2nthrt (problem 3.4.6)

Percentage Accurate: 58.4% → 87.9%

Time: 34.7s

Alternatives: 9

Speedup: 1.9×

• could not determine a ground truth

  1. x = -4.972735463973333e-295
  2. n = 1.912165338721757e+30

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: 58.4% 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: 87.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-77}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n} + -1\right)}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 0.0005:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{x}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -4e-77)
   (/ (pow x (+ (/ 1.0 n) -1.0)) n)
   (if (<= (/ 1.0 n) 0.0005)
     (/ (- (log1p x) (log x)) n)
     (- (exp (/ x n)) (pow x (/ 1.0 n))))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -4e-77) {
		tmp = pow(x, ((1.0 / n) + -1.0)) / n;
	} else if ((1.0 / n) <= 0.0005) {
		tmp = (log1p(x) - log(x)) / n;
	} else {
		tmp = exp((x / n)) - pow(x, (1.0 / n));
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -4e-77) {
		tmp = Math.pow(x, ((1.0 / n) + -1.0)) / n;
	} else if ((1.0 / n) <= 0.0005) {
		tmp = (Math.log1p(x) - Math.log(x)) / n;
	} else {
		tmp = Math.exp((x / n)) - Math.pow(x, (1.0 / n));
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -4e-77:
		tmp = math.pow(x, ((1.0 / n) + -1.0)) / n
	elif (1.0 / n) <= 0.0005:
		tmp = (math.log1p(x) - math.log(x)) / n
	else:
		tmp = math.exp((x / n)) - math.pow(x, (1.0 / n))
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -4e-77)
		tmp = Float64((x ^ Float64(Float64(1.0 / n) + -1.0)) / n);
	elseif (Float64(1.0 / n) <= 0.0005)
		tmp = Float64(Float64(log1p(x) - log(x)) / n);
	else
		tmp = Float64(exp(Float64(x / n)) - (x ^ Float64(1.0 / n)));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -4e-77], N[(N[Power[x, N[(N[(1.0 / n), $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 0.0005], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], N[(N[Exp[N[(x / n), $MachinePrecision]], $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -3.9999999999999997e-77

   1. Initial program 87.6%

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

   3. Taylor expanded in x around inf 66.3%

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

      1. log-rec 66.3%

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

      2. mul-1-neg 66.3%

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

      3. neg-mul-1 66.3%

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

      4. mul-1-neg 66.3%

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

      5. distribute-frac-neg 66.3%

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

      6. remove-double-neg 66.3%

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

      7. *-rgt-identity 66.3%

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

      8. associate-/l* 66.3%

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

      9. exp-to-pow 93.0%

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

      10. *-commutative 93.0%

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

   5. Simplified 93.0%

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

      1. *-un-lft-identity 93.0%

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

      2. associate-/r* 93.5%

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

      3. pow1 93.5%

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

      4. pow-div 93.5%

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

   7. Applied egg-rr 93.5%

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

      1. *-lft-identity 93.5%

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

      2. sub-neg 93.5%

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

      3. metadata-eval 93.5%

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

   9. Simplified 93.5%

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

   if -3.9999999999999997e-77 < (/.f64 #s(literal 1 binary64) n) < 5.0000000000000001e-4

   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 82.8%

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

      1. log1p-define 82.8%

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

   5. Simplified 82.8%

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

   if 5.0000000000000001e-4 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 46.8%

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

   3. Taylor expanded in n around 0 19.5%

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

      1. log1p-define 47.5%

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

      2. *-rgt-identity 47.5%

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

      3. associate-*l/ 47.5%

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

      4. associate-/l* 47.5%

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

      5. exp-to-pow 93.4%

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

   5. Simplified 93.4%

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

   6. Taylor expanded in x around 0 93.4%

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

Alternative 2: 78.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{n} \leq -4 \cdot 10^{-77}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n} + -1\right)}}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 0.0005:\\ \;\;\;\;\frac{\mathsf{log1p}\left(x\right) - \log x}{n}\\ \mathbf{elif}\;\frac{1}{n} \leq 2 \cdot 10^{+228}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(\frac{x}{n}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= (/ 1.0 n) -4e-77)
   (/ (pow x (+ (/ 1.0 n) -1.0)) n)
   (if (<= (/ 1.0 n) 0.0005)
     (/ (- (log1p x) (log x)) n)
     (if (<= (/ 1.0 n) 2e+228)
       (- (+ 1.0 (/ x n)) (pow x (/ 1.0 n)))
       (expm1 (/ x n))))))

C

double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -4e-77) {
		tmp = pow(x, ((1.0 / n) + -1.0)) / n;
	} else if ((1.0 / n) <= 0.0005) {
		tmp = (log1p(x) - log(x)) / n;
	} else if ((1.0 / n) <= 2e+228) {
		tmp = (1.0 + (x / n)) - pow(x, (1.0 / n));
	} else {
		tmp = expm1((x / n));
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double tmp;
	if ((1.0 / n) <= -4e-77) {
		tmp = Math.pow(x, ((1.0 / n) + -1.0)) / n;
	} else if ((1.0 / n) <= 0.0005) {
		tmp = (Math.log1p(x) - Math.log(x)) / n;
	} else if ((1.0 / n) <= 2e+228) {
		tmp = (1.0 + (x / n)) - Math.pow(x, (1.0 / n));
	} else {
		tmp = Math.expm1((x / n));
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if (1.0 / n) <= -4e-77:
		tmp = math.pow(x, ((1.0 / n) + -1.0)) / n
	elif (1.0 / n) <= 0.0005:
		tmp = (math.log1p(x) - math.log(x)) / n
	elif (1.0 / n) <= 2e+228:
		tmp = (1.0 + (x / n)) - math.pow(x, (1.0 / n))
	else:
		tmp = math.expm1((x / n))
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (Float64(1.0 / n) <= -4e-77)
		tmp = Float64((x ^ Float64(Float64(1.0 / n) + -1.0)) / n);
	elseif (Float64(1.0 / n) <= 0.0005)
		tmp = Float64(Float64(log1p(x) - log(x)) / n);
	elseif (Float64(1.0 / n) <= 2e+228)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - (x ^ Float64(1.0 / n)));
	else
		tmp = expm1(Float64(x / n));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[N[(1.0 / n), $MachinePrecision], -4e-77], N[(N[Power[x, N[(N[(1.0 / n), $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 0.0005], N[(N[(N[Log[1 + x], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], If[LessEqual[N[(1.0 / n), $MachinePrecision], 2e+228], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(Exp[N[(x / n), $MachinePrecision]] - 1), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 #s(literal 1 binary64) n) < -3.9999999999999997e-77

   1. Initial program 87.6%

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

   3. Taylor expanded in x around inf 66.3%

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

      1. log-rec 66.3%

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

      2. mul-1-neg 66.3%

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

      3. neg-mul-1 66.3%

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

      4. mul-1-neg 66.3%

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

      5. distribute-frac-neg 66.3%

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

      6. remove-double-neg 66.3%

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

      7. *-rgt-identity 66.3%

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

      8. associate-/l* 66.3%

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

      9. exp-to-pow 93.0%

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

      10. *-commutative 93.0%

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

   5. Simplified 93.0%

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

      1. *-un-lft-identity 93.0%

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

      2. associate-/r* 93.5%

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

      3. pow1 93.5%

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

      4. pow-div 93.5%

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

   7. Applied egg-rr 93.5%

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

      1. *-lft-identity 93.5%

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

      2. sub-neg 93.5%

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

      3. metadata-eval 93.5%

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

   9. Simplified 93.5%

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

   if -3.9999999999999997e-77 < (/.f64 #s(literal 1 binary64) n) < 5.0000000000000001e-4

   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 82.8%

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

      1. log1p-define 82.8%

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

   5. Simplified 82.8%

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

   if 5.0000000000000001e-4 < (/.f64 #s(literal 1 binary64) n) < 1.9999999999999998e228

   1. Initial program 64.5%

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

   3. Taylor expanded in x around 0 61.9%

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

   if 1.9999999999999998e228 < (/.f64 #s(literal 1 binary64) n)

   1. Initial program 7.7%

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

      1. log1p-expm1-u 7.7%

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

   4. Applied egg-rr 7.7%

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

   5. Taylor expanded in n around inf 32.7%

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

   6. Taylor expanded in n around 0 32.7%

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

      1. expm1-define 32.7%

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

      2. log1p-define 64.2%

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

   8. Simplified 64.2%

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

   9. Taylor expanded in x around 0 64.2%

      \[\leadsto \mathsf{expm1}\left(\frac{\color{blue}{x}}{n}\right) \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 3: 62.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;x \leq -1 \cdot 10^{-159}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{-280}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 3 \cdot 10^{-44}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (- 1.0 (pow x (/ 1.0 n)))))
   (if (<= x -1e-159)
     0.0
     (if (<= x 6.2e-280)
       t_0
       (if (<= x 3e-44) (/ (log x) (- n)) (if (<= x 1.0) t_0 0.0))))))

C

double code(double x, double n) {
	double t_0 = 1.0 - pow(x, (1.0 / n));
	double tmp;
	if (x <= -1e-159) {
		tmp = 0.0;
	} else if (x <= 6.2e-280) {
		tmp = t_0;
	} else if (x <= 3e-44) {
		tmp = log(x) / -n;
	} else if (x <= 1.0) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 - (x ** (1.0d0 / n))
    if (x <= (-1d-159)) then
        tmp = 0.0d0
    else if (x <= 6.2d-280) then
        tmp = t_0
    else if (x <= 3d-44) then
        tmp = log(x) / -n
    else if (x <= 1.0d0) then
        tmp = t_0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double t_0 = 1.0 - Math.pow(x, (1.0 / n));
	double tmp;
	if (x <= -1e-159) {
		tmp = 0.0;
	} else if (x <= 6.2e-280) {
		tmp = t_0;
	} else if (x <= 3e-44) {
		tmp = Math.log(x) / -n;
	} else if (x <= 1.0) {
		tmp = t_0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = 1.0 - math.pow(x, (1.0 / n))
	tmp = 0
	if x <= -1e-159:
		tmp = 0.0
	elif x <= 6.2e-280:
		tmp = t_0
	elif x <= 3e-44:
		tmp = math.log(x) / -n
	elif x <= 1.0:
		tmp = t_0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, n)
	t_0 = Float64(1.0 - (x ^ Float64(1.0 / n)))
	tmp = 0.0
	if (x <= -1e-159)
		tmp = 0.0;
	elseif (x <= 6.2e-280)
		tmp = t_0;
	elseif (x <= 3e-44)
		tmp = Float64(log(x) / Float64(-n));
	elseif (x <= 1.0)
		tmp = t_0;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = 1.0 - (x ^ (1.0 / n));
	tmp = 0.0;
	if (x <= -1e-159)
		tmp = 0.0;
	elseif (x <= 6.2e-280)
		tmp = t_0;
	elseif (x <= 3e-44)
		tmp = log(x) / -n;
	elseif (x <= 1.0)
		tmp = t_0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(1.0 - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1e-159], 0.0, If[LessEqual[x, 6.2e-280], t$95$0, If[LessEqual[x, 3e-44], N[(N[Log[x], $MachinePrecision] / (-n)), $MachinePrecision], If[LessEqual[x, 1.0], t$95$0, 0.0]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -9.99999999999999989e-160 or 1 < x

   1. Initial program 67.8%

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

   3. Taylor expanded in x around inf 42.1%

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

      1. log-rec 42.1%

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

      2. mul-1-neg 42.1%

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

      3. neg-mul-1 42.1%

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

      4. mul-1-neg 42.1%

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

      5. distribute-frac-neg 42.1%

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

      6. remove-double-neg 42.1%

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

      7. *-rgt-identity 42.1%

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

      8. associate-/l* 42.1%

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

      9. exp-to-pow 73.5%

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

   5. Simplified 73.5%

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

   6. Taylor expanded in x around 0 77.0%

      \[\leadsto \color{blue}{0} \]

   if -9.99999999999999989e-160 < x < 6.20000000000000042e-280 or 3.0000000000000002e-44 < x < 1

   1. Initial program 70.6%

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

   3. Taylor expanded in x around 0 40.5%

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

      1. *-rgt-identity 40.5%

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

      2. associate-*l/ 40.5%

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

      3. associate-/l* 40.5%

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

      4. exp-to-pow 68.7%

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

   5. Simplified 68.7%

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

   if 6.20000000000000042e-280 < x < 3.0000000000000002e-44

   1. Initial program 33.8%

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

   3. Taylor expanded in x around 0 33.8%

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

      1. *-rgt-identity 33.8%

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

      2. associate-*l/ 33.8%

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

      3. associate-/l* 33.8%

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

      4. exp-to-pow 33.8%

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

   5. Simplified 33.8%

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

   6. Taylor expanded in n around inf 54.8%

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

      1. associate-*r/ 54.8%

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

      2. mul-1-neg 54.8%

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

   8. Simplified 54.8%

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

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-159}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{-280}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 3 \cdot 10^{-44}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;x \leq -1.52 \cdot 10^{-167}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 8.6 \cdot 10^{-280}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t\_0\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{-41}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t\_0}{n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= x -1.52e-167)
     0.0
     (if (<= x 8.6e-280)
       (- (+ 1.0 (/ x n)) t_0)
       (if (<= x 4.2e-41) (/ (log x) (- n)) (/ (/ t_0 n) x))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if (x <= -1.52e-167) {
		tmp = 0.0;
	} else if (x <= 8.6e-280) {
		tmp = (1.0 + (x / n)) - t_0;
	} else if (x <= 4.2e-41) {
		tmp = log(x) / -n;
	} else {
		tmp = (t_0 / n) / x;
	}
	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 (x <= (-1.52d-167)) then
        tmp = 0.0d0
    else if (x <= 8.6d-280) then
        tmp = (1.0d0 + (x / n)) - t_0
    else if (x <= 4.2d-41) then
        tmp = log(x) / -n
    else
        tmp = (t_0 / n) / x
    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 (x <= -1.52e-167) {
		tmp = 0.0;
	} else if (x <= 8.6e-280) {
		tmp = (1.0 + (x / n)) - t_0;
	} else if (x <= 4.2e-41) {
		tmp = Math.log(x) / -n;
	} else {
		tmp = (t_0 / n) / x;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if x <= -1.52e-167:
		tmp = 0.0
	elif x <= 8.6e-280:
		tmp = (1.0 + (x / n)) - t_0
	elif x <= 4.2e-41:
		tmp = math.log(x) / -n
	else:
		tmp = (t_0 / n) / x
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (x <= -1.52e-167)
		tmp = 0.0;
	elseif (x <= 8.6e-280)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_0);
	elseif (x <= 4.2e-41)
		tmp = Float64(log(x) / Float64(-n));
	else
		tmp = Float64(Float64(t_0 / n) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	tmp = 0.0;
	if (x <= -1.52e-167)
		tmp = 0.0;
	elseif (x <= 8.6e-280)
		tmp = (1.0 + (x / n)) - t_0;
	elseif (x <= 4.2e-41)
		tmp = log(x) / -n;
	else
		tmp = (t_0 / n) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -1.52e-167], 0.0, If[LessEqual[x, 8.6e-280], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[x, 4.2e-41], N[(N[Log[x], $MachinePrecision] / (-n)), $MachinePrecision], N[(N[(t$95$0 / n), $MachinePrecision] / x), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.52e-167

   1. Initial program 65.8%

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

   3. Taylor expanded in x around inf 0.0%

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

      1. log-rec 0.0%

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

      2. mul-1-neg 0.0%

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

      3. neg-mul-1 0.0%

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

      4. mul-1-neg 0.0%

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

      5. distribute-frac-neg 0.0%

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

      6. remove-double-neg 0.0%

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

      7. *-rgt-identity 0.0%

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

      8. associate-/l* 0.0%

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

      9. exp-to-pow 80.5%

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

   5. Simplified 80.5%

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

   6. Taylor expanded in x around 0 89.5%

      \[\leadsto \color{blue}{0} \]

   if -1.52e-167 < x < 8.5999999999999997e-280

   1. Initial program 71.7%

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

   3. Taylor expanded in x around 0 73.6%

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

   if 8.5999999999999997e-280 < x < 4.20000000000000025e-41

   1. Initial program 34.5%

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

   3. Taylor expanded in x around 0 34.5%

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

      1. *-rgt-identity 34.5%

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

      2. associate-*l/ 34.5%

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

      3. associate-/l* 34.5%

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

      4. exp-to-pow 34.5%

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

   5. Simplified 34.5%

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

   6. Taylor expanded in n around inf 54.3%

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

      1. associate-*r/ 54.3%

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

      2. mul-1-neg 54.3%

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

   8. Simplified 54.3%

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

   if 4.20000000000000025e-41 < x

   1. Initial program 68.5%

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

   3. Taylor expanded in x around inf 91.5%

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

      1. log-rec 91.5%

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

      2. mul-1-neg 91.5%

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

      3. neg-mul-1 91.5%

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

      4. mul-1-neg 91.5%

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

      5. distribute-frac-neg 91.5%

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

      6. remove-double-neg 91.5%

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

      7. *-rgt-identity 91.5%

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

      8. associate-/l* 91.5%

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

      9. exp-to-pow 91.5%

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

      10. *-commutative 91.5%

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

   5. Simplified 91.5%

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

      1. sqr-pow 91.5%

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

      2. times-frac 92.4%

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

      3. sqrt-pow1 92.4%

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

      4. sqrt-pow1 92.4%

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

   7. Applied egg-rr 92.4%

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

      1. associate-*l/ 92.5%

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

      2. associate-*r/ 92.5%

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

      3. add-sqr-sqrt 92.5%

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

   9. Applied egg-rr 92.5%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.52 \cdot 10^{-167}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 8.6 \cdot 10^{-280}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{-41}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;x \leq -8 \cdot 10^{-171}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 4.1 \cdot 10^{-280}:\\ \;\;\;\;1 - t\_0\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{-40}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t\_0}{n}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= x -8e-171)
     0.0
     (if (<= x 4.1e-280)
       (- 1.0 t_0)
       (if (<= x 3.3e-40) (/ (log x) (- n)) (/ (/ t_0 n) x))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if (x <= -8e-171) {
		tmp = 0.0;
	} else if (x <= 4.1e-280) {
		tmp = 1.0 - t_0;
	} else if (x <= 3.3e-40) {
		tmp = log(x) / -n;
	} else {
		tmp = (t_0 / n) / x;
	}
	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 (x <= (-8d-171)) then
        tmp = 0.0d0
    else if (x <= 4.1d-280) then
        tmp = 1.0d0 - t_0
    else if (x <= 3.3d-40) then
        tmp = log(x) / -n
    else
        tmp = (t_0 / n) / x
    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 (x <= -8e-171) {
		tmp = 0.0;
	} else if (x <= 4.1e-280) {
		tmp = 1.0 - t_0;
	} else if (x <= 3.3e-40) {
		tmp = Math.log(x) / -n;
	} else {
		tmp = (t_0 / n) / x;
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if x <= -8e-171:
		tmp = 0.0
	elif x <= 4.1e-280:
		tmp = 1.0 - t_0
	elif x <= 3.3e-40:
		tmp = math.log(x) / -n
	else:
		tmp = (t_0 / n) / x
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (x <= -8e-171)
		tmp = 0.0;
	elseif (x <= 4.1e-280)
		tmp = Float64(1.0 - t_0);
	elseif (x <= 3.3e-40)
		tmp = Float64(log(x) / Float64(-n));
	else
		tmp = Float64(Float64(t_0 / n) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	tmp = 0.0;
	if (x <= -8e-171)
		tmp = 0.0;
	elseif (x <= 4.1e-280)
		tmp = 1.0 - t_0;
	elseif (x <= 3.3e-40)
		tmp = log(x) / -n;
	else
		tmp = (t_0 / n) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -8e-171], 0.0, If[LessEqual[x, 4.1e-280], N[(1.0 - t$95$0), $MachinePrecision], If[LessEqual[x, 3.3e-40], N[(N[Log[x], $MachinePrecision] / (-n)), $MachinePrecision], N[(N[(t$95$0 / n), $MachinePrecision] / x), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -7.9999999999999999e-171

   1. Initial program 65.8%

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

   3. Taylor expanded in x around inf 0.0%

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

      1. log-rec 0.0%

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

      2. mul-1-neg 0.0%

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

      3. neg-mul-1 0.0%

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

      4. mul-1-neg 0.0%

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

      5. distribute-frac-neg 0.0%

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

      6. remove-double-neg 0.0%

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

      7. *-rgt-identity 0.0%

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

      8. associate-/l* 0.0%

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

      9. exp-to-pow 80.5%

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

   5. Simplified 80.5%

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

   6. Taylor expanded in x around 0 89.5%

      \[\leadsto \color{blue}{0} \]

   if -7.9999999999999999e-171 < x < 4.1000000000000002e-280

   1. Initial program 71.7%

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

   3. Taylor expanded in x around 0 31.7%

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

      1. *-rgt-identity 31.7%

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

      2. associate-*l/ 31.7%

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

      3. associate-/l* 31.7%

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

      4. exp-to-pow 71.7%

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

   5. Simplified 71.7%

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

   if 4.1000000000000002e-280 < x < 3.29999999999999993e-40

   1. Initial program 34.5%

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

   3. Taylor expanded in x around 0 34.5%

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

      1. *-rgt-identity 34.5%

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

      2. associate-*l/ 34.5%

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

      3. associate-/l* 34.5%

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

      4. exp-to-pow 34.5%

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

   5. Simplified 34.5%

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

   6. Taylor expanded in n around inf 54.3%

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

      1. associate-*r/ 54.3%

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

      2. mul-1-neg 54.3%

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

   8. Simplified 54.3%

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

   if 3.29999999999999993e-40 < x

   1. Initial program 68.5%

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

   3. Taylor expanded in x around inf 91.5%

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

      1. log-rec 91.5%

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

      2. mul-1-neg 91.5%

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

      3. neg-mul-1 91.5%

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

      4. mul-1-neg 91.5%

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

      5. distribute-frac-neg 91.5%

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

      6. remove-double-neg 91.5%

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

      7. *-rgt-identity 91.5%

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

      8. associate-/l* 91.5%

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

      9. exp-to-pow 91.5%

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

      10. *-commutative 91.5%

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

   5. Simplified 91.5%

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

      1. sqr-pow 91.5%

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

      2. times-frac 92.4%

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

      3. sqrt-pow1 92.4%

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

      4. sqrt-pow1 92.4%

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

   7. Applied egg-rr 92.4%

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

      1. associate-*l/ 92.5%

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

      2. associate-*r/ 92.5%

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

      3. add-sqr-sqrt 92.5%

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

   9. Applied egg-rr 92.5%

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

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8 \cdot 10^{-171}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 4.1 \cdot 10^{-280}:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{-40}:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{x}^{\left(\frac{1}{n}\right)}}{n}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 61.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-310}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 54:\\ \;\;\;\;\frac{x - \log x}{n}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x -2e-310) 0.0 (if (<= x 54.0) (/ (- x (log x)) n) 0.0)))

C

double code(double x, double n) {
	double tmp;
	if (x <= -2e-310) {
		tmp = 0.0;
	} else if (x <= 54.0) {
		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 <= (-2d-310)) then
        tmp = 0.0d0
    else if (x <= 54.0d0) 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 <= -2e-310) {
		tmp = 0.0;
	} else if (x <= 54.0) {
		tmp = (x - Math.log(x)) / n;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= -2e-310:
		tmp = 0.0
	elif x <= 54.0:
		tmp = (x - math.log(x)) / n
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= -2e-310)
		tmp = 0.0;
	elseif (x <= 54.0)
		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 <= -2e-310)
		tmp = 0.0;
	elseif (x <= 54.0)
		tmp = (x - log(x)) / n;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, -2e-310], 0.0, If[LessEqual[x, 54.0], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision], 0.0]]

Derivation

1. Split input into 2 regimes

2. if x < -1.999999999999994e-310 or 54 < x

   1. Initial program 69.8%

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

   3. Taylor expanded in x around inf 36.8%

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

      1. log-rec 36.8%

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

      2. mul-1-neg 36.8%

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

      3. neg-mul-1 36.8%

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

      4. mul-1-neg 36.8%

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

      5. distribute-frac-neg 36.8%

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

      6. remove-double-neg 36.8%

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

      7. *-rgt-identity 36.8%

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

      8. associate-/l* 36.7%

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

      9. exp-to-pow 66.8%

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

   5. Simplified 66.8%

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

   6. Taylor expanded in x around 0 69.8%

      \[\leadsto \color{blue}{0} \]

   if -1.999999999999994e-310 < x < 54

   1. Initial program 42.2%

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

   3. Taylor expanded in x around 0 41.8%

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

   4. Taylor expanded in n around inf 48.8%

      \[\leadsto \color{blue}{\frac{x - \log x}{n}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 61.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-310}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x -2e-310) 0.0 (if (<= x 1.0) (/ (log x) (- n)) 0.0)))

C

double code(double x, double n) {
	double tmp;
	if (x <= -2e-310) {
		tmp = 0.0;
	} else if (x <= 1.0) {
		tmp = 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 <= (-2d-310)) then
        tmp = 0.0d0
    else if (x <= 1.0d0) then
        tmp = 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 <= -2e-310) {
		tmp = 0.0;
	} else if (x <= 1.0) {
		tmp = Math.log(x) / -n;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= -2e-310:
		tmp = 0.0
	elif x <= 1.0:
		tmp = math.log(x) / -n
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= -2e-310)
		tmp = 0.0;
	elseif (x <= 1.0)
		tmp = Float64(log(x) / Float64(-n));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= -2e-310)
		tmp = 0.0;
	elseif (x <= 1.0)
		tmp = log(x) / -n;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, -2e-310], 0.0, If[LessEqual[x, 1.0], N[(N[Log[x], $MachinePrecision] / (-n)), $MachinePrecision], 0.0]]

Derivation

1. Split input into 2 regimes

2. if x < -1.999999999999994e-310 or 1 < x

   1. Initial program 69.4%

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

   3. Taylor expanded in x around inf 36.5%

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

      1. log-rec 36.5%

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

      2. mul-1-neg 36.5%

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

      3. neg-mul-1 36.5%

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

      4. mul-1-neg 36.5%

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

      5. distribute-frac-neg 36.5%

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

      6. remove-double-neg 36.5%

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

      7. *-rgt-identity 36.5%

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

      8. associate-/l* 36.5%

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

      9. exp-to-pow 66.3%

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

   5. Simplified 66.3%

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

   6. Taylor expanded in x around 0 69.4%

      \[\leadsto \color{blue}{0} \]

   if -1.999999999999994e-310 < x < 1

   1. Initial program 42.5%

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

   3. Taylor expanded in x around 0 41.7%

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

      1. *-rgt-identity 41.7%

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

      2. associate-*l/ 41.7%

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

      3. associate-/l* 41.7%

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

      4. exp-to-pow 41.7%

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

   5. Simplified 41.7%

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

   6. Taylor expanded in n around inf 49.1%

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

      1. associate-*r/ 49.1%

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

      2. mul-1-neg 49.1%

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

   8. Simplified 49.1%

      \[\leadsto \color{blue}{\frac{-\log x}{n}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-310}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\frac{\log x}{-n}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 49.7% accurate, 14.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.5 \cdot 10^{-304}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{+128}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 1.5e-304) 0.0 (if (<= x 1.8e+128) (/ 1.0 (* n x)) 0.0)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 1.5e-304) {
		tmp = 0.0;
	} else if (x <= 1.8e+128) {
		tmp = 1.0 / (n * x);
	} 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 <= 1.5d-304) then
        tmp = 0.0d0
    else if (x <= 1.8d+128) then
        tmp = 1.0d0 / (n * x)
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 1.5e-304) {
		tmp = 0.0;
	} else if (x <= 1.8e+128) {
		tmp = 1.0 / (n * x);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 1.5e-304:
		tmp = 0.0
	elif x <= 1.8e+128:
		tmp = 1.0 / (n * x)
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 1.5e-304)
		tmp = 0.0;
	elseif (x <= 1.8e+128)
		tmp = Float64(1.0 / Float64(n * x));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 1.5e-304)
		tmp = 0.0;
	elseif (x <= 1.8e+128)
		tmp = 1.0 / (n * x);
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 1.5e-304], 0.0, If[LessEqual[x, 1.8e+128], N[(1.0 / N[(n * x), $MachinePrecision]), $MachinePrecision], 0.0]]

Derivation

1. Split input into 2 regimes

2. if x < 1.5000000000000001e-304 or 1.80000000000000014e128 < x

   1. Initial program 72.1%

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

   3. Taylor expanded in x around inf 33.3%

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

      1. log-rec 33.3%

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

      2. mul-1-neg 33.3%

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

      3. neg-mul-1 33.3%

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

      4. mul-1-neg 33.3%

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

      5. distribute-frac-neg 33.3%

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

      6. remove-double-neg 33.3%

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

      7. *-rgt-identity 33.3%

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

      8. associate-/l* 33.2%

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

      9. exp-to-pow 68.5%

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

   5. Simplified 68.5%

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

   6. Taylor expanded in x around 0 72.1%

      \[\leadsto \color{blue}{0} \]

   if 1.5000000000000001e-304 < x < 1.80000000000000014e128

   1. Initial program 44.3%

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

   3. Taylor expanded in x around inf 43.3%

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

      1. log-rec 43.3%

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

      2. mul-1-neg 43.3%

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

      3. neg-mul-1 43.3%

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

      4. mul-1-neg 43.3%

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

      5. distribute-frac-neg 43.3%

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

      6. remove-double-neg 43.3%

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

      7. *-rgt-identity 43.3%

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

      8. associate-/l* 43.3%

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

      9. exp-to-pow 43.3%

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

      10. *-commutative 43.3%

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

   5. Simplified 43.3%

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

   6. Taylor expanded in n around inf 33.5%

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

      1. *-commutative 33.5%

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

   8. Simplified 33.5%

      \[\leadsto \color{blue}{\frac{1}{x \cdot n}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 50.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.5 \cdot 10^{-304}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{+128}:\\ \;\;\;\;\frac{1}{n \cdot x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 40.2% 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 56.8%

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

3. Taylor expanded in x around inf 20.9%

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

   1. log-rec 20.9%

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

   2. mul-1-neg 20.9%

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

   3. neg-mul-1 20.9%

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

   4. mul-1-neg 20.9%

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

   5. distribute-frac-neg 20.9%

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

   6. remove-double-neg 20.9%

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

   7. *-rgt-identity 20.9%

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

   8. associate-/l* 20.8%

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

   9. exp-to-pow 36.7%

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

5. Simplified 36.7%

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

6. Taylor expanded in x around 0 38.5%

   \[\leadsto \color{blue}{0} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024181 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))