2nthrt (problem 3.4.6)

Percentage Accurate: 54.2% → 86.2%

Time: 38.0s

Alternatives: 14

Speedup: 1.9×

• 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: 54.2% 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: 86.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -0.002:\\ \;\;\;\;\frac{1}{x} \cdot \frac{t\_0}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-15}:\\ \;\;\;\;\frac{\frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right)}{n} + \left(\mathsf{log1p}\left(x\right) - \log x\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{n}} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -0.002)
     (* (/ 1.0 x) (/ t_0 n))
     (if (<= (/ 1.0 n) 1e-15)
       (/
        (+
         (/ (* 0.5 (- (pow (log1p x) 2.0) (pow (log x) 2.0))) n)
         (- (log1p x) (log x)))
        n)
       (- (exp (/ x n)) t_0)))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -0.002) {
		tmp = (1.0 / x) * (t_0 / n);
	} else if ((1.0 / n) <= 1e-15) {
		tmp = (((0.5 * (pow(log1p(x), 2.0) - pow(log(x), 2.0))) / n) + (log1p(x) - log(x))) / n;
	} else {
		tmp = exp((x / n)) - t_0;
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -0.002) {
		tmp = (1.0 / x) * (t_0 / n);
	} else if ((1.0 / n) <= 1e-15) {
		tmp = (((0.5 * (Math.pow(Math.log1p(x), 2.0) - Math.pow(Math.log(x), 2.0))) / n) + (Math.log1p(x) - Math.log(x))) / n;
	} else {
		tmp = Math.exp((x / n)) - t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -0.002:
		tmp = (1.0 / x) * (t_0 / n)
	elif (1.0 / n) <= 1e-15:
		tmp = (((0.5 * (math.pow(math.log1p(x), 2.0) - math.pow(math.log(x), 2.0))) / n) + (math.log1p(x) - math.log(x))) / n
	else:
		tmp = math.exp((x / n)) - t_0
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -0.002)
		tmp = Float64(Float64(1.0 / x) * Float64(t_0 / n));
	elseif (Float64(1.0 / n) <= 1e-15)
		tmp = Float64(Float64(Float64(Float64(0.5 * Float64((log1p(x) ^ 2.0) - (log(x) ^ 2.0))) / n) + Float64(log1p(x) - log(x))) / n);
	else
		tmp = Float64(exp(Float64(x / n)) - t_0);
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -0.002], N[(N[(1.0 / x), $MachinePrecision] * N[(t$95$0 / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-15], N[(N[(N[(N[(0.5 * N[(N[Power[N[Log[1 + x], $MachinePrecision], 2.0], $MachinePrecision] - N[Power[N[Log[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] + N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[Exp[N[(x / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -2e-3

   1. Initial program 98.9%

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

   3. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}\right), \color{blue}{\left(n \cdot x\right)}\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{\mathsf{neg}\left(\frac{\log \left(\frac{1}{x}\right)}{n}\right)}\right), \left(n \cdot x\right)\right) \]

      3. log-rec N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{\mathsf{neg}\left(\frac{\mathsf{neg}\left(\log x\right)}{n}\right)}\right), \left(n \cdot x\right)\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{\mathsf{neg}\left(\frac{-1 \cdot \log x}{n}\right)}\right), \left(n \cdot x\right)\right) \]

      5. exp-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{e^{\frac{-1 \cdot \log x}{n}}}\right), \left(\color{blue}{n} \cdot x\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(e^{\frac{-1 \cdot \log x}{n}}\right)\right), \left(\color{blue}{n} \cdot x\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(e^{\frac{\log x \cdot -1}{n}}\right)\right), \left(n \cdot x\right)\right) \]

      8. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(e^{\log x \cdot \frac{-1}{n}}\right)\right), \left(n \cdot x\right)\right) \]

      9. exp-to-pow N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left({x}^{\left(\frac{-1}{n}\right)}\right)\right), \left(n \cdot x\right)\right) \]

      10. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{pow.f64}\left(x, \left(\frac{-1}{n}\right)\right)\right), \left(n \cdot x\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(-1, n\right)\right)\right), \left(n \cdot x\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(-1, n\right)\right)\right), \left(x \cdot \color{blue}{n}\right)\right) \]

      13. *-lowering-*.f64 98.9%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(-1, n\right)\right)\right), \mathsf{*.f64}\left(x, \color{blue}{n}\right)\right) \]

   5. Simplified 98.9%

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

      1. div-inv N/A

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

      2. times-frac N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{x}\right), \color{blue}{\left(\frac{\frac{1}{{x}^{\left(\frac{-1}{n}\right)}}}{n}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\frac{\color{blue}{\frac{1}{{x}^{\left(\frac{-1}{n}\right)}}}}{n}\right)\right) \]

      5. pow-flip N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\frac{{x}^{\left(\mathsf{neg}\left(\frac{-1}{n}\right)\right)}}{n}\right)\right) \]

      6. distribute-neg-frac N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\frac{{x}^{\left(\frac{\mathsf{neg}\left(-1\right)}{n}\right)}}{n}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\frac{{x}^{\left(\frac{1}{n}\right)}}{n}\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{/.f64}\left(\left({x}^{\left(\frac{1}{n}\right)}\right), \color{blue}{n}\right)\right) \]

      9. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{/.f64}\left(\mathsf{pow.f64}\left(x, \left(\frac{1}{n}\right)\right), n\right)\right) \]

      10. /-lowering-/.f64 98.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{/.f64}\left(\mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right), n\right)\right) \]

   7. Applied egg-rr 98.9%

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

   if -2e-3 < (/.f64 #s(literal 1 binary64) n) < 1.0000000000000001e-15

   1. Initial program 27.5%

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

   3. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\frac{\left(\log \left(1 + x\right) + \frac{1}{2} \cdot \frac{{\log \left(1 + x\right)}^{2}}{n}\right) - \left(\log x + \frac{1}{2} \cdot \frac{{\log x}^{2}}{n}\right)}{n}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\log \left(1 + x\right) + \frac{1}{2} \cdot \frac{{\log \left(1 + x\right)}^{2}}{n}\right) - \left(\log x + \frac{1}{2} \cdot \frac{{\log x}^{2}}{n}\right)\right), \color{blue}{n}\right) \]

   5. Simplified 77.4%

      \[\leadsto \color{blue}{\frac{\frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right)}{n} + \left(\mathsf{log1p}\left(x\right) - \log x\right)}{n}} \]

   if 1.0000000000000001e-15 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 64.6%

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

      1. pow-to-exp N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(e^{\log \left(x + 1\right) \cdot \frac{1}{n}}\right), \mathsf{pow.f64}\left(\color{blue}{x}, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      2. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{exp.f64}\left(\left(\log \left(x + 1\right) \cdot \frac{1}{n}\right)\right), \mathsf{pow.f64}\left(\color{blue}{x}, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      3. un-div-inv N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{exp.f64}\left(\left(\frac{\log \left(x + 1\right)}{n}\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{exp.f64}\left(\mathsf{/.f64}\left(\log \left(x + 1\right), n\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{exp.f64}\left(\mathsf{/.f64}\left(\log \left(1 + x\right), n\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      6. log1p-define N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{log1p}\left(x\right)\right), n\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      7. log1p-lowering-log1p.f64 96.7%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{log1p.f64}\left(x\right), n\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

   4. Applied egg-rr 96.7%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{exp.f64}\left(\color{blue}{\left(\frac{x}{n}\right)}\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 96.7%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{exp.f64}\left(\mathsf{/.f64}\left(x, n\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

   7. Simplified 96.7%

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

Alternative 2: 86.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -0.002:\\ \;\;\;\;\frac{1}{x} \cdot \frac{t\_0}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-15}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{n}} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -0.002)
     (* (/ 1.0 x) (/ t_0 n))
     (if (<= (/ 1.0 n) 1e-15)
       (/ (log (/ (+ 1.0 x) x)) n)
       (- (exp (/ x n)) t_0)))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -0.002) {
		tmp = (1.0 / x) * (t_0 / n);
	} else if ((1.0 / n) <= 1e-15) {
		tmp = log(((1.0 + x) / x)) / n;
	} else {
		tmp = exp((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) :: tmp
    t_0 = x ** (1.0d0 / n)
    if ((1.0d0 / n) <= (-0.002d0)) then
        tmp = (1.0d0 / x) * (t_0 / n)
    else if ((1.0d0 / n) <= 1d-15) then
        tmp = log(((1.0d0 + x) / x)) / n
    else
        tmp = exp((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 tmp;
	if ((1.0 / n) <= -0.002) {
		tmp = (1.0 / x) * (t_0 / n);
	} else if ((1.0 / n) <= 1e-15) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else {
		tmp = Math.exp((x / n)) - t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -0.002:
		tmp = (1.0 / x) * (t_0 / n)
	elif (1.0 / n) <= 1e-15:
		tmp = math.log(((1.0 + x) / x)) / n
	else:
		tmp = math.exp((x / n)) - t_0
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -0.002)
		tmp = Float64(Float64(1.0 / x) * Float64(t_0 / n));
	elseif (Float64(1.0 / n) <= 1e-15)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	else
		tmp = Float64(exp(Float64(x / n)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	tmp = 0.0;
	if ((1.0 / n) <= -0.002)
		tmp = (1.0 / x) * (t_0 / n);
	elseif ((1.0 / n) <= 1e-15)
		tmp = log(((1.0 + x) / x)) / n;
	else
		tmp = exp((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]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -0.002], N[(N[(1.0 / x), $MachinePrecision] * N[(t$95$0 / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-15], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(N[Exp[N[(x / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -2e-3

   1. Initial program 98.9%

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

   3. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}\right), \color{blue}{\left(n \cdot x\right)}\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{\mathsf{neg}\left(\frac{\log \left(\frac{1}{x}\right)}{n}\right)}\right), \left(n \cdot x\right)\right) \]

      3. log-rec N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{\mathsf{neg}\left(\frac{\mathsf{neg}\left(\log x\right)}{n}\right)}\right), \left(n \cdot x\right)\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{\mathsf{neg}\left(\frac{-1 \cdot \log x}{n}\right)}\right), \left(n \cdot x\right)\right) \]

      5. exp-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{e^{\frac{-1 \cdot \log x}{n}}}\right), \left(\color{blue}{n} \cdot x\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(e^{\frac{-1 \cdot \log x}{n}}\right)\right), \left(\color{blue}{n} \cdot x\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(e^{\frac{\log x \cdot -1}{n}}\right)\right), \left(n \cdot x\right)\right) \]

      8. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(e^{\log x \cdot \frac{-1}{n}}\right)\right), \left(n \cdot x\right)\right) \]

      9. exp-to-pow N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left({x}^{\left(\frac{-1}{n}\right)}\right)\right), \left(n \cdot x\right)\right) \]

      10. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{pow.f64}\left(x, \left(\frac{-1}{n}\right)\right)\right), \left(n \cdot x\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(-1, n\right)\right)\right), \left(n \cdot x\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(-1, n\right)\right)\right), \left(x \cdot \color{blue}{n}\right)\right) \]

      13. *-lowering-*.f64 98.9%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(-1, n\right)\right)\right), \mathsf{*.f64}\left(x, \color{blue}{n}\right)\right) \]

   5. Simplified 98.9%

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

      1. div-inv N/A

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

      2. times-frac N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{x}\right), \color{blue}{\left(\frac{\frac{1}{{x}^{\left(\frac{-1}{n}\right)}}}{n}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\frac{\color{blue}{\frac{1}{{x}^{\left(\frac{-1}{n}\right)}}}}{n}\right)\right) \]

      5. pow-flip N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\frac{{x}^{\left(\mathsf{neg}\left(\frac{-1}{n}\right)\right)}}{n}\right)\right) \]

      6. distribute-neg-frac N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\frac{{x}^{\left(\frac{\mathsf{neg}\left(-1\right)}{n}\right)}}{n}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\frac{{x}^{\left(\frac{1}{n}\right)}}{n}\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{/.f64}\left(\left({x}^{\left(\frac{1}{n}\right)}\right), \color{blue}{n}\right)\right) \]

      9. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{/.f64}\left(\mathsf{pow.f64}\left(x, \left(\frac{1}{n}\right)\right), n\right)\right) \]

      10. /-lowering-/.f64 98.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{/.f64}\left(\mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right), n\right)\right) \]

   7. Applied egg-rr 98.9%

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

   if -2e-3 < (/.f64 #s(literal 1 binary64) n) < 1.0000000000000001e-15

   1. Initial program 27.5%

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

   3. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\frac{\left(\log \left(1 + x\right) + \frac{1}{2} \cdot \frac{{\log \left(1 + x\right)}^{2}}{n}\right) - \left(\log x + \frac{1}{2} \cdot \frac{{\log x}^{2}}{n}\right)}{n}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\log \left(1 + x\right) + \frac{1}{2} \cdot \frac{{\log \left(1 + x\right)}^{2}}{n}\right) - \left(\log x + \frac{1}{2} \cdot \frac{{\log x}^{2}}{n}\right)\right), \color{blue}{n}\right) \]

   5. Simplified 77.4%

      \[\leadsto \color{blue}{\frac{\frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right)}{n} + \left(\mathsf{log1p}\left(x\right) - \log x\right)}{n}} \]
   6. Step-by-step derivation

      1. clear-num N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{n}{\frac{\frac{1}{2} \cdot \left({\log \left(1 + x\right)}^{2} - {\log x}^{2}\right)}{n} + \left(\log \left(1 + x\right) - \log x\right)}\right)}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(n, \color{blue}{\left(\frac{\frac{1}{2} \cdot \left({\log \left(1 + x\right)}^{2} - {\log x}^{2}\right)}{n} + \left(\log \left(1 + x\right) - \log x\right)\right)}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(n, \mathsf{+.f64}\left(\left(\frac{\frac{1}{2} \cdot \left({\log \left(1 + x\right)}^{2} - {\log x}^{2}\right)}{n}\right), \color{blue}{\left(\log \left(1 + x\right) - \log x\right)}\right)\right)\right) \]

   7. Applied egg-rr 60.0%

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

   8. Taylor expanded in n around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\log \left(\frac{1 + x}{x}\right), \color{blue}{n}\right) \]

      2. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\left(\frac{1 + x}{x}\right)\right), n\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(\left(1 + x\right), x\right)\right), n\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(\left(x + 1\right), x\right)\right), n\right) \]

      5. +-lowering-+.f64 77.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, 1\right), x\right)\right), n\right) \]

   10. Simplified 77.3%

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

   if 1.0000000000000001e-15 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 64.6%

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

      1. pow-to-exp N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(e^{\log \left(x + 1\right) \cdot \frac{1}{n}}\right), \mathsf{pow.f64}\left(\color{blue}{x}, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      2. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{exp.f64}\left(\left(\log \left(x + 1\right) \cdot \frac{1}{n}\right)\right), \mathsf{pow.f64}\left(\color{blue}{x}, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      3. un-div-inv N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{exp.f64}\left(\left(\frac{\log \left(x + 1\right)}{n}\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{exp.f64}\left(\mathsf{/.f64}\left(\log \left(x + 1\right), n\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{exp.f64}\left(\mathsf{/.f64}\left(\log \left(1 + x\right), n\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      6. log1p-define N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{exp.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{log1p}\left(x\right)\right), n\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      7. log1p-lowering-log1p.f64 96.7%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{exp.f64}\left(\mathsf{/.f64}\left(\mathsf{log1p.f64}\left(x\right), n\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

   4. Applied egg-rr 96.7%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{exp.f64}\left(\color{blue}{\left(\frac{x}{n}\right)}\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 96.7%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{exp.f64}\left(\mathsf{/.f64}\left(x, n\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

   7. Simplified 96.7%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -0.002:\\ \;\;\;\;\frac{1}{x} \cdot \frac{{x}^{\left(\frac{1}{n}\right)}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-15}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -0.002:\\ \;\;\;\;\frac{1}{x} \cdot \frac{t\_0}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-15}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+120}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{0.5 \cdot \left(1 - \log x\right)}{n} + -0.5}{x \cdot x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -0.002)
     (* (/ 1.0 x) (/ t_0 n))
     (if (<= (/ 1.0 n) 1e-15)
       (/ (log (/ (+ 1.0 x) x)) n)
       (if (<= (/ 1.0 n) 1e+120)
         (- (+ 1.0 (/ x n)) t_0)
         (/ (/ (+ (/ (* 0.5 (- 1.0 (log x))) n) -0.5) (* x x)) n))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -0.002) {
		tmp = (1.0 / x) * (t_0 / n);
	} else if ((1.0 / n) <= 1e-15) {
		tmp = log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 1e+120) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = ((((0.5 * (1.0 - log(x))) / n) + -0.5) / (x * 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) :: tmp
    t_0 = x ** (1.0d0 / n)
    if ((1.0d0 / n) <= (-0.002d0)) then
        tmp = (1.0d0 / x) * (t_0 / n)
    else if ((1.0d0 / n) <= 1d-15) then
        tmp = log(((1.0d0 + x) / x)) / n
    else if ((1.0d0 / n) <= 1d+120) then
        tmp = (1.0d0 + (x / n)) - t_0
    else
        tmp = ((((0.5d0 * (1.0d0 - log(x))) / n) + (-0.5d0)) / (x * x)) / n
    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 tmp;
	if ((1.0 / n) <= -0.002) {
		tmp = (1.0 / x) * (t_0 / n);
	} else if ((1.0 / n) <= 1e-15) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 1e+120) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = ((((0.5 * (1.0 - Math.log(x))) / n) + -0.5) / (x * x)) / n;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -0.002:
		tmp = (1.0 / x) * (t_0 / n)
	elif (1.0 / n) <= 1e-15:
		tmp = math.log(((1.0 + x) / x)) / n
	elif (1.0 / n) <= 1e+120:
		tmp = (1.0 + (x / n)) - t_0
	else:
		tmp = ((((0.5 * (1.0 - math.log(x))) / n) + -0.5) / (x * x)) / n
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -0.002)
		tmp = Float64(Float64(1.0 / x) * Float64(t_0 / n));
	elseif (Float64(1.0 / n) <= 1e-15)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	elseif (Float64(1.0 / n) <= 1e+120)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_0);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(0.5 * Float64(1.0 - log(x))) / n) + -0.5) / Float64(x * x)) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	tmp = 0.0;
	if ((1.0 / n) <= -0.002)
		tmp = (1.0 / x) * (t_0 / n);
	elseif ((1.0 / n) <= 1e-15)
		tmp = log(((1.0 + x) / x)) / n;
	elseif ((1.0 / n) <= 1e+120)
		tmp = (1.0 + (x / n)) - t_0;
	else
		tmp = ((((0.5 * (1.0 - log(x))) / n) + -0.5) / (x * x)) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -0.002], N[(N[(1.0 / x), $MachinePrecision] * N[(t$95$0 / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-15], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e+120], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(N[(N[(N[(0.5 * N[(1.0 - N[Log[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision] + -0.5), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -2e-3

   1. Initial program 98.9%

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

   3. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}\right), \color{blue}{\left(n \cdot x\right)}\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{\mathsf{neg}\left(\frac{\log \left(\frac{1}{x}\right)}{n}\right)}\right), \left(n \cdot x\right)\right) \]

      3. log-rec N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{\mathsf{neg}\left(\frac{\mathsf{neg}\left(\log x\right)}{n}\right)}\right), \left(n \cdot x\right)\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{\mathsf{neg}\left(\frac{-1 \cdot \log x}{n}\right)}\right), \left(n \cdot x\right)\right) \]

      5. exp-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{e^{\frac{-1 \cdot \log x}{n}}}\right), \left(\color{blue}{n} \cdot x\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(e^{\frac{-1 \cdot \log x}{n}}\right)\right), \left(\color{blue}{n} \cdot x\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(e^{\frac{\log x \cdot -1}{n}}\right)\right), \left(n \cdot x\right)\right) \]

      8. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(e^{\log x \cdot \frac{-1}{n}}\right)\right), \left(n \cdot x\right)\right) \]

      9. exp-to-pow N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left({x}^{\left(\frac{-1}{n}\right)}\right)\right), \left(n \cdot x\right)\right) \]

      10. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{pow.f64}\left(x, \left(\frac{-1}{n}\right)\right)\right), \left(n \cdot x\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(-1, n\right)\right)\right), \left(n \cdot x\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(-1, n\right)\right)\right), \left(x \cdot \color{blue}{n}\right)\right) \]

      13. *-lowering-*.f64 98.9%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(-1, n\right)\right)\right), \mathsf{*.f64}\left(x, \color{blue}{n}\right)\right) \]

   5. Simplified 98.9%

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

      1. div-inv N/A

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

      2. times-frac N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{x}\right), \color{blue}{\left(\frac{\frac{1}{{x}^{\left(\frac{-1}{n}\right)}}}{n}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\frac{\color{blue}{\frac{1}{{x}^{\left(\frac{-1}{n}\right)}}}}{n}\right)\right) \]

      5. pow-flip N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\frac{{x}^{\left(\mathsf{neg}\left(\frac{-1}{n}\right)\right)}}{n}\right)\right) \]

      6. distribute-neg-frac N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\frac{{x}^{\left(\frac{\mathsf{neg}\left(-1\right)}{n}\right)}}{n}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\frac{{x}^{\left(\frac{1}{n}\right)}}{n}\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{/.f64}\left(\left({x}^{\left(\frac{1}{n}\right)}\right), \color{blue}{n}\right)\right) \]

      9. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{/.f64}\left(\mathsf{pow.f64}\left(x, \left(\frac{1}{n}\right)\right), n\right)\right) \]

      10. /-lowering-/.f64 98.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{/.f64}\left(\mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right), n\right)\right) \]

   7. Applied egg-rr 98.9%

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

   if -2e-3 < (/.f64 #s(literal 1 binary64) n) < 1.0000000000000001e-15

   1. Initial program 27.5%

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

   3. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\frac{\left(\log \left(1 + x\right) + \frac{1}{2} \cdot \frac{{\log \left(1 + x\right)}^{2}}{n}\right) - \left(\log x + \frac{1}{2} \cdot \frac{{\log x}^{2}}{n}\right)}{n}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\log \left(1 + x\right) + \frac{1}{2} \cdot \frac{{\log \left(1 + x\right)}^{2}}{n}\right) - \left(\log x + \frac{1}{2} \cdot \frac{{\log x}^{2}}{n}\right)\right), \color{blue}{n}\right) \]

   5. Simplified 77.4%

      \[\leadsto \color{blue}{\frac{\frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right)}{n} + \left(\mathsf{log1p}\left(x\right) - \log x\right)}{n}} \]
   6. Step-by-step derivation

      1. clear-num N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{n}{\frac{\frac{1}{2} \cdot \left({\log \left(1 + x\right)}^{2} - {\log x}^{2}\right)}{n} + \left(\log \left(1 + x\right) - \log x\right)}\right)}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(n, \color{blue}{\left(\frac{\frac{1}{2} \cdot \left({\log \left(1 + x\right)}^{2} - {\log x}^{2}\right)}{n} + \left(\log \left(1 + x\right) - \log x\right)\right)}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(n, \mathsf{+.f64}\left(\left(\frac{\frac{1}{2} \cdot \left({\log \left(1 + x\right)}^{2} - {\log x}^{2}\right)}{n}\right), \color{blue}{\left(\log \left(1 + x\right) - \log x\right)}\right)\right)\right) \]

   7. Applied egg-rr 60.0%

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

   8. Taylor expanded in n around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\log \left(\frac{1 + x}{x}\right), \color{blue}{n}\right) \]

      2. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\left(\frac{1 + x}{x}\right)\right), n\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(\left(1 + x\right), x\right)\right), n\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(\left(x + 1\right), x\right)\right), n\right) \]

      5. +-lowering-+.f64 77.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, 1\right), x\right)\right), n\right) \]

   10. Simplified 77.3%

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

   if 1.0000000000000001e-15 < (/.f64 #s(literal 1 binary64) n) < 9.9999999999999998e119

   1. Initial program 79.8%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(1 + \frac{x}{n}\right)}, \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]
   4. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(1 + \frac{x \cdot 1}{n}\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(1 + x \cdot \frac{1}{n}\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \left(x \cdot \frac{1}{n}\right)\right), \mathsf{pow.f64}\left(\color{blue}{x}, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      4. associate-*r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{x \cdot 1}{n}\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      5. *-rgt-identity N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{x}{n}\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      6. /-lowering-/.f64 81.2%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(x, n\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

   5. Simplified 81.2%

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

   if 9.9999999999999998e119 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 39.1%

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

   3. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\frac{\left(\log \left(1 + x\right) + \frac{1}{2} \cdot \frac{{\log \left(1 + x\right)}^{2}}{n}\right) - \left(\log x + \frac{1}{2} \cdot \frac{{\log x}^{2}}{n}\right)}{n}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\log \left(1 + x\right) + \frac{1}{2} \cdot \frac{{\log \left(1 + x\right)}^{2}}{n}\right) - \left(\log x + \frac{1}{2} \cdot \frac{{\log x}^{2}}{n}\right)\right), \color{blue}{n}\right) \]

   5. Simplified 0.2%

      \[\leadsto \color{blue}{\frac{\frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right)}{n} + \left(\mathsf{log1p}\left(x\right) - \log x\right)}{n}} \]

   6. Taylor expanded in x around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{\left(1 + \left(-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n} + \frac{1}{2} \cdot \frac{\frac{1}{n} + \frac{\log \left(\frac{1}{x}\right)}{n}}{x}\right)\right) - \frac{1}{2} \cdot \frac{1}{x}}{x}\right)}, n\right) \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(1 + \left(-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n} + \frac{1}{2} \cdot \frac{\frac{1}{n} + \frac{\log \left(\frac{1}{x}\right)}{n}}{x}\right)\right) - \frac{1}{2} \cdot \frac{1}{x}\right), x\right), n\right) \]

   8. Simplified 59.8%

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

   9. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{\frac{1}{2} \cdot \left(\frac{1}{n} - \frac{\log x}{n}\right) - \frac{1}{2}}{{x}^{2}}\right)}, n\right) \]
   10. Step-by-step derivation

       1. fmsub-define N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\mathsf{fmsub}\left(\frac{1}{2}, \left(\frac{1}{n} - \frac{\log x}{n}\right), \frac{1}{2}\right)}{{x}^{2}}\right), n\right) \]

       2. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\mathsf{fmsub}\left(\frac{1}{2}, \left(\frac{1}{n} + \left(\mathsf{neg}\left(\frac{\log x}{n}\right)\right)\right), \frac{1}{2}\right)}{{x}^{2}}\right), n\right) \]

       3. distribute-frac-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\mathsf{fmsub}\left(\frac{1}{2}, \left(\frac{1}{n} + \frac{\mathsf{neg}\left(\log x\right)}{n}\right), \frac{1}{2}\right)}{{x}^{2}}\right), n\right) \]

       4. log-rec N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\mathsf{fmsub}\left(\frac{1}{2}, \left(\frac{1}{n} + \frac{\log \left(\frac{1}{x}\right)}{n}\right), \frac{1}{2}\right)}{{x}^{2}}\right), n\right) \]

       5. fmsub-define N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{1}{2} \cdot \left(\frac{1}{n} + \frac{\log \left(\frac{1}{x}\right)}{n}\right) - \frac{1}{2}}{{x}^{2}}\right), n\right) \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{2} \cdot \left(\frac{1}{n} + \frac{\log \left(\frac{1}{x}\right)}{n}\right) - \frac{1}{2}\right), \left({x}^{2}\right)\right), n\right) \]

   11. Simplified 68.2%

       \[\leadsto \frac{\color{blue}{\frac{\frac{0.5 \cdot \left(1 - \log x\right)}{n} + -0.5}{x \cdot x}}}{n} \]
3. Recombined 4 regimes into one program.

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -0.002:\\ \;\;\;\;\frac{1}{x} \cdot \frac{{x}^{\left(\frac{1}{n}\right)}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-15}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{+120}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{0.5 \cdot \left(1 - \log x\right)}{n} + -0.5}{x \cdot x}}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 82.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -0.002:\\ \;\;\;\;\frac{1}{x} \cdot \frac{t\_0}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-15}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + x \cdot \left(\frac{1}{n} + \frac{x \cdot \frac{0.5}{n}}{n}\right)\right) - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -0.002)
     (* (/ 1.0 x) (/ t_0 n))
     (if (<= (/ 1.0 n) 1e-15)
       (/ (log (/ (+ 1.0 x) x)) n)
       (- (+ 1.0 (* x (+ (/ 1.0 n) (/ (* x (/ 0.5 n)) n)))) t_0)))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -0.002) {
		tmp = (1.0 / x) * (t_0 / n);
	} else if ((1.0 / n) <= 1e-15) {
		tmp = log(((1.0 + x) / x)) / n;
	} else {
		tmp = (1.0 + (x * ((1.0 / n) + ((x * (0.5 / n)) / 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) :: tmp
    t_0 = x ** (1.0d0 / n)
    if ((1.0d0 / n) <= (-0.002d0)) then
        tmp = (1.0d0 / x) * (t_0 / n)
    else if ((1.0d0 / n) <= 1d-15) then
        tmp = log(((1.0d0 + x) / x)) / n
    else
        tmp = (1.0d0 + (x * ((1.0d0 / n) + ((x * (0.5d0 / n)) / 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 tmp;
	if ((1.0 / n) <= -0.002) {
		tmp = (1.0 / x) * (t_0 / n);
	} else if ((1.0 / n) <= 1e-15) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else {
		tmp = (1.0 + (x * ((1.0 / n) + ((x * (0.5 / n)) / n)))) - t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -0.002:
		tmp = (1.0 / x) * (t_0 / n)
	elif (1.0 / n) <= 1e-15:
		tmp = math.log(((1.0 + x) / x)) / n
	else:
		tmp = (1.0 + (x * ((1.0 / n) + ((x * (0.5 / n)) / n)))) - t_0
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -0.002)
		tmp = Float64(Float64(1.0 / x) * Float64(t_0 / n));
	elseif (Float64(1.0 / n) <= 1e-15)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	else
		tmp = Float64(Float64(1.0 + Float64(x * Float64(Float64(1.0 / n) + Float64(Float64(x * Float64(0.5 / n)) / n)))) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	tmp = 0.0;
	if ((1.0 / n) <= -0.002)
		tmp = (1.0 / x) * (t_0 / n);
	elseif ((1.0 / n) <= 1e-15)
		tmp = log(((1.0 + x) / x)) / n;
	else
		tmp = (1.0 + (x * ((1.0 / n) + ((x * (0.5 / n)) / 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]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -0.002], N[(N[(1.0 / x), $MachinePrecision] * N[(t$95$0 / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-15], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], N[(N[(1.0 + N[(x * N[(N[(1.0 / n), $MachinePrecision] + N[(N[(x * N[(0.5 / n), $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -2e-3

   1. Initial program 98.9%

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

   3. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}\right), \color{blue}{\left(n \cdot x\right)}\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{\mathsf{neg}\left(\frac{\log \left(\frac{1}{x}\right)}{n}\right)}\right), \left(n \cdot x\right)\right) \]

      3. log-rec N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{\mathsf{neg}\left(\frac{\mathsf{neg}\left(\log x\right)}{n}\right)}\right), \left(n \cdot x\right)\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{\mathsf{neg}\left(\frac{-1 \cdot \log x}{n}\right)}\right), \left(n \cdot x\right)\right) \]

      5. exp-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{e^{\frac{-1 \cdot \log x}{n}}}\right), \left(\color{blue}{n} \cdot x\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(e^{\frac{-1 \cdot \log x}{n}}\right)\right), \left(\color{blue}{n} \cdot x\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(e^{\frac{\log x \cdot -1}{n}}\right)\right), \left(n \cdot x\right)\right) \]

      8. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(e^{\log x \cdot \frac{-1}{n}}\right)\right), \left(n \cdot x\right)\right) \]

      9. exp-to-pow N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left({x}^{\left(\frac{-1}{n}\right)}\right)\right), \left(n \cdot x\right)\right) \]

      10. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{pow.f64}\left(x, \left(\frac{-1}{n}\right)\right)\right), \left(n \cdot x\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(-1, n\right)\right)\right), \left(n \cdot x\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(-1, n\right)\right)\right), \left(x \cdot \color{blue}{n}\right)\right) \]

      13. *-lowering-*.f64 98.9%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(-1, n\right)\right)\right), \mathsf{*.f64}\left(x, \color{blue}{n}\right)\right) \]

   5. Simplified 98.9%

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

      1. div-inv N/A

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

      2. times-frac N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{x}\right), \color{blue}{\left(\frac{\frac{1}{{x}^{\left(\frac{-1}{n}\right)}}}{n}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\frac{\color{blue}{\frac{1}{{x}^{\left(\frac{-1}{n}\right)}}}}{n}\right)\right) \]

      5. pow-flip N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\frac{{x}^{\left(\mathsf{neg}\left(\frac{-1}{n}\right)\right)}}{n}\right)\right) \]

      6. distribute-neg-frac N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\frac{{x}^{\left(\frac{\mathsf{neg}\left(-1\right)}{n}\right)}}{n}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\frac{{x}^{\left(\frac{1}{n}\right)}}{n}\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{/.f64}\left(\left({x}^{\left(\frac{1}{n}\right)}\right), \color{blue}{n}\right)\right) \]

      9. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{/.f64}\left(\mathsf{pow.f64}\left(x, \left(\frac{1}{n}\right)\right), n\right)\right) \]

      10. /-lowering-/.f64 98.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{/.f64}\left(\mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right), n\right)\right) \]

   7. Applied egg-rr 98.9%

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

   if -2e-3 < (/.f64 #s(literal 1 binary64) n) < 1.0000000000000001e-15

   1. Initial program 27.5%

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

   3. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\frac{\left(\log \left(1 + x\right) + \frac{1}{2} \cdot \frac{{\log \left(1 + x\right)}^{2}}{n}\right) - \left(\log x + \frac{1}{2} \cdot \frac{{\log x}^{2}}{n}\right)}{n}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\log \left(1 + x\right) + \frac{1}{2} \cdot \frac{{\log \left(1 + x\right)}^{2}}{n}\right) - \left(\log x + \frac{1}{2} \cdot \frac{{\log x}^{2}}{n}\right)\right), \color{blue}{n}\right) \]

   5. Simplified 77.4%

      \[\leadsto \color{blue}{\frac{\frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right)}{n} + \left(\mathsf{log1p}\left(x\right) - \log x\right)}{n}} \]
   6. Step-by-step derivation

      1. clear-num N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{n}{\frac{\frac{1}{2} \cdot \left({\log \left(1 + x\right)}^{2} - {\log x}^{2}\right)}{n} + \left(\log \left(1 + x\right) - \log x\right)}\right)}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(n, \color{blue}{\left(\frac{\frac{1}{2} \cdot \left({\log \left(1 + x\right)}^{2} - {\log x}^{2}\right)}{n} + \left(\log \left(1 + x\right) - \log x\right)\right)}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(n, \mathsf{+.f64}\left(\left(\frac{\frac{1}{2} \cdot \left({\log \left(1 + x\right)}^{2} - {\log x}^{2}\right)}{n}\right), \color{blue}{\left(\log \left(1 + x\right) - \log x\right)}\right)\right)\right) \]

   7. Applied egg-rr 60.0%

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

   8. Taylor expanded in n around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\log \left(\frac{1 + x}{x}\right), \color{blue}{n}\right) \]

      2. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\left(\frac{1 + x}{x}\right)\right), n\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(\left(1 + x\right), x\right)\right), n\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(\left(x + 1\right), x\right)\right), n\right) \]

      5. +-lowering-+.f64 77.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, 1\right), x\right)\right), n\right) \]

   10. Simplified 77.3%

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

   if 1.0000000000000001e-15 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 64.6%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(1 + x \cdot \left(x \cdot \left(\frac{1}{2} \cdot \frac{1}{{n}^{2}} - \frac{1}{2} \cdot \frac{1}{n}\right) + \frac{1}{n}\right)\right)}, \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]
   4. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \left(x \cdot \left(x \cdot \left(\frac{1}{2} \cdot \frac{1}{{n}^{2}} - \frac{1}{2} \cdot \frac{1}{n}\right) + \frac{1}{n}\right)\right)\right), \mathsf{pow.f64}\left(\color{blue}{x}, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{2} \cdot \frac{1}{{n}^{2}} - \frac{1}{2} \cdot \frac{1}{n}\right) + \frac{1}{n}\right)\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\frac{1}{n} + x \cdot \left(\frac{1}{2} \cdot \frac{1}{{n}^{2}} - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\frac{1}{n}\right), \left(x \cdot \left(\frac{1}{2} \cdot \frac{1}{{n}^{2}} - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \left(x \cdot \left(\frac{1}{2} \cdot \frac{1}{{n}^{2}} - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{*.f64}\left(x, \left(\frac{1}{2} \cdot \frac{1}{{n}^{2}} - \frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{*.f64}\left(x, \left(\frac{1}{2} \cdot \frac{1}{{n}^{2}} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\frac{1}{2} \cdot \frac{1}{{n}^{2}}\right), \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\frac{\frac{1}{2} \cdot 1}{{n}^{2}}\right), \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\frac{\frac{1}{2}}{{n}^{2}}\right), \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \left({n}^{2}\right)\right), \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \left(n \cdot n\right)\right), \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(n, n\right)\right), \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{n}\right)\right)\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      14. associate-*r/ N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(n, n\right)\right), \left(\mathsf{neg}\left(\frac{\frac{1}{2} \cdot 1}{n}\right)\right)\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(n, n\right)\right), \left(\mathsf{neg}\left(\frac{\frac{1}{2}}{n}\right)\right)\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      16. distribute-neg-frac N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(n, n\right)\right), \left(\frac{\mathsf{neg}\left(\frac{1}{2}\right)}{n}\right)\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      17. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(n, n\right)\right), \left(\frac{\frac{-1}{2}}{n}\right)\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      18. /-lowering-/.f64 73.4%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(n, n\right)\right), \mathsf{/.f64}\left(\frac{-1}{2}, n\right)\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

   5. Simplified 73.4%

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

   6. Taylor expanded in n around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \color{blue}{\left(\frac{1}{2} \cdot \frac{x}{{n}^{2}}\right)}\right)\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]
   7. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \left(\frac{\frac{1}{2} \cdot x}{{n}^{2}}\right)\right)\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \left(\frac{\frac{1}{2} \cdot x}{n \cdot n}\right)\right)\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      3. associate-/r* N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \left(\frac{\frac{\frac{1}{2} \cdot x}{n}}{n}\right)\right)\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      4. associate-*r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \left(\frac{\frac{1}{2} \cdot \frac{x}{n}}{n}\right)\right)\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot \frac{x}{n}\right), n\right)\right)\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{/.f64}\left(\left(\frac{\frac{1}{2} \cdot x}{n}\right), n\right)\right)\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{/.f64}\left(\left(\frac{x \cdot \frac{1}{2}}{n}\right), n\right)\right)\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      8. associate-/l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{/.f64}\left(\left(x \cdot \frac{\frac{1}{2}}{n}\right), n\right)\right)\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{/.f64}\left(\left(x \cdot \frac{\frac{1}{2} \cdot 1}{n}\right), n\right)\right)\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      10. associate-*r/ N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{/.f64}\left(\left(x \cdot \left(\frac{1}{2} \cdot \frac{1}{n}\right)\right), n\right)\right)\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{1}{2} \cdot \frac{1}{n}\right)\right), n\right)\right)\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      12. associate-*r/ N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{\frac{1}{2} \cdot 1}{n}\right)\right), n\right)\right)\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{\frac{1}{2}}{n}\right)\right), n\right)\right)\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      14. /-lowering-/.f64 82.5%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, n\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\frac{1}{2}, n\right)\right), n\right)\right)\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

   8. Simplified 82.5%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -0.002:\\ \;\;\;\;\frac{1}{x} \cdot \frac{{x}^{\left(\frac{1}{n}\right)}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-15}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + x \cdot \left(\frac{1}{n} + \frac{x \cdot \frac{0.5}{n}}{n}\right)\right) - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -0.002:\\ \;\;\;\;\frac{1}{x} \cdot \frac{t\_0}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-15}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+233}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -0.002)
     (* (/ 1.0 x) (/ t_0 n))
     (if (<= (/ 1.0 n) 1e-15)
       (/ (log (/ (+ 1.0 x) x)) n)
       (if (<= (/ 1.0 n) 2e+233) (- (+ 1.0 (/ x n)) t_0) (/ (/ 1.0 x) n))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -0.002) {
		tmp = (1.0 / x) * (t_0 / n);
	} else if ((1.0 / n) <= 1e-15) {
		tmp = log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 2e+233) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = (1.0 / 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) :: tmp
    t_0 = x ** (1.0d0 / n)
    if ((1.0d0 / n) <= (-0.002d0)) then
        tmp = (1.0d0 / x) * (t_0 / n)
    else if ((1.0d0 / n) <= 1d-15) then
        tmp = log(((1.0d0 + x) / x)) / n
    else if ((1.0d0 / n) <= 2d+233) then
        tmp = (1.0d0 + (x / n)) - t_0
    else
        tmp = (1.0d0 / x) / n
    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 tmp;
	if ((1.0 / n) <= -0.002) {
		tmp = (1.0 / x) * (t_0 / n);
	} else if ((1.0 / n) <= 1e-15) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 2e+233) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = (1.0 / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -0.002:
		tmp = (1.0 / x) * (t_0 / n)
	elif (1.0 / n) <= 1e-15:
		tmp = math.log(((1.0 + x) / x)) / n
	elif (1.0 / n) <= 2e+233:
		tmp = (1.0 + (x / n)) - t_0
	else:
		tmp = (1.0 / x) / n
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -0.002)
		tmp = Float64(Float64(1.0 / x) * Float64(t_0 / n));
	elseif (Float64(1.0 / n) <= 1e-15)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	elseif (Float64(1.0 / n) <= 2e+233)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_0);
	else
		tmp = Float64(Float64(1.0 / x) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	tmp = 0.0;
	if ((1.0 / n) <= -0.002)
		tmp = (1.0 / x) * (t_0 / n);
	elseif ((1.0 / n) <= 1e-15)
		tmp = log(((1.0 + x) / x)) / n;
	elseif ((1.0 / n) <= 2e+233)
		tmp = (1.0 + (x / n)) - t_0;
	else
		tmp = (1.0 / x) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -0.002], N[(N[(1.0 / x), $MachinePrecision] * N[(t$95$0 / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-15], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e+233], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -2e-3

   1. Initial program 98.9%

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

   3. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}\right), \color{blue}{\left(n \cdot x\right)}\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{\mathsf{neg}\left(\frac{\log \left(\frac{1}{x}\right)}{n}\right)}\right), \left(n \cdot x\right)\right) \]

      3. log-rec N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{\mathsf{neg}\left(\frac{\mathsf{neg}\left(\log x\right)}{n}\right)}\right), \left(n \cdot x\right)\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{\mathsf{neg}\left(\frac{-1 \cdot \log x}{n}\right)}\right), \left(n \cdot x\right)\right) \]

      5. exp-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{e^{\frac{-1 \cdot \log x}{n}}}\right), \left(\color{blue}{n} \cdot x\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(e^{\frac{-1 \cdot \log x}{n}}\right)\right), \left(\color{blue}{n} \cdot x\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(e^{\frac{\log x \cdot -1}{n}}\right)\right), \left(n \cdot x\right)\right) \]

      8. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(e^{\log x \cdot \frac{-1}{n}}\right)\right), \left(n \cdot x\right)\right) \]

      9. exp-to-pow N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left({x}^{\left(\frac{-1}{n}\right)}\right)\right), \left(n \cdot x\right)\right) \]

      10. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{pow.f64}\left(x, \left(\frac{-1}{n}\right)\right)\right), \left(n \cdot x\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(-1, n\right)\right)\right), \left(n \cdot x\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(-1, n\right)\right)\right), \left(x \cdot \color{blue}{n}\right)\right) \]

      13. *-lowering-*.f64 98.9%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(-1, n\right)\right)\right), \mathsf{*.f64}\left(x, \color{blue}{n}\right)\right) \]

   5. Simplified 98.9%

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

      1. div-inv N/A

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

      2. times-frac N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{x}\right), \color{blue}{\left(\frac{\frac{1}{{x}^{\left(\frac{-1}{n}\right)}}}{n}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\frac{\color{blue}{\frac{1}{{x}^{\left(\frac{-1}{n}\right)}}}}{n}\right)\right) \]

      5. pow-flip N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\frac{{x}^{\left(\mathsf{neg}\left(\frac{-1}{n}\right)\right)}}{n}\right)\right) \]

      6. distribute-neg-frac N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\frac{{x}^{\left(\frac{\mathsf{neg}\left(-1\right)}{n}\right)}}{n}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\frac{{x}^{\left(\frac{1}{n}\right)}}{n}\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{/.f64}\left(\left({x}^{\left(\frac{1}{n}\right)}\right), \color{blue}{n}\right)\right) \]

      9. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{/.f64}\left(\mathsf{pow.f64}\left(x, \left(\frac{1}{n}\right)\right), n\right)\right) \]

      10. /-lowering-/.f64 98.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{/.f64}\left(\mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right), n\right)\right) \]

   7. Applied egg-rr 98.9%

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

   if -2e-3 < (/.f64 #s(literal 1 binary64) n) < 1.0000000000000001e-15

   1. Initial program 27.5%

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

   3. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\frac{\left(\log \left(1 + x\right) + \frac{1}{2} \cdot \frac{{\log \left(1 + x\right)}^{2}}{n}\right) - \left(\log x + \frac{1}{2} \cdot \frac{{\log x}^{2}}{n}\right)}{n}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\log \left(1 + x\right) + \frac{1}{2} \cdot \frac{{\log \left(1 + x\right)}^{2}}{n}\right) - \left(\log x + \frac{1}{2} \cdot \frac{{\log x}^{2}}{n}\right)\right), \color{blue}{n}\right) \]

   5. Simplified 77.4%

      \[\leadsto \color{blue}{\frac{\frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right)}{n} + \left(\mathsf{log1p}\left(x\right) - \log x\right)}{n}} \]
   6. Step-by-step derivation

      1. clear-num N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{n}{\frac{\frac{1}{2} \cdot \left({\log \left(1 + x\right)}^{2} - {\log x}^{2}\right)}{n} + \left(\log \left(1 + x\right) - \log x\right)}\right)}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(n, \color{blue}{\left(\frac{\frac{1}{2} \cdot \left({\log \left(1 + x\right)}^{2} - {\log x}^{2}\right)}{n} + \left(\log \left(1 + x\right) - \log x\right)\right)}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(n, \mathsf{+.f64}\left(\left(\frac{\frac{1}{2} \cdot \left({\log \left(1 + x\right)}^{2} - {\log x}^{2}\right)}{n}\right), \color{blue}{\left(\log \left(1 + x\right) - \log x\right)}\right)\right)\right) \]

   7. Applied egg-rr 60.0%

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

   8. Taylor expanded in n around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\log \left(\frac{1 + x}{x}\right), \color{blue}{n}\right) \]

      2. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\left(\frac{1 + x}{x}\right)\right), n\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(\left(1 + x\right), x\right)\right), n\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(\left(x + 1\right), x\right)\right), n\right) \]

      5. +-lowering-+.f64 77.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, 1\right), x\right)\right), n\right) \]

   10. Simplified 77.3%

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

   if 1.0000000000000001e-15 < (/.f64 #s(literal 1 binary64) n) < 1.99999999999999995e233

   1. Initial program 80.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(1 + \frac{x}{n}\right)}, \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]
   4. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(1 + \frac{x \cdot 1}{n}\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(1 + x \cdot \frac{1}{n}\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \left(x \cdot \frac{1}{n}\right)\right), \mathsf{pow.f64}\left(\color{blue}{x}, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      4. associate-*r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{x \cdot 1}{n}\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      5. *-rgt-identity N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{x}{n}\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

      6. /-lowering-/.f64 73.5%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(x, n\right)\right), \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]

   5. Simplified 73.5%

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

   if 1.99999999999999995e233 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 9.4%

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

   3. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}\right), \color{blue}{\left(n \cdot x\right)}\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{\mathsf{neg}\left(\frac{\log \left(\frac{1}{x}\right)}{n}\right)}\right), \left(n \cdot x\right)\right) \]

      3. log-rec N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{\mathsf{neg}\left(\frac{\mathsf{neg}\left(\log x\right)}{n}\right)}\right), \left(n \cdot x\right)\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{\mathsf{neg}\left(\frac{-1 \cdot \log x}{n}\right)}\right), \left(n \cdot x\right)\right) \]

      5. exp-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{e^{\frac{-1 \cdot \log x}{n}}}\right), \left(\color{blue}{n} \cdot x\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(e^{\frac{-1 \cdot \log x}{n}}\right)\right), \left(\color{blue}{n} \cdot x\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(e^{\frac{\log x \cdot -1}{n}}\right)\right), \left(n \cdot x\right)\right) \]

      8. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(e^{\log x \cdot \frac{-1}{n}}\right)\right), \left(n \cdot x\right)\right) \]

      9. exp-to-pow N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left({x}^{\left(\frac{-1}{n}\right)}\right)\right), \left(n \cdot x\right)\right) \]

      10. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{pow.f64}\left(x, \left(\frac{-1}{n}\right)\right)\right), \left(n \cdot x\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(-1, n\right)\right)\right), \left(n \cdot x\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(-1, n\right)\right)\right), \left(x \cdot \color{blue}{n}\right)\right) \]

      13. *-lowering-*.f64 0.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(-1, n\right)\right)\right), \mathsf{*.f64}\left(x, \color{blue}{n}\right)\right) \]

   5. Simplified 0.0%

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

      1. associate-/r* N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{1}{{x}^{\left(\frac{-1}{n}\right)}}}{x}\right), \color{blue}{n}\right) \]

      3. pow-flip N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{{x}^{\left(\mathsf{neg}\left(\frac{-1}{n}\right)\right)}}{x}\right), n\right) \]

      4. distribute-neg-frac N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{{x}^{\left(\frac{\mathsf{neg}\left(-1\right)}{n}\right)}}{x}\right), n\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{{x}^{\left(\frac{1}{n}\right)}}{x}\right), n\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left({x}^{\left(\frac{1}{n}\right)}\right), x\right), n\right) \]

      7. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(x, \left(\frac{1}{n}\right)\right), x\right), n\right) \]

      8. /-lowering-/.f64 1.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right), x\right), n\right) \]

   7. Applied egg-rr 1.8%

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

   8. Taylor expanded in n around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{1}{x}\right)}, n\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 86.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), n\right) \]

   10. Simplified 86.2%

       \[\leadsto \frac{\color{blue}{\frac{1}{x}}}{n} \]
3. Recombined 4 regimes into one program.

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -0.002:\\ \;\;\;\;\frac{1}{x} \cdot \frac{{x}^{\left(\frac{1}{n}\right)}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-15}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+233}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 81.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -0.002:\\ \;\;\;\;\frac{1}{x} \cdot \frac{t\_0}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-15}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+233}:\\ \;\;\;\;1 - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -0.002)
     (* (/ 1.0 x) (/ t_0 n))
     (if (<= (/ 1.0 n) 1e-15)
       (/ (log (/ (+ 1.0 x) x)) n)
       (if (<= (/ 1.0 n) 2e+233) (- 1.0 t_0) (/ (/ 1.0 x) n))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -0.002) {
		tmp = (1.0 / x) * (t_0 / n);
	} else if ((1.0 / n) <= 1e-15) {
		tmp = log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 2e+233) {
		tmp = 1.0 - t_0;
	} else {
		tmp = (1.0 / 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) :: tmp
    t_0 = x ** (1.0d0 / n)
    if ((1.0d0 / n) <= (-0.002d0)) then
        tmp = (1.0d0 / x) * (t_0 / n)
    else if ((1.0d0 / n) <= 1d-15) then
        tmp = log(((1.0d0 + x) / x)) / n
    else if ((1.0d0 / n) <= 2d+233) then
        tmp = 1.0d0 - t_0
    else
        tmp = (1.0d0 / x) / n
    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 tmp;
	if ((1.0 / n) <= -0.002) {
		tmp = (1.0 / x) * (t_0 / n);
	} else if ((1.0 / n) <= 1e-15) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 2e+233) {
		tmp = 1.0 - t_0;
	} else {
		tmp = (1.0 / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -0.002:
		tmp = (1.0 / x) * (t_0 / n)
	elif (1.0 / n) <= 1e-15:
		tmp = math.log(((1.0 + x) / x)) / n
	elif (1.0 / n) <= 2e+233:
		tmp = 1.0 - t_0
	else:
		tmp = (1.0 / x) / n
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -0.002)
		tmp = Float64(Float64(1.0 / x) * Float64(t_0 / n));
	elseif (Float64(1.0 / n) <= 1e-15)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	elseif (Float64(1.0 / n) <= 2e+233)
		tmp = Float64(1.0 - t_0);
	else
		tmp = Float64(Float64(1.0 / x) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	tmp = 0.0;
	if ((1.0 / n) <= -0.002)
		tmp = (1.0 / x) * (t_0 / n);
	elseif ((1.0 / n) <= 1e-15)
		tmp = log(((1.0 + x) / x)) / n;
	elseif ((1.0 / n) <= 2e+233)
		tmp = 1.0 - t_0;
	else
		tmp = (1.0 / x) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -0.002], N[(N[(1.0 / x), $MachinePrecision] * N[(t$95$0 / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-15], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e+233], N[(1.0 - t$95$0), $MachinePrecision], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -2e-3

   1. Initial program 98.9%

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

   3. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}\right), \color{blue}{\left(n \cdot x\right)}\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{\mathsf{neg}\left(\frac{\log \left(\frac{1}{x}\right)}{n}\right)}\right), \left(n \cdot x\right)\right) \]

      3. log-rec N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{\mathsf{neg}\left(\frac{\mathsf{neg}\left(\log x\right)}{n}\right)}\right), \left(n \cdot x\right)\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{\mathsf{neg}\left(\frac{-1 \cdot \log x}{n}\right)}\right), \left(n \cdot x\right)\right) \]

      5. exp-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{e^{\frac{-1 \cdot \log x}{n}}}\right), \left(\color{blue}{n} \cdot x\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(e^{\frac{-1 \cdot \log x}{n}}\right)\right), \left(\color{blue}{n} \cdot x\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(e^{\frac{\log x \cdot -1}{n}}\right)\right), \left(n \cdot x\right)\right) \]

      8. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(e^{\log x \cdot \frac{-1}{n}}\right)\right), \left(n \cdot x\right)\right) \]

      9. exp-to-pow N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left({x}^{\left(\frac{-1}{n}\right)}\right)\right), \left(n \cdot x\right)\right) \]

      10. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{pow.f64}\left(x, \left(\frac{-1}{n}\right)\right)\right), \left(n \cdot x\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(-1, n\right)\right)\right), \left(n \cdot x\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(-1, n\right)\right)\right), \left(x \cdot \color{blue}{n}\right)\right) \]

      13. *-lowering-*.f64 98.9%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(-1, n\right)\right)\right), \mathsf{*.f64}\left(x, \color{blue}{n}\right)\right) \]

   5. Simplified 98.9%

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

      1. div-inv N/A

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

      2. times-frac N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{x}\right), \color{blue}{\left(\frac{\frac{1}{{x}^{\left(\frac{-1}{n}\right)}}}{n}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\frac{\color{blue}{\frac{1}{{x}^{\left(\frac{-1}{n}\right)}}}}{n}\right)\right) \]

      5. pow-flip N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\frac{{x}^{\left(\mathsf{neg}\left(\frac{-1}{n}\right)\right)}}{n}\right)\right) \]

      6. distribute-neg-frac N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\frac{{x}^{\left(\frac{\mathsf{neg}\left(-1\right)}{n}\right)}}{n}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \left(\frac{{x}^{\left(\frac{1}{n}\right)}}{n}\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{/.f64}\left(\left({x}^{\left(\frac{1}{n}\right)}\right), \color{blue}{n}\right)\right) \]

      9. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{/.f64}\left(\mathsf{pow.f64}\left(x, \left(\frac{1}{n}\right)\right), n\right)\right) \]

      10. /-lowering-/.f64 98.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, x\right), \mathsf{/.f64}\left(\mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right), n\right)\right) \]

   7. Applied egg-rr 98.9%

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

   if -2e-3 < (/.f64 #s(literal 1 binary64) n) < 1.0000000000000001e-15

   1. Initial program 27.5%

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

   3. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\frac{\left(\log \left(1 + x\right) + \frac{1}{2} \cdot \frac{{\log \left(1 + x\right)}^{2}}{n}\right) - \left(\log x + \frac{1}{2} \cdot \frac{{\log x}^{2}}{n}\right)}{n}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\log \left(1 + x\right) + \frac{1}{2} \cdot \frac{{\log \left(1 + x\right)}^{2}}{n}\right) - \left(\log x + \frac{1}{2} \cdot \frac{{\log x}^{2}}{n}\right)\right), \color{blue}{n}\right) \]

   5. Simplified 77.4%

      \[\leadsto \color{blue}{\frac{\frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right)}{n} + \left(\mathsf{log1p}\left(x\right) - \log x\right)}{n}} \]
   6. Step-by-step derivation

      1. clear-num N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{n}{\frac{\frac{1}{2} \cdot \left({\log \left(1 + x\right)}^{2} - {\log x}^{2}\right)}{n} + \left(\log \left(1 + x\right) - \log x\right)}\right)}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(n, \color{blue}{\left(\frac{\frac{1}{2} \cdot \left({\log \left(1 + x\right)}^{2} - {\log x}^{2}\right)}{n} + \left(\log \left(1 + x\right) - \log x\right)\right)}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(n, \mathsf{+.f64}\left(\left(\frac{\frac{1}{2} \cdot \left({\log \left(1 + x\right)}^{2} - {\log x}^{2}\right)}{n}\right), \color{blue}{\left(\log \left(1 + x\right) - \log x\right)}\right)\right)\right) \]

   7. Applied egg-rr 60.0%

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

   8. Taylor expanded in n around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\log \left(\frac{1 + x}{x}\right), \color{blue}{n}\right) \]

      2. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\left(\frac{1 + x}{x}\right)\right), n\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(\left(1 + x\right), x\right)\right), n\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(\left(x + 1\right), x\right)\right), n\right) \]

      5. +-lowering-+.f64 77.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, 1\right), x\right)\right), n\right) \]

   10. Simplified 77.3%

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

   if 1.0000000000000001e-15 < (/.f64 #s(literal 1 binary64) n) < 1.99999999999999995e233

   1. Initial program 80.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{1}, \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 72.3%

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

   if 1.99999999999999995e233 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 9.4%

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

   3. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}\right), \color{blue}{\left(n \cdot x\right)}\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{\mathsf{neg}\left(\frac{\log \left(\frac{1}{x}\right)}{n}\right)}\right), \left(n \cdot x\right)\right) \]

      3. log-rec N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{\mathsf{neg}\left(\frac{\mathsf{neg}\left(\log x\right)}{n}\right)}\right), \left(n \cdot x\right)\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{\mathsf{neg}\left(\frac{-1 \cdot \log x}{n}\right)}\right), \left(n \cdot x\right)\right) \]

      5. exp-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{e^{\frac{-1 \cdot \log x}{n}}}\right), \left(\color{blue}{n} \cdot x\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(e^{\frac{-1 \cdot \log x}{n}}\right)\right), \left(\color{blue}{n} \cdot x\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(e^{\frac{\log x \cdot -1}{n}}\right)\right), \left(n \cdot x\right)\right) \]

      8. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(e^{\log x \cdot \frac{-1}{n}}\right)\right), \left(n \cdot x\right)\right) \]

      9. exp-to-pow N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left({x}^{\left(\frac{-1}{n}\right)}\right)\right), \left(n \cdot x\right)\right) \]

      10. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{pow.f64}\left(x, \left(\frac{-1}{n}\right)\right)\right), \left(n \cdot x\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(-1, n\right)\right)\right), \left(n \cdot x\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(-1, n\right)\right)\right), \left(x \cdot \color{blue}{n}\right)\right) \]

      13. *-lowering-*.f64 0.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(-1, n\right)\right)\right), \mathsf{*.f64}\left(x, \color{blue}{n}\right)\right) \]

   5. Simplified 0.0%

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

      1. associate-/r* N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{1}{{x}^{\left(\frac{-1}{n}\right)}}}{x}\right), \color{blue}{n}\right) \]

      3. pow-flip N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{{x}^{\left(\mathsf{neg}\left(\frac{-1}{n}\right)\right)}}{x}\right), n\right) \]

      4. distribute-neg-frac N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{{x}^{\left(\frac{\mathsf{neg}\left(-1\right)}{n}\right)}}{x}\right), n\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{{x}^{\left(\frac{1}{n}\right)}}{x}\right), n\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left({x}^{\left(\frac{1}{n}\right)}\right), x\right), n\right) \]

      7. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(x, \left(\frac{1}{n}\right)\right), x\right), n\right) \]

      8. /-lowering-/.f64 1.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right), x\right), n\right) \]

   7. Applied egg-rr 1.8%

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

   8. Taylor expanded in n around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{1}{x}\right)}, n\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 86.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), n\right) \]

   10. Simplified 86.2%

       \[\leadsto \frac{\color{blue}{\frac{1}{x}}}{n} \]
3. Recombined 4 regimes into one program.

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -0.002:\\ \;\;\;\;\frac{1}{x} \cdot \frac{{x}^{\left(\frac{1}{n}\right)}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-15}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+233}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 81.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;\frac{1}{n} \leq -0.002:\\ \;\;\;\;\frac{\frac{t\_0}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-15}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+233}:\\ \;\;\;\;1 - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= (/ 1.0 n) -0.002)
     (/ (/ t_0 n) x)
     (if (<= (/ 1.0 n) 1e-15)
       (/ (log (/ (+ 1.0 x) x)) n)
       (if (<= (/ 1.0 n) 2e+233) (- 1.0 t_0) (/ (/ 1.0 x) n))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if ((1.0 / n) <= -0.002) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 1e-15) {
		tmp = log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 2e+233) {
		tmp = 1.0 - t_0;
	} else {
		tmp = (1.0 / 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) :: tmp
    t_0 = x ** (1.0d0 / n)
    if ((1.0d0 / n) <= (-0.002d0)) then
        tmp = (t_0 / n) / x
    else if ((1.0d0 / n) <= 1d-15) then
        tmp = log(((1.0d0 + x) / x)) / n
    else if ((1.0d0 / n) <= 2d+233) then
        tmp = 1.0d0 - t_0
    else
        tmp = (1.0d0 / x) / n
    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 tmp;
	if ((1.0 / n) <= -0.002) {
		tmp = (t_0 / n) / x;
	} else if ((1.0 / n) <= 1e-15) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 2e+233) {
		tmp = 1.0 - t_0;
	} else {
		tmp = (1.0 / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if (1.0 / n) <= -0.002:
		tmp = (t_0 / n) / x
	elif (1.0 / n) <= 1e-15:
		tmp = math.log(((1.0 + x) / x)) / n
	elif (1.0 / n) <= 2e+233:
		tmp = 1.0 - t_0
	else:
		tmp = (1.0 / x) / n
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -0.002)
		tmp = Float64(Float64(t_0 / n) / x);
	elseif (Float64(1.0 / n) <= 1e-15)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	elseif (Float64(1.0 / n) <= 2e+233)
		tmp = Float64(1.0 - t_0);
	else
		tmp = Float64(Float64(1.0 / x) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	tmp = 0.0;
	if ((1.0 / n) <= -0.002)
		tmp = (t_0 / n) / x;
	elseif ((1.0 / n) <= 1e-15)
		tmp = log(((1.0 + x) / x)) / n;
	elseif ((1.0 / n) <= 2e+233)
		tmp = 1.0 - t_0;
	else
		tmp = (1.0 / x) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(1.0 / n), $MachinePrecision], -0.002], N[(N[(t$95$0 / n), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-15], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e+233], N[(1.0 - t$95$0), $MachinePrecision], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -2e-3

   1. Initial program 98.9%

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

   3. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}\right), \color{blue}{\left(n \cdot x\right)}\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{\mathsf{neg}\left(\frac{\log \left(\frac{1}{x}\right)}{n}\right)}\right), \left(n \cdot x\right)\right) \]

      3. log-rec N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{\mathsf{neg}\left(\frac{\mathsf{neg}\left(\log x\right)}{n}\right)}\right), \left(n \cdot x\right)\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{\mathsf{neg}\left(\frac{-1 \cdot \log x}{n}\right)}\right), \left(n \cdot x\right)\right) \]

      5. exp-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{e^{\frac{-1 \cdot \log x}{n}}}\right), \left(\color{blue}{n} \cdot x\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(e^{\frac{-1 \cdot \log x}{n}}\right)\right), \left(\color{blue}{n} \cdot x\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(e^{\frac{\log x \cdot -1}{n}}\right)\right), \left(n \cdot x\right)\right) \]

      8. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(e^{\log x \cdot \frac{-1}{n}}\right)\right), \left(n \cdot x\right)\right) \]

      9. exp-to-pow N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left({x}^{\left(\frac{-1}{n}\right)}\right)\right), \left(n \cdot x\right)\right) \]

      10. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{pow.f64}\left(x, \left(\frac{-1}{n}\right)\right)\right), \left(n \cdot x\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(-1, n\right)\right)\right), \left(n \cdot x\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(-1, n\right)\right)\right), \left(x \cdot \color{blue}{n}\right)\right) \]

      13. *-lowering-*.f64 98.9%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(-1, n\right)\right)\right), \mathsf{*.f64}\left(x, \color{blue}{n}\right)\right) \]

   5. Simplified 98.9%

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

      1. *-commutative N/A

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

      2. associate-/r* N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{1}{{x}^{\left(\frac{-1}{n}\right)}}}{n}\right), \color{blue}{x}\right) \]

      4. pow-flip N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{{x}^{\left(\mathsf{neg}\left(\frac{-1}{n}\right)\right)}}{n}\right), x\right) \]

      5. distribute-neg-frac N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{{x}^{\left(\frac{\mathsf{neg}\left(-1\right)}{n}\right)}}{n}\right), x\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{{x}^{\left(\frac{1}{n}\right)}}{n}\right), x\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left({x}^{\left(\frac{1}{n}\right)}\right), n\right), x\right) \]

      8. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(x, \left(\frac{1}{n}\right)\right), n\right), x\right) \]

      9. /-lowering-/.f64 98.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right), n\right), x\right) \]

   7. Applied egg-rr 98.9%

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

   if -2e-3 < (/.f64 #s(literal 1 binary64) n) < 1.0000000000000001e-15

   1. Initial program 27.5%

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

   3. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\frac{\left(\log \left(1 + x\right) + \frac{1}{2} \cdot \frac{{\log \left(1 + x\right)}^{2}}{n}\right) - \left(\log x + \frac{1}{2} \cdot \frac{{\log x}^{2}}{n}\right)}{n}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\log \left(1 + x\right) + \frac{1}{2} \cdot \frac{{\log \left(1 + x\right)}^{2}}{n}\right) - \left(\log x + \frac{1}{2} \cdot \frac{{\log x}^{2}}{n}\right)\right), \color{blue}{n}\right) \]

   5. Simplified 77.4%

      \[\leadsto \color{blue}{\frac{\frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right)}{n} + \left(\mathsf{log1p}\left(x\right) - \log x\right)}{n}} \]
   6. Step-by-step derivation

      1. clear-num N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{n}{\frac{\frac{1}{2} \cdot \left({\log \left(1 + x\right)}^{2} - {\log x}^{2}\right)}{n} + \left(\log \left(1 + x\right) - \log x\right)}\right)}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(n, \color{blue}{\left(\frac{\frac{1}{2} \cdot \left({\log \left(1 + x\right)}^{2} - {\log x}^{2}\right)}{n} + \left(\log \left(1 + x\right) - \log x\right)\right)}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(n, \mathsf{+.f64}\left(\left(\frac{\frac{1}{2} \cdot \left({\log \left(1 + x\right)}^{2} - {\log x}^{2}\right)}{n}\right), \color{blue}{\left(\log \left(1 + x\right) - \log x\right)}\right)\right)\right) \]

   7. Applied egg-rr 60.0%

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

   8. Taylor expanded in n around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\log \left(\frac{1 + x}{x}\right), \color{blue}{n}\right) \]

      2. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\left(\frac{1 + x}{x}\right)\right), n\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(\left(1 + x\right), x\right)\right), n\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(\left(x + 1\right), x\right)\right), n\right) \]

      5. +-lowering-+.f64 77.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, 1\right), x\right)\right), n\right) \]

   10. Simplified 77.3%

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

   if 1.0000000000000001e-15 < (/.f64 #s(literal 1 binary64) n) < 1.99999999999999995e233

   1. Initial program 80.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{1}, \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 72.3%

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

   if 1.99999999999999995e233 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 9.4%

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

   3. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}\right), \color{blue}{\left(n \cdot x\right)}\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{\mathsf{neg}\left(\frac{\log \left(\frac{1}{x}\right)}{n}\right)}\right), \left(n \cdot x\right)\right) \]

      3. log-rec N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{\mathsf{neg}\left(\frac{\mathsf{neg}\left(\log x\right)}{n}\right)}\right), \left(n \cdot x\right)\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{\mathsf{neg}\left(\frac{-1 \cdot \log x}{n}\right)}\right), \left(n \cdot x\right)\right) \]

      5. exp-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{e^{\frac{-1 \cdot \log x}{n}}}\right), \left(\color{blue}{n} \cdot x\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(e^{\frac{-1 \cdot \log x}{n}}\right)\right), \left(\color{blue}{n} \cdot x\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(e^{\frac{\log x \cdot -1}{n}}\right)\right), \left(n \cdot x\right)\right) \]

      8. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(e^{\log x \cdot \frac{-1}{n}}\right)\right), \left(n \cdot x\right)\right) \]

      9. exp-to-pow N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left({x}^{\left(\frac{-1}{n}\right)}\right)\right), \left(n \cdot x\right)\right) \]

      10. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{pow.f64}\left(x, \left(\frac{-1}{n}\right)\right)\right), \left(n \cdot x\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(-1, n\right)\right)\right), \left(n \cdot x\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(-1, n\right)\right)\right), \left(x \cdot \color{blue}{n}\right)\right) \]

      13. *-lowering-*.f64 0.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(-1, n\right)\right)\right), \mathsf{*.f64}\left(x, \color{blue}{n}\right)\right) \]

   5. Simplified 0.0%

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

      1. associate-/r* N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{1}{{x}^{\left(\frac{-1}{n}\right)}}}{x}\right), \color{blue}{n}\right) \]

      3. pow-flip N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{{x}^{\left(\mathsf{neg}\left(\frac{-1}{n}\right)\right)}}{x}\right), n\right) \]

      4. distribute-neg-frac N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{{x}^{\left(\frac{\mathsf{neg}\left(-1\right)}{n}\right)}}{x}\right), n\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{{x}^{\left(\frac{1}{n}\right)}}{x}\right), n\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left({x}^{\left(\frac{1}{n}\right)}\right), x\right), n\right) \]

      7. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(x, \left(\frac{1}{n}\right)\right), x\right), n\right) \]

      8. /-lowering-/.f64 1.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right), x\right), n\right) \]

   7. Applied egg-rr 1.8%

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

   8. Taylor expanded in n around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{1}{x}\right)}, n\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 86.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), n\right) \]

   10. Simplified 86.2%

       \[\leadsto \frac{\color{blue}{\frac{1}{x}}}{n} \]
3. Recombined 4 regimes into one program.

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -0.002:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \mathbf{elif}\;\frac{1}{n} \leq 10^{-15}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+233}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 66.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq 10^{-15}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+233}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) 1e-15)
   (/ (log (/ (+ 1.0 x) x)) n)
   (if (<= (/ 1.0 n) 2e+233) (- 1.0 (pow x (/ 1.0 n))) (/ (/ 1.0 x) n))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= 1e-15) {
		tmp = log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 2e+233) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else {
		tmp = (1.0 / x) / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((1.0d0 / n) <= 1d-15) then
        tmp = log(((1.0d0 + x) / x)) / n
    else if ((1.0d0 / n) <= 2d+233) then
        tmp = 1.0d0 - (x ** (1.0d0 / n))
    else
        tmp = (1.0d0 / x) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= 1e-15) {
		tmp = Math.log(((1.0 + x) / x)) / n;
	} else if ((1.0 / n) <= 2e+233) {
		tmp = 1.0 - Math.pow(x, (1.0 / n));
	} else {
		tmp = (1.0 / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= 1e-15:
		tmp = math.log(((1.0 + x) / x)) / n
	elif (1.0 / n) <= 2e+233:
		tmp = 1.0 - math.pow(x, (1.0 / n))
	else:
		tmp = (1.0 / x) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= 1e-15)
		tmp = Float64(log(Float64(Float64(1.0 + x) / x)) / n);
	elseif (Float64(1.0 / n) <= 2e+233)
		tmp = Float64(1.0 - (x ^ Float64(1.0 / n)));
	else
		tmp = Float64(Float64(1.0 / x) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((1.0 / n) <= 1e-15)
		tmp = log(((1.0 + x) / x)) / n;
	elseif ((1.0 / n) <= 2e+233)
		tmp = 1.0 - (x ^ (1.0 / n));
	else
		tmp = (1.0 / x) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], 1e-15], N[(N[Log[N[(N[(1.0 + x), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e+233], N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < 1.0000000000000001e-15

   1. Initial program 56.2%

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

   3. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\frac{\left(\log \left(1 + x\right) + \frac{1}{2} \cdot \frac{{\log \left(1 + x\right)}^{2}}{n}\right) - \left(\log x + \frac{1}{2} \cdot \frac{{\log x}^{2}}{n}\right)}{n}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\log \left(1 + x\right) + \frac{1}{2} \cdot \frac{{\log \left(1 + x\right)}^{2}}{n}\right) - \left(\log x + \frac{1}{2} \cdot \frac{{\log x}^{2}}{n}\right)\right), \color{blue}{n}\right) \]

   5. Simplified 78.4%

      \[\leadsto \color{blue}{\frac{\frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right)}{n} + \left(\mathsf{log1p}\left(x\right) - \log x\right)}{n}} \]
   6. Step-by-step derivation

      1. clear-num N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{n}{\frac{\frac{1}{2} \cdot \left({\log \left(1 + x\right)}^{2} - {\log x}^{2}\right)}{n} + \left(\log \left(1 + x\right) - \log x\right)}\right)}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(n, \color{blue}{\left(\frac{\frac{1}{2} \cdot \left({\log \left(1 + x\right)}^{2} - {\log x}^{2}\right)}{n} + \left(\log \left(1 + x\right) - \log x\right)\right)}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(n, \mathsf{+.f64}\left(\left(\frac{\frac{1}{2} \cdot \left({\log \left(1 + x\right)}^{2} - {\log x}^{2}\right)}{n}\right), \color{blue}{\left(\log \left(1 + x\right) - \log x\right)}\right)\right)\right) \]

   7. Applied egg-rr 56.0%

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

   8. Taylor expanded in n around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\log \left(\frac{1 + x}{x}\right), \color{blue}{n}\right) \]

      2. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\left(\frac{1 + x}{x}\right)\right), n\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(\left(1 + x\right), x\right)\right), n\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(\left(x + 1\right), x\right)\right), n\right) \]

      5. +-lowering-+.f64 70.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, 1\right), x\right)\right), n\right) \]

   10. Simplified 70.7%

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

   if 1.0000000000000001e-15 < (/.f64 #s(literal 1 binary64) n) < 1.99999999999999995e233

   1. Initial program 80.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{1}, \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 72.3%

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

   if 1.99999999999999995e233 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 9.4%

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

   3. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}\right), \color{blue}{\left(n \cdot x\right)}\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{\mathsf{neg}\left(\frac{\log \left(\frac{1}{x}\right)}{n}\right)}\right), \left(n \cdot x\right)\right) \]

      3. log-rec N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{\mathsf{neg}\left(\frac{\mathsf{neg}\left(\log x\right)}{n}\right)}\right), \left(n \cdot x\right)\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{\mathsf{neg}\left(\frac{-1 \cdot \log x}{n}\right)}\right), \left(n \cdot x\right)\right) \]

      5. exp-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{e^{\frac{-1 \cdot \log x}{n}}}\right), \left(\color{blue}{n} \cdot x\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(e^{\frac{-1 \cdot \log x}{n}}\right)\right), \left(\color{blue}{n} \cdot x\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(e^{\frac{\log x \cdot -1}{n}}\right)\right), \left(n \cdot x\right)\right) \]

      8. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(e^{\log x \cdot \frac{-1}{n}}\right)\right), \left(n \cdot x\right)\right) \]

      9. exp-to-pow N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left({x}^{\left(\frac{-1}{n}\right)}\right)\right), \left(n \cdot x\right)\right) \]

      10. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{pow.f64}\left(x, \left(\frac{-1}{n}\right)\right)\right), \left(n \cdot x\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(-1, n\right)\right)\right), \left(n \cdot x\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(-1, n\right)\right)\right), \left(x \cdot \color{blue}{n}\right)\right) \]

      13. *-lowering-*.f64 0.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(-1, n\right)\right)\right), \mathsf{*.f64}\left(x, \color{blue}{n}\right)\right) \]

   5. Simplified 0.0%

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

      1. associate-/r* N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{1}{{x}^{\left(\frac{-1}{n}\right)}}}{x}\right), \color{blue}{n}\right) \]

      3. pow-flip N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{{x}^{\left(\mathsf{neg}\left(\frac{-1}{n}\right)\right)}}{x}\right), n\right) \]

      4. distribute-neg-frac N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{{x}^{\left(\frac{\mathsf{neg}\left(-1\right)}{n}\right)}}{x}\right), n\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{{x}^{\left(\frac{1}{n}\right)}}{x}\right), n\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left({x}^{\left(\frac{1}{n}\right)}\right), x\right), n\right) \]

      7. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(x, \left(\frac{1}{n}\right)\right), x\right), n\right) \]

      8. /-lowering-/.f64 1.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right), x\right), n\right) \]

   7. Applied egg-rr 1.8%

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

   8. Taylor expanded in n around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{1}{x}\right)}, n\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 86.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), n\right) \]

   10. Simplified 86.2%

       \[\leadsto \frac{\color{blue}{\frac{1}{x}}}{n} \]
3. Recombined 3 regimes into one program.

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq 10^{-15}:\\ \;\;\;\;\frac{\log \left(\frac{1 + x}{x}\right)}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+233}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 57.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n) :precision binary64 (if (<= x 1.0) (/ (- x (log x)) n) 0.0))

C

double code(double x, double n) {
	double tmp;
	if (x <= 1.0) {
		tmp = (x - log(x)) / n;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 1.0d0) then
        tmp = (x - log(x)) / n
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 1.0) {
		tmp = (x - Math.log(x)) / n;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 1.0:
		tmp = (x - math.log(x)) / n
	else:
		tmp = 0.0
	return tmp

Julia

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

MATLAB

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

Wolfram

code[x_, n_] := If[LessEqual[x, 1.0], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 44.0%

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

   3. Taylor expanded in n around inf

      \[\leadsto \color{blue}{\frac{\left(\log \left(1 + x\right) + \frac{1}{2} \cdot \frac{{\log \left(1 + x\right)}^{2}}{n}\right) - \left(\log x + \frac{1}{2} \cdot \frac{{\log x}^{2}}{n}\right)}{n}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\log \left(1 + x\right) + \frac{1}{2} \cdot \frac{{\log \left(1 + x\right)}^{2}}{n}\right) - \left(\log x + \frac{1}{2} \cdot \frac{{\log x}^{2}}{n}\right)\right), \color{blue}{n}\right) \]

   5. Simplified 65.8%

      \[\leadsto \color{blue}{\frac{\frac{0.5 \cdot \left({\left(\mathsf{log1p}\left(x\right)\right)}^{2} - {\log x}^{2}\right)}{n} + \left(\mathsf{log1p}\left(x\right) - \log x\right)}{n}} \]
   6. Step-by-step derivation

      1. clear-num N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{n}{\frac{\frac{1}{2} \cdot \left({\log \left(1 + x\right)}^{2} - {\log x}^{2}\right)}{n} + \left(\log \left(1 + x\right) - \log x\right)}\right)}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(n, \color{blue}{\left(\frac{\frac{1}{2} \cdot \left({\log \left(1 + x\right)}^{2} - {\log x}^{2}\right)}{n} + \left(\log \left(1 + x\right) - \log x\right)\right)}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(n, \mathsf{+.f64}\left(\left(\frac{\frac{1}{2} \cdot \left({\log \left(1 + x\right)}^{2} - {\log x}^{2}\right)}{n}\right), \color{blue}{\left(\log \left(1 + x\right) - \log x\right)}\right)\right)\right) \]

   7. Applied egg-rr 65.8%

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

   8. Taylor expanded in n around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\log \left(\frac{1 + x}{x}\right), \color{blue}{n}\right) \]

      2. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\left(\frac{1 + x}{x}\right)\right), n\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(\left(1 + x\right), x\right)\right), n\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(\left(x + 1\right), x\right)\right), n\right) \]

      5. +-lowering-+.f64 55.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(x, 1\right), x\right)\right), n\right) \]

   10. Simplified 55.0%

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

   11. Taylor expanded in x around 0

       \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(x + -1 \cdot \log x\right)}, n\right) \]
   12. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\left(x + \left(\mathsf{neg}\left(\log x\right)\right)\right), n\right) \]

       2. unsub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\left(x - \log x\right), n\right) \]

       3. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \log x\right), n\right) \]

       4. log-lowering-log.f64 54.8%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{log.f64}\left(x\right)\right), n\right) \]

   13. Simplified 54.8%

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

   if 1 < x

   1. Initial program 72.5%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{1}, \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 28.5%

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

   6. Taylor expanded in n around inf

      \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{1}\right) \]
   7. Step-by-step derivation

   8. Simplified 72.5%

      \[\leadsto 1 - \color{blue}{1} \]
   9. Step-by-step derivation

      1. metadata-eval 72.5%

         \[\leadsto 0 \]

   10. Applied egg-rr 72.5%

       \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 57.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;0 - \frac{\log x}{n}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n) :precision binary64 (if (<= x 1.0) (- 0.0 (/ (log x) n)) 0.0))

C

double code(double x, double n) {
	double tmp;
	if (x <= 1.0) {
		tmp = 0.0 - (log(x) / n);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if (x <= 1.0d0) then
        tmp = 0.0d0 - (log(x) / n)
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 1.0) {
		tmp = 0.0 - (Math.log(x) / n);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 1.0:
		tmp = 0.0 - (math.log(x) / n)
	else:
		tmp = 0.0
	return tmp

Julia

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

MATLAB

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

Wolfram

code[x_, n_] := If[LessEqual[x, 1.0], N[(0.0 - N[(N[Log[x], $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 44.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{1}, \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 42.6%

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

   6. Taylor expanded in n around inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{\log x}{n}\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{\frac{\log x}{n}} \]

      3. remove-double-neg N/A

         \[\leadsto 0 - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{\log x}{n}\right)\right)\right)\right) \]

      4. distribute-frac-neg N/A

         \[\leadsto 0 - \left(\mathsf{neg}\left(\frac{\mathsf{neg}\left(\log x\right)}{n}\right)\right) \]

      5. log-rec N/A

         \[\leadsto 0 - \left(\mathsf{neg}\left(\frac{\log \left(\frac{1}{x}\right)}{n}\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto 0 - -1 \cdot \color{blue}{\frac{\log \left(\frac{1}{x}\right)}{n}} \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}\right)}\right) \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \left(\mathsf{neg}\left(\frac{\log \left(\frac{1}{x}\right)}{n}\right)\right)\right) \]

      9. log-rec N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \left(\mathsf{neg}\left(\frac{\mathsf{neg}\left(\log x\right)}{n}\right)\right)\right) \]

      10. distribute-frac-neg N/A

          \[\leadsto \mathsf{\_.f64}\left(0, \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{\log x}{n}\right)\right)\right)\right)\right) \]

      11. remove-double-neg N/A

          \[\leadsto \mathsf{\_.f64}\left(0, \left(\frac{\log x}{\color{blue}{n}}\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\log x, \color{blue}{n}\right)\right) \]

      13. log-lowering-log.f64 54.1%

          \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{log.f64}\left(x\right), n\right)\right) \]

   8. Simplified 54.1%

      \[\leadsto \color{blue}{0 - \frac{\log x}{n}} \]

   if 1 < x

   1. Initial program 72.5%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{1}, \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 28.5%

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

   6. Taylor expanded in n around inf

      \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{1}\right) \]
   7. Step-by-step derivation

   8. Simplified 72.5%

      \[\leadsto 1 - \color{blue}{1} \]
   9. Step-by-step derivation

      1. metadata-eval 72.5%

         \[\leadsto 0 \]

   10. Applied egg-rr 72.5%

       \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 46.9% accurate, 14.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{n \cdot x}\\ \mathbf{if}\;n \leq -7:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq -1.4 \cdot 10^{-299}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ 1.0 (* n x))))
   (if (<= n -7.0) t_0 (if (<= n -1.4e-299) 0.0 t_0))))

C

double code(double x, double n) {
	double t_0 = 1.0 / (n * x);
	double tmp;
	if (n <= -7.0) {
		tmp = t_0;
	} else if (n <= -1.4e-299) {
		tmp = 0.0;
	} else {
		tmp = 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) :: tmp
    t_0 = 1.0d0 / (n * x)
    if (n <= (-7.0d0)) then
        tmp = t_0
    else if (n <= (-1.4d-299)) then
        tmp = 0.0d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = 1.0 / (n * x);
	double tmp;
	if (n <= -7.0) {
		tmp = t_0;
	} else if (n <= -1.4e-299) {
		tmp = 0.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = 1.0 / (n * x)
	tmp = 0
	if n <= -7.0:
		tmp = t_0
	elif n <= -1.4e-299:
		tmp = 0.0
	else:
		tmp = t_0
	return tmp

Julia

function code(x, n)
	t_0 = Float64(1.0 / Float64(n * x))
	tmp = 0.0
	if (n <= -7.0)
		tmp = t_0;
	elseif (n <= -1.4e-299)
		tmp = 0.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = 1.0 / (n * x);
	tmp = 0.0;
	if (n <= -7.0)
		tmp = t_0;
	elseif (n <= -1.4e-299)
		tmp = 0.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -7.0], t$95$0, If[LessEqual[n, -1.4e-299], 0.0, t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -7 or -1.4000000000000001e-299 < n

   1. Initial program 35.2%

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

   3. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}\right), \color{blue}{\left(n \cdot x\right)}\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{\mathsf{neg}\left(\frac{\log \left(\frac{1}{x}\right)}{n}\right)}\right), \left(n \cdot x\right)\right) \]

      3. log-rec N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{\mathsf{neg}\left(\frac{\mathsf{neg}\left(\log x\right)}{n}\right)}\right), \left(n \cdot x\right)\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{\mathsf{neg}\left(\frac{-1 \cdot \log x}{n}\right)}\right), \left(n \cdot x\right)\right) \]

      5. exp-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{e^{\frac{-1 \cdot \log x}{n}}}\right), \left(\color{blue}{n} \cdot x\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(e^{\frac{-1 \cdot \log x}{n}}\right)\right), \left(\color{blue}{n} \cdot x\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(e^{\frac{\log x \cdot -1}{n}}\right)\right), \left(n \cdot x\right)\right) \]

      8. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(e^{\log x \cdot \frac{-1}{n}}\right)\right), \left(n \cdot x\right)\right) \]

      9. exp-to-pow N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left({x}^{\left(\frac{-1}{n}\right)}\right)\right), \left(n \cdot x\right)\right) \]

      10. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{pow.f64}\left(x, \left(\frac{-1}{n}\right)\right)\right), \left(n \cdot x\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(-1, n\right)\right)\right), \left(n \cdot x\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(-1, n\right)\right)\right), \left(x \cdot \color{blue}{n}\right)\right) \]

      13. *-lowering-*.f64 40.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(-1, n\right)\right)\right), \mathsf{*.f64}\left(x, \color{blue}{n}\right)\right) \]

   5. Simplified 40.1%

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

   6. Taylor expanded in n around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(n \cdot x\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(x \cdot \color{blue}{n}\right)\right) \]

      3. *-lowering-*.f64 42.9%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{n}\right)\right) \]

   8. Simplified 42.9%

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

   if -7 < n < -1.4000000000000001e-299

   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 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{1}, \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 39.9%

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

   6. Taylor expanded in n around inf

      \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{1}\right) \]
   7. Step-by-step derivation

   8. Simplified 62.7%

      \[\leadsto 1 - \color{blue}{1} \]
   9. Step-by-step derivation

      1. metadata-eval 62.7%

         \[\leadsto 0 \]

   10. Applied egg-rr 62.7%

       \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 49.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -7:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{elif}\;n \leq -1.4 \cdot 10^{-299}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 47.3% accurate, 17.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -0.2:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n) :precision binary64 (if (<= (/ 1.0 n) -0.2) 0.0 (/ (/ 1.0 x) n)))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -0.2) {
		tmp = 0.0;
	} else {
		tmp = (1.0 / x) / n;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((1.0d0 / n) <= (-0.2d0)) then
        tmp = 0.0d0
    else
        tmp = (1.0d0 / x) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -0.2) {
		tmp = 0.0;
	} else {
		tmp = (1.0 / x) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -0.2:
		tmp = 0.0
	else:
		tmp = (1.0 / x) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -0.2)
		tmp = 0.0;
	else
		tmp = Float64(Float64(1.0 / x) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((1.0 / n) <= -0.2)
		tmp = 0.0;
	else
		tmp = (1.0 / x) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -0.2], 0.0, N[(N[(1.0 / x), $MachinePrecision] / n), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -0.20000000000000001

   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 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{1}, \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 40.6%

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

   6. Taylor expanded in n around inf

      \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{1}\right) \]
   7. Step-by-step derivation

   8. Simplified 62.0%

      \[\leadsto 1 - \color{blue}{1} \]
   9. Step-by-step derivation

      1. metadata-eval 62.0%

         \[\leadsto 0 \]

   10. Applied egg-rr 62.0%

       \[\leadsto \color{blue}{0} \]

   if -0.20000000000000001 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 34.8%

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

   3. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}\right), \color{blue}{\left(n \cdot x\right)}\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{\mathsf{neg}\left(\frac{\log \left(\frac{1}{x}\right)}{n}\right)}\right), \left(n \cdot x\right)\right) \]

      3. log-rec N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{\mathsf{neg}\left(\frac{\mathsf{neg}\left(\log x\right)}{n}\right)}\right), \left(n \cdot x\right)\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{\mathsf{neg}\left(\frac{-1 \cdot \log x}{n}\right)}\right), \left(n \cdot x\right)\right) \]

      5. exp-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{e^{\frac{-1 \cdot \log x}{n}}}\right), \left(\color{blue}{n} \cdot x\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(e^{\frac{-1 \cdot \log x}{n}}\right)\right), \left(\color{blue}{n} \cdot x\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(e^{\frac{\log x \cdot -1}{n}}\right)\right), \left(n \cdot x\right)\right) \]

      8. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(e^{\log x \cdot \frac{-1}{n}}\right)\right), \left(n \cdot x\right)\right) \]

      9. exp-to-pow N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left({x}^{\left(\frac{-1}{n}\right)}\right)\right), \left(n \cdot x\right)\right) \]

      10. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{pow.f64}\left(x, \left(\frac{-1}{n}\right)\right)\right), \left(n \cdot x\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(-1, n\right)\right)\right), \left(n \cdot x\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(-1, n\right)\right)\right), \left(x \cdot \color{blue}{n}\right)\right) \]

      13. *-lowering-*.f64 39.8%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(-1, n\right)\right)\right), \mathsf{*.f64}\left(x, \color{blue}{n}\right)\right) \]

   5. Simplified 39.8%

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

      1. associate-/r* N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{1}{{x}^{\left(\frac{-1}{n}\right)}}}{x}\right), \color{blue}{n}\right) \]

      3. pow-flip N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{{x}^{\left(\mathsf{neg}\left(\frac{-1}{n}\right)\right)}}{x}\right), n\right) \]

      4. distribute-neg-frac N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{{x}^{\left(\frac{\mathsf{neg}\left(-1\right)}{n}\right)}}{x}\right), n\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{{x}^{\left(\frac{1}{n}\right)}}{x}\right), n\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left({x}^{\left(\frac{1}{n}\right)}\right), x\right), n\right) \]

      7. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(x, \left(\frac{1}{n}\right)\right), x\right), n\right) \]

      8. /-lowering-/.f64 41.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right), x\right), n\right) \]

   7. Applied egg-rr 41.2%

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

   8. Taylor expanded in n around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{1}{x}\right)}, n\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 43.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), n\right) \]

   10. Simplified 43.8%

       \[\leadsto \frac{\color{blue}{\frac{1}{x}}}{n} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 47.4% accurate, 17.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -0.2:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n) :precision binary64 (if (<= (/ 1.0 n) -0.2) 0.0 (/ (/ 1.0 n) x)))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -0.2) {
		tmp = 0.0;
	} else {
		tmp = (1.0 / n) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((1.0d0 / n) <= (-0.2d0)) then
        tmp = 0.0d0
    else
        tmp = (1.0d0 / n) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -0.2) {
		tmp = 0.0;
	} else {
		tmp = (1.0 / n) / x;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -0.2:
		tmp = 0.0
	else:
		tmp = (1.0 / n) / x
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -0.2)
		tmp = 0.0;
	else
		tmp = Float64(Float64(1.0 / n) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if ((1.0 / n) <= -0.2)
		tmp = 0.0;
	else
		tmp = (1.0 / n) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -0.2], 0.0, N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -0.20000000000000001

   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 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{1}, \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 40.6%

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

   6. Taylor expanded in n around inf

      \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{1}\right) \]
   7. Step-by-step derivation

   8. Simplified 62.0%

      \[\leadsto 1 - \color{blue}{1} \]
   9. Step-by-step derivation

      1. metadata-eval 62.0%

         \[\leadsto 0 \]

   10. Applied egg-rr 62.0%

       \[\leadsto \color{blue}{0} \]

   if -0.20000000000000001 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 34.8%

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

   3. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{-1 \cdot \frac{\log \left(\frac{1}{x}\right)}{n}}\right), \color{blue}{\left(n \cdot x\right)}\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{\mathsf{neg}\left(\frac{\log \left(\frac{1}{x}\right)}{n}\right)}\right), \left(n \cdot x\right)\right) \]

      3. log-rec N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{\mathsf{neg}\left(\frac{\mathsf{neg}\left(\log x\right)}{n}\right)}\right), \left(n \cdot x\right)\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(e^{\mathsf{neg}\left(\frac{-1 \cdot \log x}{n}\right)}\right), \left(n \cdot x\right)\right) \]

      5. exp-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{e^{\frac{-1 \cdot \log x}{n}}}\right), \left(\color{blue}{n} \cdot x\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(e^{\frac{-1 \cdot \log x}{n}}\right)\right), \left(\color{blue}{n} \cdot x\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(e^{\frac{\log x \cdot -1}{n}}\right)\right), \left(n \cdot x\right)\right) \]

      8. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(e^{\log x \cdot \frac{-1}{n}}\right)\right), \left(n \cdot x\right)\right) \]

      9. exp-to-pow N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left({x}^{\left(\frac{-1}{n}\right)}\right)\right), \left(n \cdot x\right)\right) \]

      10. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{pow.f64}\left(x, \left(\frac{-1}{n}\right)\right)\right), \left(n \cdot x\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(-1, n\right)\right)\right), \left(n \cdot x\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(-1, n\right)\right)\right), \left(x \cdot \color{blue}{n}\right)\right) \]

      13. *-lowering-*.f64 39.8%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(-1, n\right)\right)\right), \mathsf{*.f64}\left(x, \color{blue}{n}\right)\right) \]

   5. Simplified 39.8%

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

      1. *-commutative N/A

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

      2. associate-/r* N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{1}{{x}^{\left(\frac{-1}{n}\right)}}}{n}\right), \color{blue}{x}\right) \]

      4. pow-flip N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{{x}^{\left(\mathsf{neg}\left(\frac{-1}{n}\right)\right)}}{n}\right), x\right) \]

      5. distribute-neg-frac N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{{x}^{\left(\frac{\mathsf{neg}\left(-1\right)}{n}\right)}}{n}\right), x\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{{x}^{\left(\frac{1}{n}\right)}}{n}\right), x\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left({x}^{\left(\frac{1}{n}\right)}\right), n\right), x\right) \]

      8. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(x, \left(\frac{1}{n}\right)\right), n\right), x\right) \]

      9. /-lowering-/.f64 41.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right), n\right), x\right) \]

   7. Applied egg-rr 41.1%

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

   8. Taylor expanded in n around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\color{blue}{1}, n\right), x\right) \]
   9. Step-by-step derivation

   10. Simplified 43.7%

       \[\leadsto \frac{\frac{\color{blue}{1}}{n}}{x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 31.2% accurate, 211.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x n) :precision binary64 0.0)

C

double code(double x, double n) {
	return 0.0;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    code = 0.0d0
end function

Java

public static double code(double x, double n) {
	return 0.0;
}

Python

def code(x, n):
	return 0.0

Julia

function code(x, n)
	return 0.0
end

MATLAB

function tmp = code(x, n)
	tmp = 0.0;
end

Wolfram

code[x_, n_] := 0.0

Derivation

1. Initial program 57.2%

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

3. Taylor expanded in x around 0

   \[\leadsto \mathsf{\_.f64}\left(\color{blue}{1}, \mathsf{pow.f64}\left(x, \mathsf{/.f64}\left(1, n\right)\right)\right) \]
4. Step-by-step derivation

5. Simplified 36.0%

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

6. Taylor expanded in n around inf

   \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{1}\right) \]
7. Step-by-step derivation

8. Simplified 35.7%

   \[\leadsto 1 - \color{blue}{1} \]
9. Step-by-step derivation

   1. metadata-eval 35.7%

      \[\leadsto 0 \]

10. Applied egg-rr 35.7%

    \[\leadsto \color{blue}{0} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024288 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))