2nthrt (problem 3.4.6)

Percentage Accurate: 52.0% → 98.8%

Time: 1.3min

Alternatives: 11

Speedup: 2.0×

• could not determine a ground truth

  1. x = 2.3227803098881875e+90
  2. n = -3.245943116104921e+95

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: 52.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.056:\\ \;\;\;\;\sqrt[3]{{\left(e^{3}\right)}^{\left(\frac{\mathsf{log1p}\left(x\right)}{n}\right)}} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{2 \cdot \log \left({x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)}}{x \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.056)
   (- (cbrt (pow (exp 3.0) (/ (log1p x) n))) (pow x (/ 1.0 n)))
   (/ (exp (* 2.0 (log (pow x (/ (/ 1.0 n) 2.0))))) (* x n))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.056) {
		tmp = cbrt(pow(exp(3.0), (log1p(x) / n))) - pow(x, (1.0 / n));
	} else {
		tmp = exp((2.0 * log(pow(x, ((1.0 / n) / 2.0))))) / (x * n);
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 0.056) {
		tmp = Math.cbrt(Math.pow(Math.exp(3.0), (Math.log1p(x) / n))) - Math.pow(x, (1.0 / n));
	} else {
		tmp = Math.exp((2.0 * Math.log(Math.pow(x, ((1.0 / n) / 2.0))))) / (x * n);
	}
	return tmp;
}

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 0.056)
		tmp = Float64(cbrt((exp(3.0) ^ Float64(log1p(x) / n))) - (x ^ Float64(1.0 / n)));
	else
		tmp = Float64(exp(Float64(2.0 * log((x ^ Float64(Float64(1.0 / n) / 2.0))))) / Float64(x * n));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.056], N[(N[Power[N[Power[N[Exp[3.0], $MachinePrecision], N[(N[Log[1 + x], $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Exp[N[(2.0 * N[Log[N[Power[x, N[(N[(1.0 / n), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(x * n), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0560000000000000012

   1. Initial program 78.6%

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

      1. add-cbrt-cube 78.6%

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

      2. pow3 78.6%

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

      3. pow-to-exp 78.6%

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

      4. pow-exp 78.6%

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

      5. un-div-inv 78.6%

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

      6. +-commutative 78.6%

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

      7. log1p-udef 98.9%

         \[\leadsto \sqrt[3]{e^{\frac{\color{blue}{\mathsf{log1p}\left(x\right)}}{n} \cdot 3}} - {x}^{\left(\frac{1}{n}\right)} \]

   3. Applied egg-rr 98.9%

      \[\leadsto \color{blue}{\sqrt[3]{e^{\frac{\mathsf{log1p}\left(x\right)}{n} \cdot 3}}} - {x}^{\left(\frac{1}{n}\right)} \]

   4. Taylor expanded in n around 0 98.9%

      \[\leadsto \sqrt[3]{\color{blue}{e^{3 \cdot \frac{\mathsf{log1p}\left(x\right)}{n}}}} - {x}^{\left(\frac{1}{n}\right)} \]
   5. Step-by-step derivation

      1. exp-prod 98.9%

         \[\leadsto \sqrt[3]{\color{blue}{{\left(e^{3}\right)}^{\left(\frac{\mathsf{log1p}\left(x\right)}{n}\right)}}} - {x}^{\left(\frac{1}{n}\right)} \]

   6. Simplified 98.9%

      \[\leadsto \sqrt[3]{\color{blue}{{\left(e^{3}\right)}^{\left(\frac{\mathsf{log1p}\left(x\right)}{n}\right)}}} - {x}^{\left(\frac{1}{n}\right)} \]

   if 0.0560000000000000012 < x

   1. Initial program 6.7%

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

   2. Taylor expanded in x around inf 99.8%

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

      1. log-rec 99.8%

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

      2. mul-1-neg 99.8%

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

      3. associate-*r/ 99.8%

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

      4. neg-mul-1 99.8%

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

      5. mul-1-neg 99.8%

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

      6. remove-double-neg 99.8%

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

      7. *-commutative 99.8%

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

   4. Simplified 99.8%

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

      1. add-log-exp 99.8%

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

      2. div-inv 99.8%

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

      3. pow-to-exp 99.8%

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

      4. add-sqr-sqrt 99.8%

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

      5. log-prod 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. count-2 99.8%

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

   8. Simplified 99.8%

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

      1. sqrt-pow1 99.9%

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

   10. Applied egg-rr 99.9%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.056:\\ \;\;\;\;\sqrt[3]{{\left(e^{3}\right)}^{\left(\frac{\mathsf{log1p}\left(x\right)}{n}\right)}} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{2 \cdot \log \left({x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)}}{x \cdot n}\\ \end{array} \]

Alternative 2: 98.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.056:\\ \;\;\;\;\sqrt[3]{e^{3 \cdot \frac{x}{n}}} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{2 \cdot \log \left({x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)}}{x \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 0.056)
   (- (cbrt (exp (* 3.0 (/ x n)))) (pow x (/ 1.0 n)))
   (/ (exp (* 2.0 (log (pow x (/ (/ 1.0 n) 2.0))))) (* x n))))

C

double code(double x, double n) {
	double tmp;
	if (x <= 0.056) {
		tmp = cbrt(exp((3.0 * (x / n)))) - pow(x, (1.0 / n));
	} else {
		tmp = exp((2.0 * log(pow(x, ((1.0 / n) / 2.0))))) / (x * n);
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 0.056) {
		tmp = Math.cbrt(Math.exp((3.0 * (x / n)))) - Math.pow(x, (1.0 / n));
	} else {
		tmp = Math.exp((2.0 * Math.log(Math.pow(x, ((1.0 / n) / 2.0))))) / (x * n);
	}
	return tmp;
}

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 0.056)
		tmp = Float64(cbrt(exp(Float64(3.0 * Float64(x / n)))) - (x ^ Float64(1.0 / n)));
	else
		tmp = Float64(exp(Float64(2.0 * log((x ^ Float64(Float64(1.0 / n) / 2.0))))) / Float64(x * n));
	end
	return tmp
end

Wolfram

code[x_, n_] := If[LessEqual[x, 0.056], N[(N[Power[N[Exp[N[(3.0 * N[(x / n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision] - N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Exp[N[(2.0 * N[Log[N[Power[x, N[(N[(1.0 / n), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(x * n), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0560000000000000012

   1. Initial program 78.6%

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

      1. add-cbrt-cube 78.6%

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

      2. pow3 78.6%

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

      3. pow-to-exp 78.6%

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

      4. pow-exp 78.6%

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

      5. un-div-inv 78.6%

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

      6. +-commutative 78.6%

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

      7. log1p-udef 98.9%

         \[\leadsto \sqrt[3]{e^{\frac{\color{blue}{\mathsf{log1p}\left(x\right)}}{n} \cdot 3}} - {x}^{\left(\frac{1}{n}\right)} \]

   3. Applied egg-rr 98.9%

      \[\leadsto \color{blue}{\sqrt[3]{e^{\frac{\mathsf{log1p}\left(x\right)}{n} \cdot 3}}} - {x}^{\left(\frac{1}{n}\right)} \]

   4. Taylor expanded in x around 0 98.9%

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

   if 0.0560000000000000012 < x

   1. Initial program 6.7%

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

   2. Taylor expanded in x around inf 99.8%

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

      1. log-rec 99.8%

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

      2. mul-1-neg 99.8%

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

      3. associate-*r/ 99.8%

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

      4. neg-mul-1 99.8%

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

      5. mul-1-neg 99.8%

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

      6. remove-double-neg 99.8%

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

      7. *-commutative 99.8%

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

   4. Simplified 99.8%

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

      1. add-log-exp 99.8%

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

      2. div-inv 99.8%

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

      3. pow-to-exp 99.8%

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

      4. add-sqr-sqrt 99.8%

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

      5. log-prod 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. count-2 99.8%

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

   8. Simplified 99.8%

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

      1. sqrt-pow1 99.9%

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

   10. Applied egg-rr 99.9%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.056:\\ \;\;\;\;\sqrt[3]{e^{3 \cdot \frac{x}{n}}} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{2 \cdot \log \left({x}^{\left(\frac{\frac{1}{n}}{2}\right)}\right)}}{x \cdot n}\\ \end{array} \]

Alternative 3: 98.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;x \leq 0.056:\\ \;\;\;\;\sqrt[3]{e^{3 \cdot \frac{x}{n}}} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{x \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= x 0.056) (- (cbrt (exp (* 3.0 (/ x n)))) t_0) (/ t_0 (* x n)))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if (x <= 0.056) {
		tmp = cbrt(exp((3.0 * (x / n)))) - t_0;
	} else {
		tmp = t_0 / (x * n);
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if (x <= 0.056) {
		tmp = Math.cbrt(Math.exp((3.0 * (x / n)))) - t_0;
	} else {
		tmp = t_0 / (x * n);
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (x <= 0.056)
		tmp = Float64(cbrt(exp(Float64(3.0 * Float64(x / n)))) - t_0);
	else
		tmp = Float64(t_0 / Float64(x * n));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 0.056], N[(N[Power[N[Exp[N[(3.0 * N[(x / n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision] - t$95$0), $MachinePrecision], N[(t$95$0 / N[(x * n), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0560000000000000012

   1. Initial program 78.6%

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

      1. add-cbrt-cube 78.6%

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

      2. pow3 78.6%

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

      3. pow-to-exp 78.6%

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

      4. pow-exp 78.6%

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

      5. un-div-inv 78.6%

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

      6. +-commutative 78.6%

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

      7. log1p-udef 98.9%

         \[\leadsto \sqrt[3]{e^{\frac{\color{blue}{\mathsf{log1p}\left(x\right)}}{n} \cdot 3}} - {x}^{\left(\frac{1}{n}\right)} \]

   3. Applied egg-rr 98.9%

      \[\leadsto \color{blue}{\sqrt[3]{e^{\frac{\mathsf{log1p}\left(x\right)}{n} \cdot 3}}} - {x}^{\left(\frac{1}{n}\right)} \]

   4. Taylor expanded in x around 0 98.9%

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

   if 0.0560000000000000012 < x

   1. Initial program 6.7%

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

   2. Taylor expanded in n around 0 6.7%

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

      1. log1p-def 6.7%

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

   4. Simplified 6.7%

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

   5. Taylor expanded in x around inf 99.8%

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

      1. mul-1-neg 99.8%

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

      2. log-rec 99.8%

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

      3. distribute-frac-neg 99.8%

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

      4. remove-double-neg 99.8%

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

      5. *-rgt-identity 99.8%

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

      6. associate-*r/ 99.8%

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

      7. exp-to-pow 99.8%

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

      8. *-commutative 99.8%

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

   7. Simplified 99.8%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.056:\\ \;\;\;\;\sqrt[3]{e^{3 \cdot \frac{x}{n}}} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}\\ \end{array} \]

Alternative 4: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;x \leq 0.056:\\ \;\;\;\;e^{\frac{x}{n}} - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{x \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= x 0.056) (- (exp (/ x n)) t_0) (/ t_0 (* x n)))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if (x <= 0.056) {
		tmp = exp((x / n)) - t_0;
	} else {
		tmp = t_0 / (x * n);
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    if (x <= 0.056d0) then
        tmp = exp((x / n)) - t_0
    else
        tmp = t_0 / (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 (x <= 0.056) {
		tmp = Math.exp((x / n)) - t_0;
	} else {
		tmp = t_0 / (x * n);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if x <= 0.056:
		tmp = math.exp((x / n)) - t_0
	else:
		tmp = t_0 / (x * n)
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (x <= 0.056)
		tmp = Float64(exp(Float64(x / n)) - t_0);
	else
		tmp = Float64(t_0 / Float64(x * n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	tmp = 0.0;
	if (x <= 0.056)
		tmp = exp((x / n)) - t_0;
	else
		tmp = t_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[x, 0.056], N[(N[Exp[N[(x / n), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision], N[(t$95$0 / N[(x * n), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0560000000000000012

   1. Initial program 78.6%

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

   2. Taylor expanded in n around 0 78.6%

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

      1. log1p-def 98.9%

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

   4. Simplified 98.9%

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

   5. Taylor expanded in x around 0 98.9%

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

   if 0.0560000000000000012 < x

   1. Initial program 6.7%

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

   2. Taylor expanded in n around 0 6.7%

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

      1. log1p-def 6.7%

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

   4. Simplified 6.7%

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

   5. Taylor expanded in x around inf 99.8%

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

      1. mul-1-neg 99.8%

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

      2. log-rec 99.8%

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

      3. distribute-frac-neg 99.8%

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

      4. remove-double-neg 99.8%

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

      5. *-rgt-identity 99.8%

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

      6. associate-*r/ 99.8%

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

      7. exp-to-pow 99.8%

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

      8. *-commutative 99.8%

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

   7. Simplified 99.8%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.056:\\ \;\;\;\;e^{\frac{x}{n}} - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}\\ \end{array} \]

Alternative 5: 90.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;x \leq 1.5 \cdot 10^{-141}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t_0\\ \mathbf{elif}\;x \leq 0.056:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{x \cdot n}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{x \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= x 1.5e-141)
     (- (+ 1.0 (/ x n)) t_0)
     (if (<= x 0.056) (log1p (expm1 (/ 1.0 (* x n)))) (/ t_0 (* x n))))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if (x <= 1.5e-141) {
		tmp = (1.0 + (x / n)) - t_0;
	} else if (x <= 0.056) {
		tmp = log1p(expm1((1.0 / (x * n))));
	} else {
		tmp = t_0 / (x * n);
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = Math.pow(x, (1.0 / n));
	double tmp;
	if (x <= 1.5e-141) {
		tmp = (1.0 + (x / n)) - t_0;
	} else if (x <= 0.056) {
		tmp = Math.log1p(Math.expm1((1.0 / (x * n))));
	} else {
		tmp = t_0 / (x * n);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if x <= 1.5e-141:
		tmp = (1.0 + (x / n)) - t_0
	elif x <= 0.056:
		tmp = math.log1p(math.expm1((1.0 / (x * n))))
	else:
		tmp = t_0 / (x * n)
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (x <= 1.5e-141)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_0);
	elseif (x <= 0.056)
		tmp = log1p(expm1(Float64(1.0 / Float64(x * n))));
	else
		tmp = Float64(t_0 / Float64(x * n));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 1.5e-141], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[x, 0.056], N[Log[1 + N[(Exp[N[(1.0 / N[(x * n), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision], N[(t$95$0 / N[(x * n), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < 1.49999999999999992e-141

   1. Initial program 91.6%

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

   2. Taylor expanded in x around 0 91.9%

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

   if 1.49999999999999992e-141 < x < 0.0560000000000000012

   1. Initial program 60.5%

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

   2. Taylor expanded in n around 0 60.5%

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

      1. log1p-def 99.9%

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

   4. Simplified 99.9%

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

   5. Taylor expanded in x around inf 47.8%

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

      1. mul-1-neg 47.8%

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

      2. log-rec 47.8%

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

      3. distribute-frac-neg 47.8%

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

      4. remove-double-neg 47.8%

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

      5. *-rgt-identity 47.8%

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

      6. associate-*r/ 47.8%

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

      7. exp-to-pow 47.8%

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

      8. *-commutative 47.8%

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

   7. Simplified 47.8%

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

   8. Taylor expanded in n around inf 35.2%

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

      1. *-commutative 35.2%

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

   10. Simplified 35.2%

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

       1. log1p-expm1-u 91.5%

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

   12. Applied egg-rr 91.5%

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

   if 0.0560000000000000012 < x

   1. Initial program 6.7%

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

   2. Taylor expanded in n around 0 6.7%

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

      1. log1p-def 6.7%

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

   4. Simplified 6.7%

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

   5. Taylor expanded in x around inf 99.8%

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

      1. mul-1-neg 99.8%

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

      2. log-rec 99.8%

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

      3. distribute-frac-neg 99.8%

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

      4. remove-double-neg 99.8%

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

      5. *-rgt-identity 99.8%

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

      6. associate-*r/ 99.8%

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

      7. exp-to-pow 99.8%

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

      8. *-commutative 99.8%

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

   7. Simplified 99.8%

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

4. Final simplification 94.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.5 \cdot 10^{-141}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 0.056:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{x \cdot n}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}\\ \end{array} \]

Alternative 6: 87.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{1}{n}}{x}\\ t_1 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;x \leq 7 \cdot 10^{-142}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t_1\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{-5}:\\ \;\;\;\;\sqrt[3]{t_0 \cdot \frac{t_0}{x \cdot n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{x \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (/ (/ 1.0 n) x)) (t_1 (pow x (/ 1.0 n))))
   (if (<= x 7e-142)
     (- (+ 1.0 (/ x n)) t_1)
     (if (<= x 1.2e-5) (cbrt (* t_0 (/ t_0 (* x n)))) (/ t_1 (* x n))))))

C

double code(double x, double n) {
	double t_0 = (1.0 / n) / x;
	double t_1 = pow(x, (1.0 / n));
	double tmp;
	if (x <= 7e-142) {
		tmp = (1.0 + (x / n)) - t_1;
	} else if (x <= 1.2e-5) {
		tmp = cbrt((t_0 * (t_0 / (x * n))));
	} else {
		tmp = t_1 / (x * n);
	}
	return tmp;
}

Java

public static double code(double x, double n) {
	double t_0 = (1.0 / n) / x;
	double t_1 = Math.pow(x, (1.0 / n));
	double tmp;
	if (x <= 7e-142) {
		tmp = (1.0 + (x / n)) - t_1;
	} else if (x <= 1.2e-5) {
		tmp = Math.cbrt((t_0 * (t_0 / (x * n))));
	} else {
		tmp = t_1 / (x * n);
	}
	return tmp;
}

Julia

function code(x, n)
	t_0 = Float64(Float64(1.0 / n) / x)
	t_1 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (x <= 7e-142)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_1);
	elseif (x <= 1.2e-5)
		tmp = cbrt(Float64(t_0 * Float64(t_0 / Float64(x * n))));
	else
		tmp = Float64(t_1 / Float64(x * n));
	end
	return tmp
end

Wolfram

code[x_, n_] := Block[{t$95$0 = N[(N[(1.0 / n), $MachinePrecision] / x), $MachinePrecision]}, Block[{t$95$1 = N[Power[x, N[(1.0 / n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 7e-142], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], If[LessEqual[x, 1.2e-5], N[Power[N[(t$95$0 * N[(t$95$0 / N[(x * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], N[(t$95$1 / N[(x * n), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < 7.00000000000000029e-142

   1. Initial program 91.6%

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

   2. Taylor expanded in x around 0 91.9%

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

   if 7.00000000000000029e-142 < x < 1.2e-5

   1. Initial program 58.1%

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

   2. Taylor expanded in n around 0 58.1%

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

      1. log1p-def 99.9%

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

   4. Simplified 99.9%

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

   5. Taylor expanded in x around inf 46.1%

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

      1. mul-1-neg 46.1%

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

      2. log-rec 46.1%

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

      3. distribute-frac-neg 46.1%

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

      4. remove-double-neg 46.1%

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

      5. *-rgt-identity 46.1%

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

      6. associate-*r/ 46.1%

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

      7. exp-to-pow 46.1%

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

      8. *-commutative 46.1%

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

   7. Simplified 46.1%

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

   8. Taylor expanded in n around inf 37.0%

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

      1. *-commutative 37.0%

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

   10. Simplified 37.0%

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

       1. add-cbrt-cube 85.0%

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

   12. Applied egg-rr 85.0%

       \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{1}{x \cdot n} \cdot \frac{1}{x \cdot n}\right) \cdot \frac{1}{x \cdot n}}} \]
   13. Step-by-step derivation

       1. associate-*l* 85.0%

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

       2. *-commutative 85.0%

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

       3. associate-/r* 85.0%

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

       4. associate-*l/ 85.0%

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

       5. *-lft-identity 85.0%

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

       6. *-commutative 85.0%

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

       7. associate-/r* 85.0%

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

   14. Simplified 85.0%

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

   if 1.2e-5 < x

   1. Initial program 10.5%

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

   2. Taylor expanded in n around 0 10.5%

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

      1. log1p-def 10.5%

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

   4. Simplified 10.5%

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

   5. Taylor expanded in x around inf 98.8%

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

      1. mul-1-neg 98.8%

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

      2. log-rec 98.8%

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

      3. distribute-frac-neg 98.8%

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

      4. remove-double-neg 98.8%

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

      5. *-rgt-identity 98.8%

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

      6. associate-*r/ 98.8%

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

      7. exp-to-pow 98.8%

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

      8. *-commutative 98.8%

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

   7. Simplified 98.8%

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

4. Final simplification 92.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 7 \cdot 10^{-142}:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{-5}:\\ \;\;\;\;\sqrt[3]{\frac{\frac{1}{n}}{x} \cdot \frac{\frac{\frac{1}{n}}{x}}{x \cdot n}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}\\ \end{array} \]

Alternative 7: 83.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;x \leq 0.056:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{x \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= x 0.056) (- (+ 1.0 (/ x n)) t_0) (/ t_0 (* x n)))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if (x <= 0.056) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = t_0 / (x * n);
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    if (x <= 0.056d0) then
        tmp = (1.0d0 + (x / n)) - t_0
    else
        tmp = t_0 / (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 (x <= 0.056) {
		tmp = (1.0 + (x / n)) - t_0;
	} else {
		tmp = t_0 / (x * n);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if x <= 0.056:
		tmp = (1.0 + (x / n)) - t_0
	else:
		tmp = t_0 / (x * n)
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (x <= 0.056)
		tmp = Float64(Float64(1.0 + Float64(x / n)) - t_0);
	else
		tmp = Float64(t_0 / Float64(x * n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	tmp = 0.0;
	if (x <= 0.056)
		tmp = (1.0 + (x / n)) - t_0;
	else
		tmp = t_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[x, 0.056], N[(N[(1.0 + N[(x / n), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(t$95$0 / N[(x * n), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0560000000000000012

   1. Initial program 78.6%

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

   2. Taylor expanded in x around 0 76.7%

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

   if 0.0560000000000000012 < x

   1. Initial program 6.7%

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

   2. Taylor expanded in n around 0 6.7%

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

      1. log1p-def 6.7%

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

   4. Simplified 6.7%

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

   5. Taylor expanded in x around inf 99.8%

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

      1. mul-1-neg 99.8%

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

      2. log-rec 99.8%

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

      3. distribute-frac-neg 99.8%

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

      4. remove-double-neg 99.8%

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

      5. *-rgt-identity 99.8%

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

      6. associate-*r/ 99.8%

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

      7. exp-to-pow 99.8%

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

      8. *-commutative 99.8%

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

   7. Simplified 99.8%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.056:\\ \;\;\;\;\left(1 + \frac{x}{n}\right) - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}\\ \end{array} \]

Alternative 8: 83.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{if}\;x \leq 0.056:\\ \;\;\;\;1 - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{x \cdot n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (let* ((t_0 (pow x (/ 1.0 n))))
   (if (<= x 0.056) (- 1.0 t_0) (/ t_0 (* x n)))))

C

double code(double x, double n) {
	double t_0 = pow(x, (1.0 / n));
	double tmp;
	if (x <= 0.056) {
		tmp = 1.0 - t_0;
	} else {
		tmp = t_0 / (x * n);
	}
	return tmp;
}

Fortran

real(8) function code(x, n)
    real(8), intent (in) :: x
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x ** (1.0d0 / n)
    if (x <= 0.056d0) then
        tmp = 1.0d0 - t_0
    else
        tmp = t_0 / (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 (x <= 0.056) {
		tmp = 1.0 - t_0;
	} else {
		tmp = t_0 / (x * n);
	}
	return tmp;
}

Python

def code(x, n):
	t_0 = math.pow(x, (1.0 / n))
	tmp = 0
	if x <= 0.056:
		tmp = 1.0 - t_0
	else:
		tmp = t_0 / (x * n)
	return tmp

Julia

function code(x, n)
	t_0 = x ^ Float64(1.0 / n)
	tmp = 0.0
	if (x <= 0.056)
		tmp = Float64(1.0 - t_0);
	else
		tmp = Float64(t_0 / Float64(x * n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	t_0 = x ^ (1.0 / n);
	tmp = 0.0;
	if (x <= 0.056)
		tmp = 1.0 - t_0;
	else
		tmp = t_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[x, 0.056], N[(1.0 - t$95$0), $MachinePrecision], N[(t$95$0 / N[(x * n), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0560000000000000012

   1. Initial program 78.6%

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

   2. Taylor expanded in x around 0 76.2%

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

   if 0.0560000000000000012 < x

   1. Initial program 6.7%

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

   2. Taylor expanded in n around 0 6.7%

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

      1. log1p-def 6.7%

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

   4. Simplified 6.7%

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

   5. Taylor expanded in x around inf 99.8%

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

      1. mul-1-neg 99.8%

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

      2. log-rec 99.8%

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

      3. distribute-frac-neg 99.8%

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

      4. remove-double-neg 99.8%

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

      5. *-rgt-identity 99.8%

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

      6. associate-*r/ 99.8%

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

      7. exp-to-pow 99.8%

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

      8. *-commutative 99.8%

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

   7. Simplified 99.8%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.056:\\ \;\;\;\;1 - {x}^{\left(\frac{1}{n}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{\left(\frac{1}{n}\right)}}{x \cdot n}\\ \end{array} \]

Alternative 9: 65.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

double code(double x, double n) {
	double tmp;
	if (x <= 1.0) {
		tmp = 1.0 - pow(x, (1.0 / n));
	} else {
		tmp = (x - log(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 (x <= 1.0d0) then
        tmp = 1.0d0 - (x ** (1.0d0 / n))
    else
        tmp = (x - log(x)) / n
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 79.0%

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

   2. Taylor expanded in x around 0 76.6%

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

   if 1 < x

   1. Initial program 3.6%

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

   2. Taylor expanded in n around 0 3.6%

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

      1. log1p-def 3.6%

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

   4. Simplified 3.6%

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

   5. Taylor expanded in x around 0 3.5%

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

   6. Taylor expanded in n around inf 52.3%

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

4. Final simplification 68.0%

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

Alternative 10: 42.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.2 \cdot 10^{+23}:\\ \;\;\;\;\frac{1}{x \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - \log x}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (x n)
 :precision binary64
 (if (<= x 4.2e+23) (/ 1.0 (* x n)) (/ (- x (log x)) n)))

C

double code(double x, double n) {
	double tmp;
	if (x <= 4.2e+23) {
		tmp = 1.0 / (x * n);
	} else {
		tmp = (x - log(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 (x <= 4.2d+23) then
        tmp = 1.0d0 / (x * n)
    else
        tmp = (x - log(x)) / n
    end if
    code = tmp
end function

Java

public static double code(double x, double n) {
	double tmp;
	if (x <= 4.2e+23) {
		tmp = 1.0 / (x * n);
	} else {
		tmp = (x - Math.log(x)) / n;
	}
	return tmp;
}

Python

def code(x, n):
	tmp = 0
	if x <= 4.2e+23:
		tmp = 1.0 / (x * n)
	else:
		tmp = (x - math.log(x)) / n
	return tmp

Julia

function code(x, n)
	tmp = 0.0
	if (x <= 4.2e+23)
		tmp = Float64(1.0 / Float64(x * n));
	else
		tmp = Float64(Float64(x - log(x)) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, n)
	tmp = 0.0;
	if (x <= 4.2e+23)
		tmp = 1.0 / (x * n);
	else
		tmp = (x - log(x)) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, n_] := If[LessEqual[x, 4.2e+23], N[(1.0 / N[(x * n), $MachinePrecision]), $MachinePrecision], N[(N[(x - N[Log[x], $MachinePrecision]), $MachinePrecision] / n), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.2000000000000003e23

   1. Initial program 74.5%

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

   2. Taylor expanded in n around 0 74.5%

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

      1. log1p-def 92.9%

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

   4. Simplified 92.9%

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

   5. Taylor expanded in x around inf 54.9%

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

      1. mul-1-neg 54.9%

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

      2. log-rec 54.9%

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

      3. distribute-frac-neg 54.9%

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

      4. remove-double-neg 54.9%

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

      5. *-rgt-identity 54.9%

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

      6. associate-*r/ 54.9%

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

      7. exp-to-pow 54.9%

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

      8. *-commutative 54.9%

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

   7. Simplified 54.9%

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

   8. Taylor expanded in n around inf 36.3%

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

      1. *-commutative 36.3%

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

   10. Simplified 36.3%

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

   if 4.2000000000000003e23 < x

   1. Initial program 0.3%

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

   2. Taylor expanded in n around 0 0.3%

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

      1. log1p-def 0.3%

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

   4. Simplified 0.3%

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

   5. Taylor expanded in x around 0 0.1%

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

   6. Taylor expanded in n around inf 60.9%

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

4. Final simplification 43.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.2 \cdot 10^{+23}:\\ \;\;\;\;\frac{1}{x \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - \log x}{n}\\ \end{array} \]

Alternative 11: 28.1% accurate, 42.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.2%

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

2. Taylor expanded in n around 0 52.2%

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

   1. log1p-def 65.0%

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

4. Simplified 65.0%

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

5. Taylor expanded in x around inf 68.4%

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

   1. mul-1-neg 68.4%

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

   2. log-rec 68.4%

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

   3. distribute-frac-neg 68.4%

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

   4. remove-double-neg 68.4%

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

   5. *-rgt-identity 68.4%

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

   6. associate-*r/ 68.4%

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

   7. exp-to-pow 68.4%

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

   8. *-commutative 68.4%

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

7. Simplified 68.4%

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

8. Taylor expanded in n around inf 26.8%

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

   1. *-commutative 26.8%

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

10. Simplified 26.8%

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

11. Final simplification 26.8%

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

Reproduce

herbie shell --seed 2023278 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1.0) (/ 1.0 n)) (pow x (/ 1.0 n))))