333333333333333 / x)) / x) / x)) / n);
	else
		tmp = Float64((x ^ -3.0) * Float64(0.3333333333333333 / n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = 1.0 - (x ^ (1.0 / n));
	t_1 = log(x) / -n;
	tmp = 0.0;
	if (x <= 4.7e-287)
		tmp = t_0;
	elseif (x <= 2e-224)
		tmp = t_1;
	elseif (x <= 2.7e-189)
		tmp = t_0;
	elseif (x <= 4.9e-36)
		tmp = t_1;
	elseif (x <= 3.1e+127)
		tmp = ((1.0 / x) + (((-0.5 + (0.3333333333333333 / x)) / x) / x)) / n;
	else
		tmp = (x ^ -3.0) * (0.3333333333333333 / n);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Log[x], $MachinePrecision] / (-n)), $MachinePrecision]}, If[LessEqual[x, 4.7e-287], t$95$0, If[LessEqual[x, 2e-224], t$95$1, If[LessEqual[x, 2.7e-189], t$95$0, If[LessEqual[x, 4.9e-36], t$95$1, If[LessEqual[x, 3.1e+127], N[(N[(N[(1.0 / x), $MachinePrecision] + N[(N[(N[(-0.5 + N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[Power[x, -3.0], $MachinePrecision] * N[(0.3333333333333333 / n), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < 4.6999999999999999e-287 or 2e-224 < x < 2.6999999999999999e-189

   1. Initial program 72.4%

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

   3. Taylor expanded in x around 0 72.4%

      \[\leadsto \color{blue}{1 - e^{\frac{\log x}{n}}} \]
   4. Step-by-step derivation

      1. *-rgt-identity 72.4%

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

      2. associate-/l* 72.4%

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

      3. exp-to-pow 72.4%

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

   5. Simplified 72.4%

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

   if 4.6999999999999999e-287 < x < 2e-224 or 2.6999999999999999e-189 < x < 4.8999999999999997e-36

   1. Initial program 41.6%

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

   3. Taylor expanded in x around 0 41.6%

      \[\leadsto \color{blue}{1 - e^{\frac{\log x}{n}}} \]
   4. Step-by-step derivation

      1. *-rgt-identity 41.6%

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

      2. associate-/l* 41.6%

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

      3. exp-to-pow 41.6%

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

   5. Simplified 41.6%

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

   6. Taylor expanded in n around inf 58.1%

      \[\leadsto \color{blue}{-1 \cdot \frac{\log x}{n}} \]
   7. Step-by-step derivation

      1. neg-mul-1 58.1%

         \[\leadsto \color{blue}{-\frac{\log x}{n}} \]

      2. distribute-neg-frac 58.1%

         \[\leadsto \color{blue}{\frac{-\log x}{n}} \]

   8. Simplified 58.1%

      \[\leadsto \color{blue}{\frac{-\log x}{n}} \]

   if 4.8999999999999997e-36 < x < 3.1000000000000002e127

   1. Initial program 50.3%

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

   3. Taylor expanded in n around inf 44.7%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   4. Step-by-step derivation

      1. log1p-define 44.7%

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

   5. Simplified 44.7%

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

   6. Taylor expanded in x around -inf 64.8%

      \[\leadsto \frac{\color{blue}{-1 \cdot \frac{-1 \cdot \frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{x} - 1}{x}}}{n} \]
   7. Step-by-step derivation

      1. div-sub 64.8%

         \[\leadsto \frac{-1 \cdot \color{blue}{\left(\frac{-1 \cdot \frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{x}}{x} - \frac{1}{x}\right)}}{n} \]

      2. mul-1-neg 64.8%

         \[\leadsto \frac{-1 \cdot \left(\frac{\color{blue}{-\frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{x}}}{x} - \frac{1}{x}\right)}{n} \]

      3. sub-neg 64.8%

         \[\leadsto \frac{-1 \cdot \left(\frac{-\frac{\color{blue}{0.3333333333333333 \cdot \frac{1}{x} + \left(-0.5\right)}}{x}}{x} - \frac{1}{x}\right)}{n} \]

      4. un-div-inv 64.8%

         \[\leadsto \frac{-1 \cdot \left(\frac{-\frac{\color{blue}{\frac{0.3333333333333333}{x}} + \left(-0.5\right)}{x}}{x} - \frac{1}{x}\right)}{n} \]

      5. metadata-eval 64.8%

         \[\leadsto \frac{-1 \cdot \left(\frac{-\frac{\frac{0.3333333333333333}{x} + \color{blue}{-0.5}}{x}}{x} - \frac{1}{x}\right)}{n} \]

   8. Applied egg-rr 64.8%

      \[\leadsto \frac{-1 \cdot \color{blue}{\left(\frac{-\frac{\frac{0.3333333333333333}{x} + -0.5}{x}}{x} - \frac{1}{x}\right)}}{n} \]

   if 3.1000000000000002e127 < x

   1. Initial program 89.1%

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

   3. Taylor expanded in n around inf 89.1%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   4. Step-by-step derivation

      1. log1p-define 89.1%

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

   5. Simplified 89.1%

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

   6. Taylor expanded in x around -inf 67.2%

      \[\leadsto \frac{\color{blue}{-1 \cdot \frac{-1 \cdot \frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{x} - 1}{x}}}{n} \]

   7. Taylor expanded in x around 0 89.1%

      \[\leadsto \color{blue}{\frac{0.3333333333333333}{n \cdot {x}^{3}}} \]
   8. Step-by-step derivation

      1. associate-/r* 89.1%

         \[\leadsto \color{blue}{\frac{\frac{0.3333333333333333}{n}}{{x}^{3}}} \]

   9. Simplified 89.1%

      \[\leadsto \color{blue}{\frac{\frac{0.3333333333333333}{n}}{{x}^{3}}} \]
   10. Step-by-step derivation

       1. clear-num 89.1%

          \[\leadsto \color{blue}{\frac{1}{\frac{{x}^{3}}{\frac{0.3333333333333333}{n}}}} \]

       2. associate-/r/ 89.1%

          \[\leadsto \color{blue}{\frac{1}{{x}^{3}} \cdot \frac{0.3333333333333333}{n}} \]

       3. pow-flip 89.1%

          \[\leadsto \color{blue}{{x}^{\left(-3\right)}} \cdot \frac{0.3333333333333333}{n} \]

       4. metadata-eval 89.1%

          \[\leadsto {x}^{\color{blue}{-3}} \cdot \frac{0.3333333333333333}{n} \]

   11. Applied egg-rr 89.1%

       \[\leadsto \color{blue}{{x}^{-3} \cdot \frac{0.3333333333333333}{n}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.7 \cdot 10^{-287}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-224}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{-189}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 4.9 \cdot 10^{-36}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{+127}:\\ \;\;\;\;\frac{\frac{1}{x} + \frac{\frac{-0.5 + \frac{0.3333333333333333}{x}}{x}}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-3} \cdot \frac{0.3333333333333333}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 57.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-0.5 + \frac{0.3333333333333333}{x}}{x}\\ t_1 := \frac{\log x}{-n}\\ \mathbf{if}\;x \leq 2 \cdot 10^{-224}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 6.4 \cdot 10^{-189}:\\ \;\;\;\;\frac{\frac{1 + t\_0}{x}}{n}\\ \mathbf{elif}\;x \leq 4.9 \cdot 10^{-36}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{+128}:\\ \;\;\;\;\frac{\frac{1}{x} + \frac{t\_0}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-3} \cdot \frac{0.3333333333333333}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (+ -0.5 (/ 0.3333333333333333 x)) x)) (t_1 (/ (log x) (- n))))
   (if (<= x 2e-224)
     t_1
     (if (<= x 6.4e-189)
       (/ (/ (+ 1.0 t_0) x) n)
       (if (<= x 4.9e-36)
         t_1
         (if (<= x 1.9e+128)
           (/ (+ (/ 1.0 x) (/ t_0 x)) n)
           (* (pow x -3.0) (/ 0.3333333333333333 n))))))))

C

double code(double x, double n) {
	double t_0 = (-0.5 + (0.3333333333333333 / x)) / x;
	double t_1 = log(x) / -n;
	double tmp;
	if (x <= 2e-224) {
		tmp = t_1;
	} else if (x <= 6.4e-189) {
		tmp = ((1.0 + t_0) / x) / n;
	} else if (x <= 4.9e-36) {
		tmp = t_1;
	} else if (x <= 1.9e+128) {
		tmp = ((1.0 / x) + (t_0 / x)) / n;
	} else {
		tmp = pow(x, -3.0) * (0.3333333333333333 / n);
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((-0.5d0) + (0.3333333333333333d0 / x)) / x
    t_1 = log(x) / -n
    if (x <= 2d-224) then
        tmp = t_1
    else if (x <= 6.4d-189) then
        tmp = ((1.0d0 + t_0) / x) / n
    else if (x <= 4.9d-36) then
        tmp = t_1
    else if (x <= 1.9d+128) then
        tmp = ((1.0d0 / x) + (t_0 / x)) / n
    else
        tmp = (x ** (-3.0d0)) * (0.3333333333333333d0 / n)
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = (-0.5 + (0.3333333333333333 / x)) / x;
	double t_1 = Math.log(x) / -n;
	double tmp;
	if (x <= 2e-224) {
		tmp = t_1;
	} else if (x <= 6.4e-189) {
		tmp = ((1.0 + t_0) / x) / n;
	} else if (x <= 4.9e-36) {
		tmp = t_1;
	} else if (x <= 1.9e+128) {
		tmp = ((1.0 / x) + (t_0 / x)) / n;
	} else {
		tmp = Math.pow(x, -3.0) * (0.3333333333333333 / n);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = (-0.5 + (0.3333333333333333 / x)) / x
	t_1 = math.log(x) / -n
	tmp = 0
	if x <= 2e-224:
		tmp = t_1
	elif x <= 6.4e-189:
		tmp = ((1.0 + t_0) / x) / n
	elif x <= 4.9e-36:
		tmp = t_1
	elif x <= 1.9e+128:
		tmp = ((1.0 / x) + (t_0 / x)) / n
	else:
		tmp = math.pow(x, -3.0) * (0.3333333333333333 / n)
	return tmp

Julia

function code(x, n)
	t_0 = Float64(Float64(-0.5 + Float64(0.3333333333333333 / x)) / x)
	t_1 = Float64(log(x) / Float64(-n))
	tmp = 0.0
	if (x <= 2e-224)
		tmp = t_1;
	elseif (x <= 6.4e-189)
		tmp = Float64(Float64(Float64(1.0 + t_0) / x) / n);
	elseif (x <= 4.9e-36)
		tmp = t_1;
	elseif (x <= 1.9e+128)
		tmp = Float64(Float64(Float64(1.0 / x) + Float64(t_0 / x)) / n);
	else
		tmp = Float64((x ^ -3.0) * Float64(0.3333333333333333 / n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = (-0.5 + (0.3333333333333333 / x)) / x;
	t_1 = log(x) / -n;
	tmp = 0.0;
	if (x <= 2e-224)
		tmp = t_1;
	elseif (x <= 6.4e-189)
		tmp = ((1.0 + t_0) / x) / n;
	elseif (x <= 4.9e-36)
		tmp = t_1;
	elseif (x <= 1.9e+128)
		tmp = ((1.0 / x) + (t_0 / x)) / n;
	else
		tmp = (x ^ -3.0) * (0.3333333333333333 / n);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[(-0.5 + N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, Block[{t$95$1 = N[(N[Log[x], $MachinePrecision] / (-n)), $MachinePrecision]}, If[LessEqual[x, 2e-224], t$95$1, If[LessEqual[x, 6.4e-189], N[(N[(N[(1.0 + t$95$0), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[x, 4.9e-36], t$95$1, If[LessEqual[x, 1.9e+128], N[(N[(N[(1.0 / x), $MachinePrecision] + N[(t$95$0 / x), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[Power[x, -3.0], $MachinePrecision] * N[(0.3333333333333333 / n), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < 2e-224 or 6.4000000000000001e-189 < x < 4.8999999999999997e-36

   1. Initial program 44.6%

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

   3. Taylor expanded in x around 0 44.6%

      \[\leadsto \color{blue}{1 - e^{\frac{\log x}{n}}} \]
   4. Step-by-step derivation

      1. *-rgt-identity 44.6%

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

      2. associate-/l* 44.6%

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

      3. exp-to-pow 44.6%

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

   5. Simplified 44.6%

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

   6. Taylor expanded in n around inf 56.0%

      \[\leadsto \color{blue}{-1 \cdot \frac{\log x}{n}} \]
   7. Step-by-step derivation

      1. neg-mul-1 56.0%

         \[\leadsto \color{blue}{-\frac{\log x}{n}} \]

      2. distribute-neg-frac 56.0%

         \[\leadsto \color{blue}{\frac{-\log x}{n}} \]

   8. Simplified 56.0%

      \[\leadsto \color{blue}{\frac{-\log x}{n}} \]

   if 2e-224 < x < 6.4000000000000001e-189

   1. Initial program 74.8%

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

   3. Taylor expanded in n around inf 31.4%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   4. Step-by-step derivation

      1. log1p-define 31.4%

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

   5. Simplified 31.4%

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

   6. Taylor expanded in x around -inf 67.4%

      \[\leadsto \frac{\color{blue}{-1 \cdot \frac{-1 \cdot \frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{x} - 1}{x}}}{n} \]

   7. Taylor expanded in n around 0 67.4%

      \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{x} - 1}{n \cdot x}} \]

   9. Simplified 67.4%

      \[\leadsto \color{blue}{\frac{\frac{1 + \frac{-0.5 + \frac{0.3333333333333333}{x}}{x}}{x}}{n}} \]

   if 4.8999999999999997e-36 < x < 1.89999999999999995e128

   1. Initial program 50.3%

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

   3. Taylor expanded in n around inf 44.7%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   4. Step-by-step derivation

      1. log1p-define 44.7%

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

   5. Simplified 44.7%

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

   6. Taylor expanded in x around -inf 64.8%

      \[\leadsto \frac{\color{blue}{-1 \cdot \frac{-1 \cdot \frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{x} - 1}{x}}}{n} \]
   7. Step-by-step derivation

      1. div-sub 64.8%

         \[\leadsto \frac{-1 \cdot \color{blue}{\left(\frac{-1 \cdot \frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{x}}{x} - \frac{1}{x}\right)}}{n} \]

      2. mul-1-neg 64.8%

         \[\leadsto \frac{-1 \cdot \left(\frac{\color{blue}{-\frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{x}}}{x} - \frac{1}{x}\right)}{n} \]

      3. sub-neg 64.8%

         \[\leadsto \frac{-1 \cdot \left(\frac{-\frac{\color{blue}{0.3333333333333333 \cdot \frac{1}{x} + \left(-0.5\right)}}{x}}{x} - \frac{1}{x}\right)}{n} \]

      4. un-div-inv 64.8%

         \[\leadsto \frac{-1 \cdot \left(\frac{-\frac{\color{blue}{\frac{0.3333333333333333}{x}} + \left(-0.5\right)}{x}}{x} - \frac{1}{x}\right)}{n} \]

      5. metadata-eval 64.8%

         \[\leadsto \frac{-1 \cdot \left(\frac{-\frac{\frac{0.3333333333333333}{x} + \color{blue}{-0.5}}{x}}{x} - \frac{1}{x}\right)}{n} \]

   8. Applied egg-rr 64.8%

      \[\leadsto \frac{-1 \cdot \color{blue}{\left(\frac{-\frac{\frac{0.3333333333333333}{x} + -0.5}{x}}{x} - \frac{1}{x}\right)}}{n} \]

   if 1.89999999999999995e128 < x

   1. Initial program 89.1%

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

   3. Taylor expanded in n around inf 89.1%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   4. Step-by-step derivation

      1. log1p-define 89.1%

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

   5. Simplified 89.1%

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

   6. Taylor expanded in x around -inf 67.2%

      \[\leadsto \frac{\color{blue}{-1 \cdot \frac{-1 \cdot \frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{x} - 1}{x}}}{n} \]

   7. Taylor expanded in x around 0 89.1%

      \[\leadsto \color{blue}{\frac{0.3333333333333333}{n \cdot {x}^{3}}} \]
   8. Step-by-step derivation

      1. associate-/r* 89.1%

         \[\leadsto \color{blue}{\frac{\frac{0.3333333333333333}{n}}{{x}^{3}}} \]

   9. Simplified 89.1%

      \[\leadsto \color{blue}{\frac{\frac{0.3333333333333333}{n}}{{x}^{3}}} \]
   10. Step-by-step derivation

       1. clear-num 89.1%

          \[\leadsto \color{blue}{\frac{1}{\frac{{x}^{3}}{\frac{0.3333333333333333}{n}}}} \]

       2. associate-/r/ 89.1%

          \[\leadsto \color{blue}{\frac{1}{{x}^{3}} \cdot \frac{0.3333333333333333}{n}} \]

       3. pow-flip 89.1%

          \[\leadsto \color{blue}{{x}^{\left(-3\right)}} \cdot \frac{0.3333333333333333}{n} \]

       4. metadata-eval 89.1%

          \[\leadsto {x}^{\color{blue}{-3}} \cdot \frac{0.3333333333333333}{n} \]

   11. Applied egg-rr 89.1%

       \[\leadsto \color{blue}{{x}^{-3} \cdot \frac{0.3333333333333333}{n}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{-224}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 6.4 \cdot 10^{-189}:\\ \;\;\;\;\frac{\frac{1 + \frac{-0.5 + \frac{0.3333333333333333}{x}}{x}}{x}}{n}\\ \mathbf{elif}\;x \leq 4.9 \cdot 10^{-36}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{+128}:\\ \;\;\;\;\frac{\frac{1}{x} + \frac{\frac{-0.5 + \frac{0.3333333333333333}{x}}{x}}{x}}{n}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-3} \cdot \frac{0.3333333333333333}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 54.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-0.5 + \frac{0.3333333333333333}{x}}{x}\\ t_1 := \frac{\log x}{-n}\\ \mathbf{if}\;x \leq 1.6 \cdot 10^{-224}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{-188}:\\ \;\;\;\;\frac{\frac{1 + t\_0}{x}}{n}\\ \mathbf{elif}\;x \leq 4.9 \cdot 10^{-36}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x} + \frac{t\_0}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (+ -0.5 (/ 0.3333333333333333 x)) x)) (t_1 (/ (log x) (- n))))
   (if (<= x 1.6e-224)
     t_1
     (if (<= x 1.1e-188)
       (/ (/ (+ 1.0 t_0) x) n)
       (if (<= x 4.9e-36) t_1 (/ (+ (/ 1.0 x) (/ t_0 x)) n))))))

C

double code(double x, double n) {
	double t_0 = (-0.5 + (0.3333333333333333 / x)) / x;
	double t_1 = log(x) / -n;
	double tmp;
	if (x <= 1.6e-224) {
		tmp = t_1;
	} else if (x <= 1.1e-188) {
		tmp = ((1.0 + t_0) / x) / n;
	} else if (x <= 4.9e-36) {
		tmp = t_1;
	} else {
		tmp = ((1.0 / x) + (t_0 / x)) / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((-0.5d0) + (0.3333333333333333d0 / x)) / x
    t_1 = log(x) / -n
    if (x <= 1.6d-224) then
        tmp = t_1
    else if (x <= 1.1d-188) then
        tmp = ((1.0d0 + t_0) / x) / n
    else if (x <= 4.9d-36) then
        tmp = t_1
    else
        tmp = ((1.0d0 / x) + (t_0 / x)) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = (-0.5 + (0.3333333333333333 / x)) / x;
	double t_1 = Math.log(x) / -n;
	double tmp;
	if (x <= 1.6e-224) {
		tmp = t_1;
	} else if (x <= 1.1e-188) {
		tmp = ((1.0 + t_0) / x) / n;
	} else if (x <= 4.9e-36) {
		tmp = t_1;
	} else {
		tmp = ((1.0 / x) + (t_0 / x)) / n;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = (-0.5 + (0.3333333333333333 / x)) / x
	t_1 = math.log(x) / -n
	tmp = 0
	if x <= 1.6e-224:
		tmp = t_1
	elif x <= 1.1e-188:
		tmp = ((1.0 + t_0) / x) / n
	elif x <= 4.9e-36:
		tmp = t_1
	else:
		tmp = ((1.0 / x) + (t_0 / x)) / n
	return tmp

Julia

function code(x, n)
	t_0 = Float64(Float64(-0.5 + Float64(0.3333333333333333 / x)) / x)
	t_1 = Float64(log(x) / Float64(-n))
	tmp = 0.0
	if (x <= 1.6e-224)
		tmp = t_1;
	elseif (x <= 1.1e-188)
		tmp = Float64(Float64(Float64(1.0 + t_0) / x) / n);
	elseif (x <= 4.9e-36)
		tmp = t_1;
	else
		tmp = Float64(Float64(Float64(1.0 / x) + Float64(t_0 / x)) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = (-0.5 + (0.3333333333333333 / x)) / x;
	t_1 = log(x) / -n;
	tmp = 0.0;
	if (x <= 1.6e-224)
		tmp = t_1;
	elseif (x <= 1.1e-188)
		tmp = ((1.0 + t_0) / x) / n;
	elseif (x <= 4.9e-36)
		tmp = t_1;
	else
		tmp = ((1.0 / x) + (t_0 / x)) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[(-0.5 + N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]}, Block[{t$95$1 = N[(N[Log[x], $MachinePrecision] / (-n)), $MachinePrecision]}, If[LessEqual[x, 1.6e-224], t$95$1, If[LessEqual[x, 1.1e-188], N[(N[(N[(1.0 + t$95$0), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[x, 4.9e-36], t$95$1, N[(N[(N[(1.0 / x), $MachinePrecision] + N[(t$95$0 / x), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < 1.5999999999999999e-224 or 1.1e-188 < x < 4.8999999999999997e-36

   1. Initial program 44.6%

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

   3. Taylor expanded in x around 0 44.6%

      \[\leadsto \color{blue}{1 - e^{\frac{\log x}{n}}} \]
   4. Step-by-step derivation

      1. *-rgt-identity 44.6%

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

      2. associate-/l* 44.6%

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

      3. exp-to-pow 44.6%

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

   5. Simplified 44.6%

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

   6. Taylor expanded in n around inf 56.0%

      \[\leadsto \color{blue}{-1 \cdot \frac{\log x}{n}} \]
   7. Step-by-step derivation

      1. neg-mul-1 56.0%

         \[\leadsto \color{blue}{-\frac{\log x}{n}} \]

      2. distribute-neg-frac 56.0%

         \[\leadsto \color{blue}{\frac{-\log x}{n}} \]

   8. Simplified 56.0%

      \[\leadsto \color{blue}{\frac{-\log x}{n}} \]

   if 1.5999999999999999e-224 < x < 1.1e-188

   1. Initial program 74.8%

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

   3. Taylor expanded in n around inf 31.4%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   4. Step-by-step derivation

      1. log1p-define 31.4%

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

   5. Simplified 31.4%

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

   6. Taylor expanded in x around -inf 67.4%

      \[\leadsto \frac{\color{blue}{-1 \cdot \frac{-1 \cdot \frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{x} - 1}{x}}}{n} \]

   7. Taylor expanded in n around 0 67.4%

      \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{x} - 1}{n \cdot x}} \]

   9. Simplified 67.4%

      \[\leadsto \color{blue}{\frac{\frac{1 + \frac{-0.5 + \frac{0.3333333333333333}{x}}{x}}{x}}{n}} \]

   if 4.8999999999999997e-36 < x

   1. Initial program 68.3%

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

   3. Taylor expanded in n around inf 65.2%

      \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
   4. Step-by-step derivation

      1. log1p-define 65.2%

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

   5. Simplified 65.2%

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

   6. Taylor expanded in x around -inf 65.9%

      \[\leadsto \frac{\color{blue}{-1 \cdot \frac{-1 \cdot \frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{x} - 1}{x}}}{n} \]
   7. Step-by-step derivation

      1. div-sub 65.9%

         \[\leadsto \frac{-1 \cdot \color{blue}{\left(\frac{-1 \cdot \frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{x}}{x} - \frac{1}{x}\right)}}{n} \]

      2. mul-1-neg 65.9%

         \[\leadsto \frac{-1 \cdot \left(\frac{\color{blue}{-\frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{x}}}{x} - \frac{1}{x}\right)}{n} \]

      3. sub-neg 65.9%

         \[\leadsto \frac{-1 \cdot \left(\frac{-\frac{\color{blue}{0.3333333333333333 \cdot \frac{1}{x} + \left(-0.5\right)}}{x}}{x} - \frac{1}{x}\right)}{n} \]

      4. un-div-inv 65.9%

         \[\leadsto \frac{-1 \cdot \left(\frac{-\frac{\color{blue}{\frac{0.3333333333333333}{x}} + \left(-0.5\right)}{x}}{x} - \frac{1}{x}\right)}{n} \]

      5. metadata-eval 65.9%

         \[\leadsto \frac{-1 \cdot \left(\frac{-\frac{\frac{0.3333333333333333}{x} + \color{blue}{-0.5}}{x}}{x} - \frac{1}{x}\right)}{n} \]

   8. Applied egg-rr 65.9%

      \[\leadsto \frac{-1 \cdot \color{blue}{\left(\frac{-\frac{\frac{0.3333333333333333}{x} + -0.5}{x}}{x} - \frac{1}{x}\right)}}{n} \]
3. Recombined 3 regimes into one program.

4. Final simplification 60.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.6 \cdot 10^{-224}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{-188}:\\ \;\;\;\;\frac{\frac{1 + \frac{-0.5 + \frac{0.3333333333333333}{x}}{x}}{x}}{n}\\ \mathbf{elif}\;x \leq 4.9 \cdot 10^{-36}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x} + \frac{\frac{-0.5 + \frac{0.3333333333333333}{x}}{x}}{x}}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 47.1% accurate, 12.4× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1}{n} + \frac{\frac{0.3333333333333333}{x \cdot n} + \frac{-0.5}{n}}{x}}{x} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (/ (+ (/ 1.0 n) (/ (+ (/ 0.3333333333333333 (* x n)) (/ -0.5 n)) x)) x))

C

double code(double x, double n) {
	return ((1.0 / n) + (((0.3333333333333333 / (x * n)) + (-0.5 / n)) / x)) / x;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    code = ((1.0d0 / n) + (((0.3333333333333333d0 / (x * n)) + ((-0.5d0) / n)) / x)) / x
end function

Java

public static double code(double x, double n) {
	return ((1.0 / n) + (((0.3333333333333333 / (x * n)) + (-0.5 / n)) / x)) / x;
}

Python

def code(x, n):
	return ((1.0 / n) + (((0.3333333333333333 / (x * n)) + (-0.5 / n)) / x)) / x

Julia

function code(x, n)
	return Float64(Float64(Float64(1.0 / n) + Float64(Float64(Float64(0.3333333333333333 / Float64(x * n)) + Float64(-0.5 / n)) / x)) / x)
end

MATLAB

function tmp = code(x, n)
	tmp = ((1.0 / n) + (((0.3333333333333333 / (x * n)) + (-0.5 / n)) / x)) / x;
end

Wolfram

code[x_, n_] := N[(N[(N[(1.0 / n), $MachinePrecision] + N[(N[(N[(0.3333333333333333 / N[(x * n), $MachinePrecision]), $MachinePrecision] + N[(-0.5 / n), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]

Derivation

1. Initial program 56.4%

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

3. Taylor expanded in n around inf 58.5%

   \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
4. Step-by-step derivation

   1. log1p-define 58.5%

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

5. Simplified 58.5%

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

6. Taylor expanded in x around -inf 47.8%

   \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{0.3333333333333333 \cdot \frac{1}{n \cdot x} - 0.5 \cdot \frac{1}{n}}{x} - \frac{1}{n}}{x}} \]
7. Step-by-step derivation

   1. mul-1-neg 47.8%

      \[\leadsto \color{blue}{-\frac{-1 \cdot \frac{0.3333333333333333 \cdot \frac{1}{n \cdot x} - 0.5 \cdot \frac{1}{n}}{x} - \frac{1}{n}}{x}} \]

   2. distribute-neg-frac2 47.8%

      \[\leadsto \color{blue}{\frac{-1 \cdot \frac{0.3333333333333333 \cdot \frac{1}{n \cdot x} - 0.5 \cdot \frac{1}{n}}{x} - \frac{1}{n}}{-x}} \]

8. Simplified 47.8%

   \[\leadsto \color{blue}{\frac{\frac{\frac{0.3333333333333333}{x \cdot n} + \frac{-0.5}{n}}{-x} + \frac{-1}{n}}{-x}} \]

9. Final simplification 47.8%

   \[\leadsto \frac{\frac{1}{n} + \frac{\frac{0.3333333333333333}{x \cdot n} + \frac{-0.5}{n}}{x}}{x} \]
10. Add Preprocessing

Alternative 10: 47.1% accurate, 14.1× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1}{x} + \frac{\frac{-0.5 + \frac{0.3333333333333333}{x}}{x}}{x}}{n} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (/ (+ (/ 1.0 x) (/ (/ (+ -0.5 (/ 0.3333333333333333 x)) x) x)) n))

C

double code(double x, double n) {
	return ((1.0 / x) + (((-0.5 + (0.3333333333333333 / x)) / x) / x)) / n;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    code = ((1.0d0 / x) + ((((-0.5d0) + (0.3333333333333333d0 / x)) / x) / x)) / n
end function

Java

public static double code(double x, double n) {
	return ((1.0 / x) + (((-0.5 + (0.3333333333333333 / x)) / x) / x)) / n;
}

Python

def code(x, n):
	return ((1.0 / x) + (((-0.5 + (0.3333333333333333 / x)) / x) / x)) / n

Julia

function code(x, n)
	return Float64(Float64(Float64(1.0 / x) + Float64(Float64(Float64(-0.5 + Float64(0.3333333333333333 / x)) / x) / x)) / n)
end

MATLAB

function tmp = code(x, n)
	tmp = ((1.0 / x) + (((-0.5 + (0.3333333333333333 / x)) / x) / x)) / n;
end

Wolfram

code[x_, n_] := N[(N[(N[(1.0 / x), $MachinePrecision] + N[(N[(N[(-0.5 + N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]

Derivation

1. Initial program 56.4%

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

3. Taylor expanded in n around inf 58.5%

   \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
4. Step-by-step derivation

   1. log1p-define 58.5%

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

5. Simplified 58.5%

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

6. Taylor expanded in x around -inf 47.8%

   \[\leadsto \frac{\color{blue}{-1 \cdot \frac{-1 \cdot \frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{x} - 1}{x}}}{n} \]
7. Step-by-step derivation

   1. div-sub 47.8%

      \[\leadsto \frac{-1 \cdot \color{blue}{\left(\frac{-1 \cdot \frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{x}}{x} - \frac{1}{x}\right)}}{n} \]

   2. mul-1-neg 47.8%

      \[\leadsto \frac{-1 \cdot \left(\frac{\color{blue}{-\frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{x}}}{x} - \frac{1}{x}\right)}{n} \]

   3. sub-neg 47.8%

      \[\leadsto \frac{-1 \cdot \left(\frac{-\frac{\color{blue}{0.3333333333333333 \cdot \frac{1}{x} + \left(-0.5\right)}}{x}}{x} - \frac{1}{x}\right)}{n} \]

   4. un-div-inv 47.8%

      \[\leadsto \frac{-1 \cdot \left(\frac{-\frac{\color{blue}{\frac{0.3333333333333333}{x}} + \left(-0.5\right)}{x}}{x} - \frac{1}{x}\right)}{n} \]

   5. metadata-eval 47.8%

      \[\leadsto \frac{-1 \cdot \left(\frac{-\frac{\frac{0.3333333333333333}{x} + \color{blue}{-0.5}}{x}}{x} - \frac{1}{x}\right)}{n} \]

8. Applied egg-rr 47.8%

   \[\leadsto \frac{-1 \cdot \color{blue}{\left(\frac{-\frac{\frac{0.3333333333333333}{x} + -0.5}{x}}{x} - \frac{1}{x}\right)}}{n} \]

9. Final simplification 47.8%

   \[\leadsto \frac{\frac{1}{x} + \frac{\frac{-0.5 + \frac{0.3333333333333333}{x}}{x}}{x}}{n} \]
10. Add Preprocessing

Alternative 11: 47.1% accurate, 16.2× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1 + \frac{-0.5 + \frac{0.3333333333333333}{x}}{x}}{x}}{n} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (/ (/ (+ 1.0 (/ (+ -0.5 (/ 0.3333333333333333 x)) x)) x) n))

C

double code(double x, double n) {
	return ((1.0 + ((-0.5 + (0.3333333333333333 / x)) / x)) / x) / n;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    code = ((1.0d0 + (((-0.5d0) + (0.3333333333333333d0 / x)) / x)) / x) / n
end function

Java

public static double code(double x, double n) {
	return ((1.0 + ((-0.5 + (0.3333333333333333 / x)) / x)) / x) / n;
}

Python

def code(x, n):
	return ((1.0 + ((-0.5 + (0.3333333333333333 / x)) / x)) / x) / n

Julia

function code(x, n)
	return Float64(Float64(Float64(1.0 + Float64(Float64(-0.5 + Float64(0.3333333333333333 / x)) / x)) / x) / n)
end

MATLAB

function tmp = code(x, n)
	tmp = ((1.0 + ((-0.5 + (0.3333333333333333 / x)) / x)) / x) / n;
end

Wolfram

code[x_, n_] := N[(N[(N[(1.0 + N[(N[(-0.5 + N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / n), $MachinePrecision]

Derivation

1. Initial program 56.4%

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

3. Taylor expanded in n around inf 58.5%

   \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
4. Step-by-step derivation

   1. log1p-define 58.5%

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

5. Simplified 58.5%

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

6. Taylor expanded in x around -inf 47.8%

   \[\leadsto \frac{\color{blue}{-1 \cdot \frac{-1 \cdot \frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{x} - 1}{x}}}{n} \]

7. Taylor expanded in n around 0 47.3%

   \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{0.3333333333333333 \cdot \frac{1}{x} - 0.5}{x} - 1}{n \cdot x}} \]

9. Simplified 47.8%

   \[\leadsto \color{blue}{\frac{\frac{1 + \frac{-0.5 + \frac{0.3333333333333333}{x}}{x}}{x}}{n}} \]
10. Add Preprocessing

Alternative 12: 40.9% accurate, 42.2× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1}{x}}{n} \end{array} \]

FPCore

(FPCore (x n) :precision binary64 (/ (/ 1.0 x) n))

C

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

Fortran

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

Java

public static double code(double x, double n) {
	return (1.0 / x) / n;
}

Python

def code(x, n):
	return (1.0 / x) / n

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.4%

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

3. Taylor expanded in n around inf 58.5%

   \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
4. Step-by-step derivation

   1. log1p-define 58.5%

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

5. Simplified 58.5%

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

6. Taylor expanded in x around inf 41.7%

   \[\leadsto \frac{\color{blue}{\frac{1}{x}}}{n} \]
7. Add Preprocessing

Alternative 13: 40.9% accurate, 42.2× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1}{n}}{x} \end{array} \]

FPCore

(FPCore (x n) :precision binary64 (/ (/ 1.0 n) x))

C

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

Fortran

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

Java

public static double code(double x, double n) {
	return (1.0 / n) / x;
}

Python

def code(x, n):
	return (1.0 / n) / x

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.4%

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

3. Taylor expanded in n around inf 58.5%

   \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
4. Step-by-step derivation

   1. log1p-define 58.5%

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

5. Simplified 58.5%

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

6. Taylor expanded in x around inf 26.9%

   \[\leadsto \color{blue}{\frac{\frac{1}{n} - 0.5 \cdot \frac{1}{n \cdot x}}{x}} \]
7. Step-by-step derivation

   1. sub-neg 26.9%

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

   2. associate-*r/ 26.9%

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

   3. metadata-eval 26.9%

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

   4. distribute-neg-frac 26.9%

      \[\leadsto \frac{\frac{1}{n} + \color{blue}{\frac{-0.5}{n \cdot x}}}{x} \]

   5. metadata-eval 26.9%

      \[\leadsto \frac{\frac{1}{n} + \frac{\color{blue}{-0.5}}{n \cdot x}}{x} \]

   6. *-commutative 26.9%

      \[\leadsto \frac{\frac{1}{n} + \frac{-0.5}{\color{blue}{x \cdot n}}}{x} \]

8. Simplified 26.9%

   \[\leadsto \color{blue}{\frac{\frac{1}{n} + \frac{-0.5}{x \cdot n}}{x}} \]

9. Taylor expanded in x around inf 41.7%

   \[\leadsto \frac{\color{blue}{\frac{1}{n}}}{x} \]
10. Add Preprocessing

Alternative 14: 40.3% accurate, 42.2× speedup

Math

\[\begin{array}{l} \\ \frac{1}{x \cdot n} \end{array} \]

FPCore

(FPCore (x n) :precision binary64 (/ 1.0 (* x n)))

C

double code(double x, double n) {
	return 1.0 / (x * n);
}

Fortran

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

Java

public static double code(double x, double n) {
	return 1.0 / (x * n);
}

Python

def code(x, n):
	return 1.0 / (x * n)

Julia

function code(x, n)
	return Float64(1.0 / Float64(x * n))
end

MATLAB

function tmp = code(x, n)
	tmp = 1.0 / (x * n);
end

Wolfram

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

Derivation

1. Initial program 56.4%

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

3. Taylor expanded in n around inf 58.5%

   \[\leadsto \color{blue}{\frac{\log \left(1 + x\right) - \log x}{n}} \]
4. Step-by-step derivation

   1. log1p-define 58.5%

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

5. Simplified 58.5%

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

6. Taylor expanded in x around inf 41.2%

   \[\leadsto \color{blue}{\frac{1}{n \cdot x}} \]
7. Step-by-step derivation

   1. *-commutative 41.2%

      \[\leadsto \frac{1}{\color{blue}{x \cdot n}} \]

8. Simplified 41.2%

   \[\leadsto \color{blue}{\frac{1}{x \cdot n}} \]
9. Add Preprocessing

Alternative 15: 4.5% accurate, 70.3× speedup

Math

\[\begin{array}{l} \\ \frac{x}{n} \end{array} \]

FPCore

(FPCore (x n) :precision binary64 (/ x n))

C

double code(double x, double n) {
	return x / n;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    code = x / n
end function

Java

public static double code(double x, double n) {
	return x / n;
}

Python

def code(x, n):
	return x / n

Julia

function code(x, n)
	return Float64(x / n)
end

MATLAB

function tmp = code(x, n)
	tmp = x / n;
end

Wolfram

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

Derivation

1. Initial program 56.4%

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

3. Taylor expanded in x around 0 37.0%

   \[\leadsto \color{blue}{\left(1 + \frac{x}{n}\right) - e^{\frac{\log x}{n}}} \]
4. Step-by-step derivation

   1. *-rgt-identity 37.0%

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

   2. associate-/l* 37.0%

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

   3. exp-to-pow 37.0%

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

5. Simplified 37.0%

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

6. Taylor expanded in x around inf 4.5%

   \[\leadsto \color{blue}{\frac{x}{n}} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024096 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))