2log (problem 3.3.6)

Percentage Accurate: 24.4% → 99.4%

Time: 9.6s

Alternatives: 9

Speedup: 68.3×

Specification

\[N > 1 \land N < 10^{+40}\]

Math

\[\begin{array}{l} \\ \log \left(N + 1\right) - \log N \end{array} \]

FPCore

(FPCore (N) :precision binary64 (- (log (+ N 1.0)) (log N)))

C

double code(double N) {
	return log((N + 1.0)) - log(N);
}

Fortran

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

Java

public static double code(double N) {
	return Math.log((N + 1.0)) - Math.log(N);
}

Python

def code(N):
	return math.log((N + 1.0)) - math.log(N)

Julia

function code(N)
	return Float64(log(Float64(N + 1.0)) - log(N))
end

MATLAB

function tmp = code(N)
	tmp = log((N + 1.0)) - log(N);
end

Wolfram

code[N_] := N[(N[Log[N[(N + 1.0), $MachinePrecision]], $MachinePrecision] - N[Log[N], $MachinePrecision]), $MachinePrecision]

Initial Program: 24.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \log \left(N + 1\right) - \log N \end{array} \]

FPCore

(FPCore (N) :precision binary64 (- (log (+ N 1.0)) (log N)))

C

double code(double N) {
	return log((N + 1.0)) - log(N);
}

Fortran

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

Java

public static double code(double N) {
	return Math.log((N + 1.0)) - Math.log(N);
}

Python

def code(N):
	return math.log((N + 1.0)) - math.log(N)

Julia

function code(N)
	return Float64(log(Float64(N + 1.0)) - log(N))
end

MATLAB

function tmp = code(N)
	tmp = log((N + 1.0)) - log(N);
end

Wolfram

code[N_] := N[(N[Log[N[(N + 1.0), $MachinePrecision]], $MachinePrecision] - N[Log[N], $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 0.0006:\\ \;\;\;\;\frac{1}{N} + \left(\left(\frac{-0.5}{{N}^{2}} + \frac{-0.25}{{N}^{4}}\right) + \frac{0.3333333333333333}{{N}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;-\log \left(\frac{N}{N + 1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= (- (log (+ N 1.0)) (log N)) 0.0006)
   (+
    (/ 1.0 N)
    (+
     (+ (/ -0.5 (pow N 2.0)) (/ -0.25 (pow N 4.0)))
     (/ 0.3333333333333333 (pow N 3.0))))
   (- (log (/ N (+ N 1.0))))))

C

double code(double N) {
	double tmp;
	if ((log((N + 1.0)) - log(N)) <= 0.0006) {
		tmp = (1.0 / N) + (((-0.5 / pow(N, 2.0)) + (-0.25 / pow(N, 4.0))) + (0.3333333333333333 / pow(N, 3.0)));
	} else {
		tmp = -log((N / (N + 1.0)));
	}
	return tmp;
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((log((n + 1.0d0)) - log(n)) <= 0.0006d0) then
        tmp = (1.0d0 / n) + ((((-0.5d0) / (n ** 2.0d0)) + ((-0.25d0) / (n ** 4.0d0))) + (0.3333333333333333d0 / (n ** 3.0d0)))
    else
        tmp = -log((n / (n + 1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double N) {
	double tmp;
	if ((Math.log((N + 1.0)) - Math.log(N)) <= 0.0006) {
		tmp = (1.0 / N) + (((-0.5 / Math.pow(N, 2.0)) + (-0.25 / Math.pow(N, 4.0))) + (0.3333333333333333 / Math.pow(N, 3.0)));
	} else {
		tmp = -Math.log((N / (N + 1.0)));
	}
	return tmp;
}

Python

def code(N):
	tmp = 0
	if (math.log((N + 1.0)) - math.log(N)) <= 0.0006:
		tmp = (1.0 / N) + (((-0.5 / math.pow(N, 2.0)) + (-0.25 / math.pow(N, 4.0))) + (0.3333333333333333 / math.pow(N, 3.0)))
	else:
		tmp = -math.log((N / (N + 1.0)))
	return tmp

Julia

function code(N)
	tmp = 0.0
	if (Float64(log(Float64(N + 1.0)) - log(N)) <= 0.0006)
		tmp = Float64(Float64(1.0 / N) + Float64(Float64(Float64(-0.5 / (N ^ 2.0)) + Float64(-0.25 / (N ^ 4.0))) + Float64(0.3333333333333333 / (N ^ 3.0))));
	else
		tmp = Float64(-log(Float64(N / Float64(N + 1.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(N)
	tmp = 0.0;
	if ((log((N + 1.0)) - log(N)) <= 0.0006)
		tmp = (1.0 / N) + (((-0.5 / (N ^ 2.0)) + (-0.25 / (N ^ 4.0))) + (0.3333333333333333 / (N ^ 3.0)));
	else
		tmp = -log((N / (N + 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[N_] := If[LessEqual[N[(N[Log[N[(N + 1.0), $MachinePrecision]], $MachinePrecision] - N[Log[N], $MachinePrecision]), $MachinePrecision], 0.0006], N[(N[(1.0 / N), $MachinePrecision] + N[(N[(N[(-0.5 / N[Power[N, 2.0], $MachinePrecision]), $MachinePrecision] + N[(-0.25 / N[Power[N, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.3333333333333333 / N[Power[N, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-N[Log[N[(N / N[(N + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (log.f64 (+.f64 N 1)) (log.f64 N)) < 5.99999999999999947e-4

   1. Initial program 19.5%

      \[\log \left(N + 1\right) - \log N \]
   2. Step-by-step derivation

      1. +-commutative 19.5%

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

      2. log1p-def 19.5%

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

   3. Simplified 19.5%

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

   4. Taylor expanded in N around inf 99.7%

      \[\leadsto \color{blue}{\left(0.3333333333333333 \cdot \frac{1}{{N}^{3}} + \frac{1}{N}\right) - \left(0.25 \cdot \frac{1}{{N}^{4}} + 0.5 \cdot \frac{1}{{N}^{2}}\right)} \]
   5. Step-by-step derivation

      1. associate--l+ 99.7%

         \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{1}{{N}^{3}} + \left(\frac{1}{N} - \left(0.25 \cdot \frac{1}{{N}^{4}} + 0.5 \cdot \frac{1}{{N}^{2}}\right)\right)} \]

      2. +-commutative 99.7%

         \[\leadsto \color{blue}{\left(\frac{1}{N} - \left(0.25 \cdot \frac{1}{{N}^{4}} + 0.5 \cdot \frac{1}{{N}^{2}}\right)\right) + 0.3333333333333333 \cdot \frac{1}{{N}^{3}}} \]

      3. sub-neg 99.7%

         \[\leadsto \color{blue}{\left(\frac{1}{N} + \left(-\left(0.25 \cdot \frac{1}{{N}^{4}} + 0.5 \cdot \frac{1}{{N}^{2}}\right)\right)\right)} + 0.3333333333333333 \cdot \frac{1}{{N}^{3}} \]

      4. associate-+l+ 99.7%

         \[\leadsto \color{blue}{\frac{1}{N} + \left(\left(-\left(0.25 \cdot \frac{1}{{N}^{4}} + 0.5 \cdot \frac{1}{{N}^{2}}\right)\right) + 0.3333333333333333 \cdot \frac{1}{{N}^{3}}\right)} \]

   6. Simplified 99.7%

      \[\leadsto \color{blue}{\frac{1}{N} + \left(\left(\frac{-0.5}{{N}^{2}} + \frac{-0.25}{{N}^{4}}\right) + \frac{0.3333333333333333}{{N}^{3}}\right)} \]

   if 5.99999999999999947e-4 < (-.f64 (log.f64 (+.f64 N 1)) (log.f64 N))

   1. Initial program 90.2%

      \[\log \left(N + 1\right) - \log N \]
   2. Step-by-step derivation

      1. +-commutative 90.2%

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

      2. log1p-def 90.2%

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

   3. Simplified 90.2%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(N\right) - \log N} \]
   4. Step-by-step derivation

      1. add-log-exp 90.2%

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

      2. add-cube-cbrt 90.1%

         \[\leadsto \log \color{blue}{\left(\left(\sqrt[3]{e^{\mathsf{log1p}\left(N\right) - \log N}} \cdot \sqrt[3]{e^{\mathsf{log1p}\left(N\right) - \log N}}\right) \cdot \sqrt[3]{e^{\mathsf{log1p}\left(N\right) - \log N}}\right)} \]

      3. log-prod 89.8%

         \[\leadsto \color{blue}{\log \left(\sqrt[3]{e^{\mathsf{log1p}\left(N\right) - \log N}} \cdot \sqrt[3]{e^{\mathsf{log1p}\left(N\right) - \log N}}\right) + \log \left(\sqrt[3]{e^{\mathsf{log1p}\left(N\right) - \log N}}\right)} \]

      4. pow2 89.8%

         \[\leadsto \log \color{blue}{\left({\left(\sqrt[3]{e^{\mathsf{log1p}\left(N\right) - \log N}}\right)}^{2}\right)} + \log \left(\sqrt[3]{e^{\mathsf{log1p}\left(N\right) - \log N}}\right) \]

      5. exp-diff 89.8%

         \[\leadsto \log \left({\left(\sqrt[3]{\color{blue}{\frac{e^{\mathsf{log1p}\left(N\right)}}{e^{\log N}}}}\right)}^{2}\right) + \log \left(\sqrt[3]{e^{\mathsf{log1p}\left(N\right) - \log N}}\right) \]

      6. log1p-udef 89.8%

         \[\leadsto \log \left({\left(\sqrt[3]{\frac{e^{\color{blue}{\log \left(1 + N\right)}}}{e^{\log N}}}\right)}^{2}\right) + \log \left(\sqrt[3]{e^{\mathsf{log1p}\left(N\right) - \log N}}\right) \]

      7. rem-exp-log 90.0%

         \[\leadsto \log \left({\left(\sqrt[3]{\frac{\color{blue}{1 + N}}{e^{\log N}}}\right)}^{2}\right) + \log \left(\sqrt[3]{e^{\mathsf{log1p}\left(N\right) - \log N}}\right) \]

      8. add-exp-log 90.5%

         \[\leadsto \log \left({\left(\sqrt[3]{\frac{1 + N}{\color{blue}{N}}}\right)}^{2}\right) + \log \left(\sqrt[3]{e^{\mathsf{log1p}\left(N\right) - \log N}}\right) \]

      9. +-commutative 90.5%

         \[\leadsto \log \left({\left(\sqrt[3]{\frac{\color{blue}{N + 1}}{N}}\right)}^{2}\right) + \log \left(\sqrt[3]{e^{\mathsf{log1p}\left(N\right) - \log N}}\right) \]

      10. exp-diff 90.5%

          \[\leadsto \log \left({\left(\sqrt[3]{\frac{N + 1}{N}}\right)}^{2}\right) + \log \left(\sqrt[3]{\color{blue}{\frac{e^{\mathsf{log1p}\left(N\right)}}{e^{\log N}}}}\right) \]

      11. log1p-udef 90.5%

          \[\leadsto \log \left({\left(\sqrt[3]{\frac{N + 1}{N}}\right)}^{2}\right) + \log \left(\sqrt[3]{\frac{e^{\color{blue}{\log \left(1 + N\right)}}}{e^{\log N}}}\right) \]

      12. rem-exp-log 91.1%

          \[\leadsto \log \left({\left(\sqrt[3]{\frac{N + 1}{N}}\right)}^{2}\right) + \log \left(\sqrt[3]{\frac{\color{blue}{1 + N}}{e^{\log N}}}\right) \]

      13. add-exp-log 90.7%

          \[\leadsto \log \left({\left(\sqrt[3]{\frac{N + 1}{N}}\right)}^{2}\right) + \log \left(\sqrt[3]{\frac{1 + N}{\color{blue}{N}}}\right) \]

   5. Applied egg-rr 90.7%

      \[\leadsto \color{blue}{\log \left({\left(\sqrt[3]{\frac{N + 1}{N}}\right)}^{2}\right) + \log \left(\sqrt[3]{\frac{N + 1}{N}}\right)} \]
   6. Step-by-step derivation

      1. log-pow 90.6%

         \[\leadsto \color{blue}{2 \cdot \log \left(\sqrt[3]{\frac{N + 1}{N}}\right)} + \log \left(\sqrt[3]{\frac{N + 1}{N}}\right) \]

      2. distribute-lft1-in 90.6%

         \[\leadsto \color{blue}{\left(2 + 1\right) \cdot \log \left(\sqrt[3]{\frac{N + 1}{N}}\right)} \]

      3. metadata-eval 90.6%

         \[\leadsto \color{blue}{3} \cdot \log \left(\sqrt[3]{\frac{N + 1}{N}}\right) \]

   7. Simplified 90.6%

      \[\leadsto \color{blue}{3 \cdot \log \left(\sqrt[3]{\frac{N + 1}{N}}\right)} \]
   8. Step-by-step derivation

      1. *-commutative 90.6%

         \[\leadsto \color{blue}{\log \left(\sqrt[3]{\frac{N + 1}{N}}\right) \cdot 3} \]

      2. add-log-exp 91.0%

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

      3. exp-to-pow 91.0%

         \[\leadsto \log \color{blue}{\left({\left(\sqrt[3]{\frac{N + 1}{N}}\right)}^{3}\right)} \]

      4. pow3 90.9%

         \[\leadsto \log \color{blue}{\left(\left(\sqrt[3]{\frac{N + 1}{N}} \cdot \sqrt[3]{\frac{N + 1}{N}}\right) \cdot \sqrt[3]{\frac{N + 1}{N}}\right)} \]

      5. add-cube-cbrt 93.3%

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

      6. clear-num 93.3%

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

      7. log-div 95.0%

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

      8. metadata-eval 95.0%

         \[\leadsto \color{blue}{0} - \log \left(\frac{N}{N + 1}\right) \]

   9. Applied egg-rr 95.0%

      \[\leadsto \color{blue}{0 - \log \left(\frac{N}{N + 1}\right)} \]
   10. Step-by-step derivation

       1. neg-sub0 95.0%

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

   11. Simplified 95.0%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 0.0006:\\ \;\;\;\;\frac{1}{N} + \left(\left(\frac{-0.5}{{N}^{2}} + \frac{-0.25}{{N}^{4}}\right) + \frac{0.3333333333333333}{{N}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;-\log \left(\frac{N}{N + 1}\right)\\ \end{array} \]

Alternative 2: 99.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 0.0006:\\ \;\;\;\;\frac{0.3333333333333333}{{N}^{3}} + \left(\left(\frac{1}{N} - \frac{0.25}{{N}^{4}}\right) - \frac{0.5}{{N}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;-\log \left(\frac{N}{N + 1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= (- (log (+ N 1.0)) (log N)) 0.0006)
   (+
    (/ 0.3333333333333333 (pow N 3.0))
    (- (- (/ 1.0 N) (/ 0.25 (pow N 4.0))) (/ 0.5 (pow N 2.0))))
   (- (log (/ N (+ N 1.0))))))

C

double code(double N) {
	double tmp;
	if ((log((N + 1.0)) - log(N)) <= 0.0006) {
		tmp = (0.3333333333333333 / pow(N, 3.0)) + (((1.0 / N) - (0.25 / pow(N, 4.0))) - (0.5 / pow(N, 2.0)));
	} else {
		tmp = -log((N / (N + 1.0)));
	}
	return tmp;
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((log((n + 1.0d0)) - log(n)) <= 0.0006d0) then
        tmp = (0.3333333333333333d0 / (n ** 3.0d0)) + (((1.0d0 / n) - (0.25d0 / (n ** 4.0d0))) - (0.5d0 / (n ** 2.0d0)))
    else
        tmp = -log((n / (n + 1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double N) {
	double tmp;
	if ((Math.log((N + 1.0)) - Math.log(N)) <= 0.0006) {
		tmp = (0.3333333333333333 / Math.pow(N, 3.0)) + (((1.0 / N) - (0.25 / Math.pow(N, 4.0))) - (0.5 / Math.pow(N, 2.0)));
	} else {
		tmp = -Math.log((N / (N + 1.0)));
	}
	return tmp;
}

Python

def code(N):
	tmp = 0
	if (math.log((N + 1.0)) - math.log(N)) <= 0.0006:
		tmp = (0.3333333333333333 / math.pow(N, 3.0)) + (((1.0 / N) - (0.25 / math.pow(N, 4.0))) - (0.5 / math.pow(N, 2.0)))
	else:
		tmp = -math.log((N / (N + 1.0)))
	return tmp

Julia

function code(N)
	tmp = 0.0
	if (Float64(log(Float64(N + 1.0)) - log(N)) <= 0.0006)
		tmp = Float64(Float64(0.3333333333333333 / (N ^ 3.0)) + Float64(Float64(Float64(1.0 / N) - Float64(0.25 / (N ^ 4.0))) - Float64(0.5 / (N ^ 2.0))));
	else
		tmp = Float64(-log(Float64(N / Float64(N + 1.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(N)
	tmp = 0.0;
	if ((log((N + 1.0)) - log(N)) <= 0.0006)
		tmp = (0.3333333333333333 / (N ^ 3.0)) + (((1.0 / N) - (0.25 / (N ^ 4.0))) - (0.5 / (N ^ 2.0)));
	else
		tmp = -log((N / (N + 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[N_] := If[LessEqual[N[(N[Log[N[(N + 1.0), $MachinePrecision]], $MachinePrecision] - N[Log[N], $MachinePrecision]), $MachinePrecision], 0.0006], N[(N[(0.3333333333333333 / N[Power[N, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(1.0 / N), $MachinePrecision] - N[(0.25 / N[Power[N, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.5 / N[Power[N, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-N[Log[N[(N / N[(N + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (log.f64 (+.f64 N 1)) (log.f64 N)) < 5.99999999999999947e-4

   1. Initial program 19.5%

      \[\log \left(N + 1\right) - \log N \]
   2. Step-by-step derivation

      1. +-commutative 19.5%

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

      2. log1p-def 19.5%

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

   3. Simplified 19.5%

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

   4. Taylor expanded in N around inf 99.7%

      \[\leadsto \color{blue}{\left(0.3333333333333333 \cdot \frac{1}{{N}^{3}} + \frac{1}{N}\right) - \left(0.25 \cdot \frac{1}{{N}^{4}} + 0.5 \cdot \frac{1}{{N}^{2}}\right)} \]
   5. Step-by-step derivation

      1. associate--l+ 99.7%

         \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{1}{{N}^{3}} + \left(\frac{1}{N} - \left(0.25 \cdot \frac{1}{{N}^{4}} + 0.5 \cdot \frac{1}{{N}^{2}}\right)\right)} \]

      2. +-commutative 99.7%

         \[\leadsto \color{blue}{\left(\frac{1}{N} - \left(0.25 \cdot \frac{1}{{N}^{4}} + 0.5 \cdot \frac{1}{{N}^{2}}\right)\right) + 0.3333333333333333 \cdot \frac{1}{{N}^{3}}} \]

      3. sub-neg 99.7%

         \[\leadsto \color{blue}{\left(\frac{1}{N} + \left(-\left(0.25 \cdot \frac{1}{{N}^{4}} + 0.5 \cdot \frac{1}{{N}^{2}}\right)\right)\right)} + 0.3333333333333333 \cdot \frac{1}{{N}^{3}} \]

      4. associate-+l+ 99.7%

         \[\leadsto \color{blue}{\frac{1}{N} + \left(\left(-\left(0.25 \cdot \frac{1}{{N}^{4}} + 0.5 \cdot \frac{1}{{N}^{2}}\right)\right) + 0.3333333333333333 \cdot \frac{1}{{N}^{3}}\right)} \]

   6. Simplified 99.7%

      \[\leadsto \color{blue}{\frac{1}{N} + \left(\left(\frac{-0.5}{{N}^{2}} + \frac{-0.25}{{N}^{4}}\right) + \frac{0.3333333333333333}{{N}^{3}}\right)} \]

   7. Taylor expanded in N around 0 99.7%

      \[\leadsto \color{blue}{\left(0.3333333333333333 \cdot \frac{1}{{N}^{3}} + \frac{1}{N}\right) - \left(0.5 \cdot \frac{1}{{N}^{2}} + 0.25 \cdot \frac{1}{{N}^{4}}\right)} \]
   8. Step-by-step derivation

      1. +-commutative 99.7%

         \[\leadsto \left(0.3333333333333333 \cdot \frac{1}{{N}^{3}} + \frac{1}{N}\right) - \color{blue}{\left(0.25 \cdot \frac{1}{{N}^{4}} + 0.5 \cdot \frac{1}{{N}^{2}}\right)} \]

      2. associate-*r/ 99.7%

         \[\leadsto \left(\color{blue}{\frac{0.3333333333333333 \cdot 1}{{N}^{3}}} + \frac{1}{N}\right) - \left(0.25 \cdot \frac{1}{{N}^{4}} + 0.5 \cdot \frac{1}{{N}^{2}}\right) \]

      3. metadata-eval 99.7%

         \[\leadsto \left(\frac{\color{blue}{0.3333333333333333}}{{N}^{3}} + \frac{1}{N}\right) - \left(0.25 \cdot \frac{1}{{N}^{4}} + 0.5 \cdot \frac{1}{{N}^{2}}\right) \]

      4. associate-*r/ 99.7%

         \[\leadsto \left(\frac{0.3333333333333333}{{N}^{3}} + \frac{1}{N}\right) - \left(\color{blue}{\frac{0.25 \cdot 1}{{N}^{4}}} + 0.5 \cdot \frac{1}{{N}^{2}}\right) \]

      5. metadata-eval 99.7%

         \[\leadsto \left(\frac{0.3333333333333333}{{N}^{3}} + \frac{1}{N}\right) - \left(\frac{\color{blue}{0.25}}{{N}^{4}} + 0.5 \cdot \frac{1}{{N}^{2}}\right) \]

      6. associate-*r/ 99.7%

         \[\leadsto \left(\frac{0.3333333333333333}{{N}^{3}} + \frac{1}{N}\right) - \left(\frac{0.25}{{N}^{4}} + \color{blue}{\frac{0.5 \cdot 1}{{N}^{2}}}\right) \]

      7. metadata-eval 99.7%

         \[\leadsto \left(\frac{0.3333333333333333}{{N}^{3}} + \frac{1}{N}\right) - \left(\frac{0.25}{{N}^{4}} + \frac{\color{blue}{0.5}}{{N}^{2}}\right) \]

   9. Simplified 99.7%

      \[\leadsto \color{blue}{\left(\frac{0.3333333333333333}{{N}^{3}} + \frac{1}{N}\right) - \left(\frac{0.25}{{N}^{4}} + \frac{0.5}{{N}^{2}}\right)} \]

   10. Taylor expanded in N around 0 99.7%

       \[\leadsto \color{blue}{\left(0.3333333333333333 \cdot \frac{1}{{N}^{3}} + \frac{1}{N}\right) - \left(0.5 \cdot \frac{1}{{N}^{2}} + 0.25 \cdot \frac{1}{{N}^{4}}\right)} \]
   11. Step-by-step derivation

       1. associate--l+ 99.7%

          \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{1}{{N}^{3}} + \left(\frac{1}{N} - \left(0.5 \cdot \frac{1}{{N}^{2}} + 0.25 \cdot \frac{1}{{N}^{4}}\right)\right)} \]

       2. +-commutative 99.7%

          \[\leadsto 0.3333333333333333 \cdot \frac{1}{{N}^{3}} + \left(\frac{1}{N} - \color{blue}{\left(0.25 \cdot \frac{1}{{N}^{4}} + 0.5 \cdot \frac{1}{{N}^{2}}\right)}\right) \]

       3. associate-*r/ 99.7%

          \[\leadsto \color{blue}{\frac{0.3333333333333333 \cdot 1}{{N}^{3}}} + \left(\frac{1}{N} - \left(0.25 \cdot \frac{1}{{N}^{4}} + 0.5 \cdot \frac{1}{{N}^{2}}\right)\right) \]

       4. metadata-eval 99.7%

          \[\leadsto \frac{\color{blue}{0.3333333333333333}}{{N}^{3}} + \left(\frac{1}{N} - \left(0.25 \cdot \frac{1}{{N}^{4}} + 0.5 \cdot \frac{1}{{N}^{2}}\right)\right) \]

       5. associate--r+ 99.7%

          \[\leadsto \frac{0.3333333333333333}{{N}^{3}} + \color{blue}{\left(\left(\frac{1}{N} - 0.25 \cdot \frac{1}{{N}^{4}}\right) - 0.5 \cdot \frac{1}{{N}^{2}}\right)} \]

       6. associate-*r/ 99.7%

          \[\leadsto \frac{0.3333333333333333}{{N}^{3}} + \left(\left(\frac{1}{N} - \color{blue}{\frac{0.25 \cdot 1}{{N}^{4}}}\right) - 0.5 \cdot \frac{1}{{N}^{2}}\right) \]

       7. metadata-eval 99.7%

          \[\leadsto \frac{0.3333333333333333}{{N}^{3}} + \left(\left(\frac{1}{N} - \frac{\color{blue}{0.25}}{{N}^{4}}\right) - 0.5 \cdot \frac{1}{{N}^{2}}\right) \]

       8. associate-*r/ 99.7%

          \[\leadsto \frac{0.3333333333333333}{{N}^{3}} + \left(\left(\frac{1}{N} - \frac{0.25}{{N}^{4}}\right) - \color{blue}{\frac{0.5 \cdot 1}{{N}^{2}}}\right) \]

       9. metadata-eval 99.7%

          \[\leadsto \frac{0.3333333333333333}{{N}^{3}} + \left(\left(\frac{1}{N} - \frac{0.25}{{N}^{4}}\right) - \frac{\color{blue}{0.5}}{{N}^{2}}\right) \]

   12. Simplified 99.7%

       \[\leadsto \color{blue}{\frac{0.3333333333333333}{{N}^{3}} + \left(\left(\frac{1}{N} - \frac{0.25}{{N}^{4}}\right) - \frac{0.5}{{N}^{2}}\right)} \]

   if 5.99999999999999947e-4 < (-.f64 (log.f64 (+.f64 N 1)) (log.f64 N))

   1. Initial program 90.2%

      \[\log \left(N + 1\right) - \log N \]
   2. Step-by-step derivation

      1. +-commutative 90.2%

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

      2. log1p-def 90.2%

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

   3. Simplified 90.2%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(N\right) - \log N} \]
   4. Step-by-step derivation

      1. add-log-exp 90.2%

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

      2. add-cube-cbrt 90.1%

         \[\leadsto \log \color{blue}{\left(\left(\sqrt[3]{e^{\mathsf{log1p}\left(N\right) - \log N}} \cdot \sqrt[3]{e^{\mathsf{log1p}\left(N\right) - \log N}}\right) \cdot \sqrt[3]{e^{\mathsf{log1p}\left(N\right) - \log N}}\right)} \]

      3. log-prod 89.8%

         \[\leadsto \color{blue}{\log \left(\sqrt[3]{e^{\mathsf{log1p}\left(N\right) - \log N}} \cdot \sqrt[3]{e^{\mathsf{log1p}\left(N\right) - \log N}}\right) + \log \left(\sqrt[3]{e^{\mathsf{log1p}\left(N\right) - \log N}}\right)} \]

      4. pow2 89.8%

         \[\leadsto \log \color{blue}{\left({\left(\sqrt[3]{e^{\mathsf{log1p}\left(N\right) - \log N}}\right)}^{2}\right)} + \log \left(\sqrt[3]{e^{\mathsf{log1p}\left(N\right) - \log N}}\right) \]

      5. exp-diff 89.8%

         \[\leadsto \log \left({\left(\sqrt[3]{\color{blue}{\frac{e^{\mathsf{log1p}\left(N\right)}}{e^{\log N}}}}\right)}^{2}\right) + \log \left(\sqrt[3]{e^{\mathsf{log1p}\left(N\right) - \log N}}\right) \]

      6. log1p-udef 89.8%

         \[\leadsto \log \left({\left(\sqrt[3]{\frac{e^{\color{blue}{\log \left(1 + N\right)}}}{e^{\log N}}}\right)}^{2}\right) + \log \left(\sqrt[3]{e^{\mathsf{log1p}\left(N\right) - \log N}}\right) \]

      7. rem-exp-log 90.0%

         \[\leadsto \log \left({\left(\sqrt[3]{\frac{\color{blue}{1 + N}}{e^{\log N}}}\right)}^{2}\right) + \log \left(\sqrt[3]{e^{\mathsf{log1p}\left(N\right) - \log N}}\right) \]

      8. add-exp-log 90.5%

         \[\leadsto \log \left({\left(\sqrt[3]{\frac{1 + N}{\color{blue}{N}}}\right)}^{2}\right) + \log \left(\sqrt[3]{e^{\mathsf{log1p}\left(N\right) - \log N}}\right) \]

      9. +-commutative 90.5%

         \[\leadsto \log \left({\left(\sqrt[3]{\frac{\color{blue}{N + 1}}{N}}\right)}^{2}\right) + \log \left(\sqrt[3]{e^{\mathsf{log1p}\left(N\right) - \log N}}\right) \]

      10. exp-diff 90.5%

          \[\leadsto \log \left({\left(\sqrt[3]{\frac{N + 1}{N}}\right)}^{2}\right) + \log \left(\sqrt[3]{\color{blue}{\frac{e^{\mathsf{log1p}\left(N\right)}}{e^{\log N}}}}\right) \]

      11. log1p-udef 90.5%

          \[\leadsto \log \left({\left(\sqrt[3]{\frac{N + 1}{N}}\right)}^{2}\right) + \log \left(\sqrt[3]{\frac{e^{\color{blue}{\log \left(1 + N\right)}}}{e^{\log N}}}\right) \]

      12. rem-exp-log 91.1%

          \[\leadsto \log \left({\left(\sqrt[3]{\frac{N + 1}{N}}\right)}^{2}\right) + \log \left(\sqrt[3]{\frac{\color{blue}{1 + N}}{e^{\log N}}}\right) \]

      13. add-exp-log 90.7%

          \[\leadsto \log \left({\left(\sqrt[3]{\frac{N + 1}{N}}\right)}^{2}\right) + \log \left(\sqrt[3]{\frac{1 + N}{\color{blue}{N}}}\right) \]

   5. Applied egg-rr 90.7%

      \[\leadsto \color{blue}{\log \left({\left(\sqrt[3]{\frac{N + 1}{N}}\right)}^{2}\right) + \log \left(\sqrt[3]{\frac{N + 1}{N}}\right)} \]
   6. Step-by-step derivation

      1. log-pow 90.6%

         \[\leadsto \color{blue}{2 \cdot \log \left(\sqrt[3]{\frac{N + 1}{N}}\right)} + \log \left(\sqrt[3]{\frac{N + 1}{N}}\right) \]

      2. distribute-lft1-in 90.6%

         \[\leadsto \color{blue}{\left(2 + 1\right) \cdot \log \left(\sqrt[3]{\frac{N + 1}{N}}\right)} \]

      3. metadata-eval 90.6%

         \[\leadsto \color{blue}{3} \cdot \log \left(\sqrt[3]{\frac{N + 1}{N}}\right) \]

   7. Simplified 90.6%

      \[\leadsto \color{blue}{3 \cdot \log \left(\sqrt[3]{\frac{N + 1}{N}}\right)} \]
   8. Step-by-step derivation

      1. *-commutative 90.6%

         \[\leadsto \color{blue}{\log \left(\sqrt[3]{\frac{N + 1}{N}}\right) \cdot 3} \]

      2. add-log-exp 91.0%

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

      3. exp-to-pow 91.0%

         \[\leadsto \log \color{blue}{\left({\left(\sqrt[3]{\frac{N + 1}{N}}\right)}^{3}\right)} \]

      4. pow3 90.9%

         \[\leadsto \log \color{blue}{\left(\left(\sqrt[3]{\frac{N + 1}{N}} \cdot \sqrt[3]{\frac{N + 1}{N}}\right) \cdot \sqrt[3]{\frac{N + 1}{N}}\right)} \]

      5. add-cube-cbrt 93.3%

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

      6. clear-num 93.3%

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

      7. log-div 95.0%

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

      8. metadata-eval 95.0%

         \[\leadsto \color{blue}{0} - \log \left(\frac{N}{N + 1}\right) \]

   9. Applied egg-rr 95.0%

      \[\leadsto \color{blue}{0 - \log \left(\frac{N}{N + 1}\right)} \]
   10. Step-by-step derivation

       1. neg-sub0 95.0%

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

   11. Simplified 95.0%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 0.0006:\\ \;\;\;\;\frac{0.3333333333333333}{{N}^{3}} + \left(\left(\frac{1}{N} - \frac{0.25}{{N}^{4}}\right) - \frac{0.5}{{N}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;-\log \left(\frac{N}{N + 1}\right)\\ \end{array} \]

Alternative 3: 99.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 0.0006:\\ \;\;\;\;\frac{0.3333333333333333}{{N}^{3}} + \left(\frac{1}{N} - \left(\frac{0.5}{{N}^{2}} + \frac{0.25}{{N}^{4}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-\log \left(\frac{N}{N + 1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= (- (log (+ N 1.0)) (log N)) 0.0006)
   (+
    (/ 0.3333333333333333 (pow N 3.0))
    (- (/ 1.0 N) (+ (/ 0.5 (pow N 2.0)) (/ 0.25 (pow N 4.0)))))
   (- (log (/ N (+ N 1.0))))))

C

double code(double N) {
	double tmp;
	if ((log((N + 1.0)) - log(N)) <= 0.0006) {
		tmp = (0.3333333333333333 / pow(N, 3.0)) + ((1.0 / N) - ((0.5 / pow(N, 2.0)) + (0.25 / pow(N, 4.0))));
	} else {
		tmp = -log((N / (N + 1.0)));
	}
	return tmp;
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((log((n + 1.0d0)) - log(n)) <= 0.0006d0) then
        tmp = (0.3333333333333333d0 / (n ** 3.0d0)) + ((1.0d0 / n) - ((0.5d0 / (n ** 2.0d0)) + (0.25d0 / (n ** 4.0d0))))
    else
        tmp = -log((n / (n + 1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double N) {
	double tmp;
	if ((Math.log((N + 1.0)) - Math.log(N)) <= 0.0006) {
		tmp = (0.3333333333333333 / Math.pow(N, 3.0)) + ((1.0 / N) - ((0.5 / Math.pow(N, 2.0)) + (0.25 / Math.pow(N, 4.0))));
	} else {
		tmp = -Math.log((N / (N + 1.0)));
	}
	return tmp;
}

Python

def code(N):
	tmp = 0
	if (math.log((N + 1.0)) - math.log(N)) <= 0.0006:
		tmp = (0.3333333333333333 / math.pow(N, 3.0)) + ((1.0 / N) - ((0.5 / math.pow(N, 2.0)) + (0.25 / math.pow(N, 4.0))))
	else:
		tmp = -math.log((N / (N + 1.0)))
	return tmp

Julia

function code(N)
	tmp = 0.0
	if (Float64(log(Float64(N + 1.0)) - log(N)) <= 0.0006)
		tmp = Float64(Float64(0.3333333333333333 / (N ^ 3.0)) + Float64(Float64(1.0 / N) - Float64(Float64(0.5 / (N ^ 2.0)) + Float64(0.25 / (N ^ 4.0)))));
	else
		tmp = Float64(-log(Float64(N / Float64(N + 1.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(N)
	tmp = 0.0;
	if ((log((N + 1.0)) - log(N)) <= 0.0006)
		tmp = (0.3333333333333333 / (N ^ 3.0)) + ((1.0 / N) - ((0.5 / (N ^ 2.0)) + (0.25 / (N ^ 4.0))));
	else
		tmp = -log((N / (N + 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[N_] := If[LessEqual[N[(N[Log[N[(N + 1.0), $MachinePrecision]], $MachinePrecision] - N[Log[N], $MachinePrecision]), $MachinePrecision], 0.0006], N[(N[(0.3333333333333333 / N[Power[N, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / N), $MachinePrecision] - N[(N[(0.5 / N[Power[N, 2.0], $MachinePrecision]), $MachinePrecision] + N[(0.25 / N[Power[N, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-N[Log[N[(N / N[(N + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (log.f64 (+.f64 N 1)) (log.f64 N)) < 5.99999999999999947e-4

   1. Initial program 19.5%

      \[\log \left(N + 1\right) - \log N \]
   2. Step-by-step derivation

      1. +-commutative 19.5%

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

      2. log1p-def 19.5%

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

   3. Simplified 19.5%

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

   4. Taylor expanded in N around inf 99.7%

      \[\leadsto \color{blue}{\left(0.3333333333333333 \cdot \frac{1}{{N}^{3}} + \frac{1}{N}\right) - \left(0.25 \cdot \frac{1}{{N}^{4}} + 0.5 \cdot \frac{1}{{N}^{2}}\right)} \]
   5. Step-by-step derivation

      1. associate--l+ 99.7%

         \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{1}{{N}^{3}} + \left(\frac{1}{N} - \left(0.25 \cdot \frac{1}{{N}^{4}} + 0.5 \cdot \frac{1}{{N}^{2}}\right)\right)} \]

      2. associate-*r/ 99.7%

         \[\leadsto \color{blue}{\frac{0.3333333333333333 \cdot 1}{{N}^{3}}} + \left(\frac{1}{N} - \left(0.25 \cdot \frac{1}{{N}^{4}} + 0.5 \cdot \frac{1}{{N}^{2}}\right)\right) \]

      3. metadata-eval 99.7%

         \[\leadsto \frac{\color{blue}{0.3333333333333333}}{{N}^{3}} + \left(\frac{1}{N} - \left(0.25 \cdot \frac{1}{{N}^{4}} + 0.5 \cdot \frac{1}{{N}^{2}}\right)\right) \]

      4. +-commutative 99.7%

         \[\leadsto \frac{0.3333333333333333}{{N}^{3}} + \left(\frac{1}{N} - \color{blue}{\left(0.5 \cdot \frac{1}{{N}^{2}} + 0.25 \cdot \frac{1}{{N}^{4}}\right)}\right) \]

      5. associate-*r/ 99.7%

         \[\leadsto \frac{0.3333333333333333}{{N}^{3}} + \left(\frac{1}{N} - \left(\color{blue}{\frac{0.5 \cdot 1}{{N}^{2}}} + 0.25 \cdot \frac{1}{{N}^{4}}\right)\right) \]

      6. metadata-eval 99.7%

         \[\leadsto \frac{0.3333333333333333}{{N}^{3}} + \left(\frac{1}{N} - \left(\frac{\color{blue}{0.5}}{{N}^{2}} + 0.25 \cdot \frac{1}{{N}^{4}}\right)\right) \]

      7. associate-*r/ 99.7%

         \[\leadsto \frac{0.3333333333333333}{{N}^{3}} + \left(\frac{1}{N} - \left(\frac{0.5}{{N}^{2}} + \color{blue}{\frac{0.25 \cdot 1}{{N}^{4}}}\right)\right) \]

      8. metadata-eval 99.7%

         \[\leadsto \frac{0.3333333333333333}{{N}^{3}} + \left(\frac{1}{N} - \left(\frac{0.5}{{N}^{2}} + \frac{\color{blue}{0.25}}{{N}^{4}}\right)\right) \]

   6. Simplified 99.7%

      \[\leadsto \color{blue}{\frac{0.3333333333333333}{{N}^{3}} + \left(\frac{1}{N} - \left(\frac{0.5}{{N}^{2}} + \frac{0.25}{{N}^{4}}\right)\right)} \]

   if 5.99999999999999947e-4 < (-.f64 (log.f64 (+.f64 N 1)) (log.f64 N))

   1. Initial program 90.2%

      \[\log \left(N + 1\right) - \log N \]
   2. Step-by-step derivation

      1. +-commutative 90.2%

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

      2. log1p-def 90.2%

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

   3. Simplified 90.2%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(N\right) - \log N} \]
   4. Step-by-step derivation

      1. add-log-exp 90.2%

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

      2. add-cube-cbrt 90.1%

         \[\leadsto \log \color{blue}{\left(\left(\sqrt[3]{e^{\mathsf{log1p}\left(N\right) - \log N}} \cdot \sqrt[3]{e^{\mathsf{log1p}\left(N\right) - \log N}}\right) \cdot \sqrt[3]{e^{\mathsf{log1p}\left(N\right) - \log N}}\right)} \]

      3. log-prod 89.8%

         \[\leadsto \color{blue}{\log \left(\sqrt[3]{e^{\mathsf{log1p}\left(N\right) - \log N}} \cdot \sqrt[3]{e^{\mathsf{log1p}\left(N\right) - \log N}}\right) + \log \left(\sqrt[3]{e^{\mathsf{log1p}\left(N\right) - \log N}}\right)} \]

      4. pow2 89.8%

         \[\leadsto \log \color{blue}{\left({\left(\sqrt[3]{e^{\mathsf{log1p}\left(N\right) - \log N}}\right)}^{2}\right)} + \log \left(\sqrt[3]{e^{\mathsf{log1p}\left(N\right) - \log N}}\right) \]

      5. exp-diff 89.8%

         \[\leadsto \log \left({\left(\sqrt[3]{\color{blue}{\frac{e^{\mathsf{log1p}\left(N\right)}}{e^{\log N}}}}\right)}^{2}\right) + \log \left(\sqrt[3]{e^{\mathsf{log1p}\left(N\right) - \log N}}\right) \]

      6. log1p-udef 89.8%

         \[\leadsto \log \left({\left(\sqrt[3]{\frac{e^{\color{blue}{\log \left(1 + N\right)}}}{e^{\log N}}}\right)}^{2}\right) + \log \left(\sqrt[3]{e^{\mathsf{log1p}\left(N\right) - \log N}}\right) \]

      7. rem-exp-log 90.0%

         \[\leadsto \log \left({\left(\sqrt[3]{\frac{\color{blue}{1 + N}}{e^{\log N}}}\right)}^{2}\right) + \log \left(\sqrt[3]{e^{\mathsf{log1p}\left(N\right) - \log N}}\right) \]

      8. add-exp-log 90.5%

         \[\leadsto \log \left({\left(\sqrt[3]{\frac{1 + N}{\color{blue}{N}}}\right)}^{2}\right) + \log \left(\sqrt[3]{e^{\mathsf{log1p}\left(N\right) - \log N}}\right) \]

      9. +-commutative 90.5%

         \[\leadsto \log \left({\left(\sqrt[3]{\frac{\color{blue}{N + 1}}{N}}\right)}^{2}\right) + \log \left(\sqrt[3]{e^{\mathsf{log1p}\left(N\right) - \log N}}\right) \]

      10. exp-diff 90.5%

          \[\leadsto \log \left({\left(\sqrt[3]{\frac{N + 1}{N}}\right)}^{2}\right) + \log \left(\sqrt[3]{\color{blue}{\frac{e^{\mathsf{log1p}\left(N\right)}}{e^{\log N}}}}\right) \]

      11. log1p-udef 90.5%

          \[\leadsto \log \left({\left(\sqrt[3]{\frac{N + 1}{N}}\right)}^{2}\right) + \log \left(\sqrt[3]{\frac{e^{\color{blue}{\log \left(1 + N\right)}}}{e^{\log N}}}\right) \]

      12. rem-exp-log 91.1%

          \[\leadsto \log \left({\left(\sqrt[3]{\frac{N + 1}{N}}\right)}^{2}\right) + \log \left(\sqrt[3]{\frac{\color{blue}{1 + N}}{e^{\log N}}}\right) \]

      13. add-exp-log 90.7%

          \[\leadsto \log \left({\left(\sqrt[3]{\frac{N + 1}{N}}\right)}^{2}\right) + \log \left(\sqrt[3]{\frac{1 + N}{\color{blue}{N}}}\right) \]

   5. Applied egg-rr 90.7%

      \[\leadsto \color{blue}{\log \left({\left(\sqrt[3]{\frac{N + 1}{N}}\right)}^{2}\right) + \log \left(\sqrt[3]{\frac{N + 1}{N}}\right)} \]
   6. Step-by-step derivation

      1. log-pow 90.6%

         \[\leadsto \color{blue}{2 \cdot \log \left(\sqrt[3]{\frac{N + 1}{N}}\right)} + \log \left(\sqrt[3]{\frac{N + 1}{N}}\right) \]

      2. distribute-lft1-in 90.6%

         \[\leadsto \color{blue}{\left(2 + 1\right) \cdot \log \left(\sqrt[3]{\frac{N + 1}{N}}\right)} \]

      3. metadata-eval 90.6%

         \[\leadsto \color{blue}{3} \cdot \log \left(\sqrt[3]{\frac{N + 1}{N}}\right) \]

   7. Simplified 90.6%

      \[\leadsto \color{blue}{3 \cdot \log \left(\sqrt[3]{\frac{N + 1}{N}}\right)} \]
   8. Step-by-step derivation

      1. *-commutative 90.6%

         \[\leadsto \color{blue}{\log \left(\sqrt[3]{\frac{N + 1}{N}}\right) \cdot 3} \]

      2. add-log-exp 91.0%

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

      3. exp-to-pow 91.0%

         \[\leadsto \log \color{blue}{\left({\left(\sqrt[3]{\frac{N + 1}{N}}\right)}^{3}\right)} \]

      4. pow3 90.9%

         \[\leadsto \log \color{blue}{\left(\left(\sqrt[3]{\frac{N + 1}{N}} \cdot \sqrt[3]{\frac{N + 1}{N}}\right) \cdot \sqrt[3]{\frac{N + 1}{N}}\right)} \]

      5. add-cube-cbrt 93.3%

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

      6. clear-num 93.3%

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

      7. log-div 95.0%

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

      8. metadata-eval 95.0%

         \[\leadsto \color{blue}{0} - \log \left(\frac{N}{N + 1}\right) \]

   9. Applied egg-rr 95.0%

      \[\leadsto \color{blue}{0 - \log \left(\frac{N}{N + 1}\right)} \]
   10. Step-by-step derivation

       1. neg-sub0 95.0%

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

   11. Simplified 95.0%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 0.0006:\\ \;\;\;\;\frac{0.3333333333333333}{{N}^{3}} + \left(\frac{1}{N} - \left(\frac{0.5}{{N}^{2}} + \frac{0.25}{{N}^{4}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-\log \left(\frac{N}{N + 1}\right)\\ \end{array} \]

Alternative 4: 98.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 0.0001:\\ \;\;\;\;\frac{0.3333333333333333}{{N}^{3}} + \left(\frac{1}{N} - \frac{0.5}{{N}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;-\log \left(\frac{N}{N + 1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= (- (log (+ N 1.0)) (log N)) 0.0001)
   (+ (/ 0.3333333333333333 (pow N 3.0)) (- (/ 1.0 N) (/ 0.5 (pow N 2.0))))
   (- (log (/ N (+ N 1.0))))))

C

double code(double N) {
	double tmp;
	if ((log((N + 1.0)) - log(N)) <= 0.0001) {
		tmp = (0.3333333333333333 / pow(N, 3.0)) + ((1.0 / N) - (0.5 / pow(N, 2.0)));
	} else {
		tmp = -log((N / (N + 1.0)));
	}
	return tmp;
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((log((n + 1.0d0)) - log(n)) <= 0.0001d0) then
        tmp = (0.3333333333333333d0 / (n ** 3.0d0)) + ((1.0d0 / n) - (0.5d0 / (n ** 2.0d0)))
    else
        tmp = -log((n / (n + 1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double N) {
	double tmp;
	if ((Math.log((N + 1.0)) - Math.log(N)) <= 0.0001) {
		tmp = (0.3333333333333333 / Math.pow(N, 3.0)) + ((1.0 / N) - (0.5 / Math.pow(N, 2.0)));
	} else {
		tmp = -Math.log((N / (N + 1.0)));
	}
	return tmp;
}

Python

def code(N):
	tmp = 0
	if (math.log((N + 1.0)) - math.log(N)) <= 0.0001:
		tmp = (0.3333333333333333 / math.pow(N, 3.0)) + ((1.0 / N) - (0.5 / math.pow(N, 2.0)))
	else:
		tmp = -math.log((N / (N + 1.0)))
	return tmp

Julia

function code(N)
	tmp = 0.0
	if (Float64(log(Float64(N + 1.0)) - log(N)) <= 0.0001)
		tmp = Float64(Float64(0.3333333333333333 / (N ^ 3.0)) + Float64(Float64(1.0 / N) - Float64(0.5 / (N ^ 2.0))));
	else
		tmp = Float64(-log(Float64(N / Float64(N + 1.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(N)
	tmp = 0.0;
	if ((log((N + 1.0)) - log(N)) <= 0.0001)
		tmp = (0.3333333333333333 / (N ^ 3.0)) + ((1.0 / N) - (0.5 / (N ^ 2.0)));
	else
		tmp = -log((N / (N + 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[N_] := If[LessEqual[N[(N[Log[N[(N + 1.0), $MachinePrecision]], $MachinePrecision] - N[Log[N], $MachinePrecision]), $MachinePrecision], 0.0001], N[(N[(0.3333333333333333 / N[Power[N, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / N), $MachinePrecision] - N[(0.5 / N[Power[N, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-N[Log[N[(N / N[(N + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (log.f64 (+.f64 N 1)) (log.f64 N)) < 1.00000000000000005e-4

   1. Initial program 18.0%

      \[\log \left(N + 1\right) - \log N \]
   2. Step-by-step derivation

      1. +-commutative 18.0%

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

      2. log1p-def 18.0%

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

   3. Simplified 18.0%

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

   4. Taylor expanded in N around inf 99.5%

      \[\leadsto \color{blue}{\left(0.3333333333333333 \cdot \frac{1}{{N}^{3}} + \frac{1}{N}\right) - 0.5 \cdot \frac{1}{{N}^{2}}} \]
   5. Step-by-step derivation

      1. associate--l+ 99.5%

         \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{1}{{N}^{3}} + \left(\frac{1}{N} - 0.5 \cdot \frac{1}{{N}^{2}}\right)} \]

      2. associate-*r/ 99.5%

         \[\leadsto \color{blue}{\frac{0.3333333333333333 \cdot 1}{{N}^{3}}} + \left(\frac{1}{N} - 0.5 \cdot \frac{1}{{N}^{2}}\right) \]

      3. metadata-eval 99.5%

         \[\leadsto \frac{\color{blue}{0.3333333333333333}}{{N}^{3}} + \left(\frac{1}{N} - 0.5 \cdot \frac{1}{{N}^{2}}\right) \]

      4. associate-*r/ 99.5%

         \[\leadsto \frac{0.3333333333333333}{{N}^{3}} + \left(\frac{1}{N} - \color{blue}{\frac{0.5 \cdot 1}{{N}^{2}}}\right) \]

      5. metadata-eval 99.5%

         \[\leadsto \frac{0.3333333333333333}{{N}^{3}} + \left(\frac{1}{N} - \frac{\color{blue}{0.5}}{{N}^{2}}\right) \]

   6. Simplified 99.5%

      \[\leadsto \color{blue}{\frac{0.3333333333333333}{{N}^{3}} + \left(\frac{1}{N} - \frac{0.5}{{N}^{2}}\right)} \]

   if 1.00000000000000005e-4 < (-.f64 (log.f64 (+.f64 N 1)) (log.f64 N))

   1. Initial program 86.8%

      \[\log \left(N + 1\right) - \log N \]
   2. Step-by-step derivation

      1. +-commutative 86.8%

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

      2. log1p-def 86.8%

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

   3. Simplified 86.8%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(N\right) - \log N} \]
   4. Step-by-step derivation

      1. add-log-exp 86.8%

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

      2. add-cube-cbrt 86.6%

         \[\leadsto \log \color{blue}{\left(\left(\sqrt[3]{e^{\mathsf{log1p}\left(N\right) - \log N}} \cdot \sqrt[3]{e^{\mathsf{log1p}\left(N\right) - \log N}}\right) \cdot \sqrt[3]{e^{\mathsf{log1p}\left(N\right) - \log N}}\right)} \]

      3. log-prod 86.4%

         \[\leadsto \color{blue}{\log \left(\sqrt[3]{e^{\mathsf{log1p}\left(N\right) - \log N}} \cdot \sqrt[3]{e^{\mathsf{log1p}\left(N\right) - \log N}}\right) + \log \left(\sqrt[3]{e^{\mathsf{log1p}\left(N\right) - \log N}}\right)} \]

      4. pow2 86.4%

         \[\leadsto \log \color{blue}{\left({\left(\sqrt[3]{e^{\mathsf{log1p}\left(N\right) - \log N}}\right)}^{2}\right)} + \log \left(\sqrt[3]{e^{\mathsf{log1p}\left(N\right) - \log N}}\right) \]

      5. exp-diff 86.4%

         \[\leadsto \log \left({\left(\sqrt[3]{\color{blue}{\frac{e^{\mathsf{log1p}\left(N\right)}}{e^{\log N}}}}\right)}^{2}\right) + \log \left(\sqrt[3]{e^{\mathsf{log1p}\left(N\right) - \log N}}\right) \]

      6. log1p-udef 86.4%

         \[\leadsto \log \left({\left(\sqrt[3]{\frac{e^{\color{blue}{\log \left(1 + N\right)}}}{e^{\log N}}}\right)}^{2}\right) + \log \left(\sqrt[3]{e^{\mathsf{log1p}\left(N\right) - \log N}}\right) \]

      7. rem-exp-log 86.8%

         \[\leadsto \log \left({\left(\sqrt[3]{\frac{\color{blue}{1 + N}}{e^{\log N}}}\right)}^{2}\right) + \log \left(\sqrt[3]{e^{\mathsf{log1p}\left(N\right) - \log N}}\right) \]

      8. add-exp-log 87.5%

         \[\leadsto \log \left({\left(\sqrt[3]{\frac{1 + N}{\color{blue}{N}}}\right)}^{2}\right) + \log \left(\sqrt[3]{e^{\mathsf{log1p}\left(N\right) - \log N}}\right) \]

      9. +-commutative 87.5%

         \[\leadsto \log \left({\left(\sqrt[3]{\frac{\color{blue}{N + 1}}{N}}\right)}^{2}\right) + \log \left(\sqrt[3]{e^{\mathsf{log1p}\left(N\right) - \log N}}\right) \]

      10. exp-diff 87.5%

          \[\leadsto \log \left({\left(\sqrt[3]{\frac{N + 1}{N}}\right)}^{2}\right) + \log \left(\sqrt[3]{\color{blue}{\frac{e^{\mathsf{log1p}\left(N\right)}}{e^{\log N}}}}\right) \]

      11. log1p-udef 87.5%

          \[\leadsto \log \left({\left(\sqrt[3]{\frac{N + 1}{N}}\right)}^{2}\right) + \log \left(\sqrt[3]{\frac{e^{\color{blue}{\log \left(1 + N\right)}}}{e^{\log N}}}\right) \]

      12. rem-exp-log 88.1%

          \[\leadsto \log \left({\left(\sqrt[3]{\frac{N + 1}{N}}\right)}^{2}\right) + \log \left(\sqrt[3]{\frac{\color{blue}{1 + N}}{e^{\log N}}}\right) \]

      13. add-exp-log 87.7%

          \[\leadsto \log \left({\left(\sqrt[3]{\frac{N + 1}{N}}\right)}^{2}\right) + \log \left(\sqrt[3]{\frac{1 + N}{\color{blue}{N}}}\right) \]

   5. Applied egg-rr 87.7%

      \[\leadsto \color{blue}{\log \left({\left(\sqrt[3]{\frac{N + 1}{N}}\right)}^{2}\right) + \log \left(\sqrt[3]{\frac{N + 1}{N}}\right)} \]
   6. Step-by-step derivation

      1. log-pow 87.7%

         \[\leadsto \color{blue}{2 \cdot \log \left(\sqrt[3]{\frac{N + 1}{N}}\right)} + \log \left(\sqrt[3]{\frac{N + 1}{N}}\right) \]

      2. distribute-lft1-in 87.7%

         \[\leadsto \color{blue}{\left(2 + 1\right) \cdot \log \left(\sqrt[3]{\frac{N + 1}{N}}\right)} \]

      3. metadata-eval 87.7%

         \[\leadsto \color{blue}{3} \cdot \log \left(\sqrt[3]{\frac{N + 1}{N}}\right) \]

   7. Simplified 87.7%

      \[\leadsto \color{blue}{3 \cdot \log \left(\sqrt[3]{\frac{N + 1}{N}}\right)} \]
   8. Step-by-step derivation

      1. *-commutative 87.7%

         \[\leadsto \color{blue}{\log \left(\sqrt[3]{\frac{N + 1}{N}}\right) \cdot 3} \]

      2. add-log-exp 87.9%

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

      3. exp-to-pow 87.9%

         \[\leadsto \log \color{blue}{\left({\left(\sqrt[3]{\frac{N + 1}{N}}\right)}^{3}\right)} \]

      4. pow3 87.8%

         \[\leadsto \log \color{blue}{\left(\left(\sqrt[3]{\frac{N + 1}{N}} \cdot \sqrt[3]{\frac{N + 1}{N}}\right) \cdot \sqrt[3]{\frac{N + 1}{N}}\right)} \]

      5. add-cube-cbrt 90.4%

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

      6. clear-num 90.4%

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

      7. log-div 92.3%

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

      8. metadata-eval 92.3%

         \[\leadsto \color{blue}{0} - \log \left(\frac{N}{N + 1}\right) \]

   9. Applied egg-rr 92.3%

      \[\leadsto \color{blue}{0 - \log \left(\frac{N}{N + 1}\right)} \]
   10. Step-by-step derivation

       1. neg-sub0 92.3%

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

   11. Simplified 92.3%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 0.0001:\\ \;\;\;\;\frac{0.3333333333333333}{{N}^{3}} + \left(\frac{1}{N} - \frac{0.5}{{N}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;-\log \left(\frac{N}{N + 1}\right)\\ \end{array} \]

Alternative 5: 98.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 0.0001:\\ \;\;\;\;\frac{1}{N} + \left(\frac{-0.5}{{N}^{2}} + \frac{0.3333333333333333}{{N}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;-\log \left(\frac{N}{N + 1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= (- (log (+ N 1.0)) (log N)) 0.0001)
   (+ (/ 1.0 N) (+ (/ -0.5 (pow N 2.0)) (/ 0.3333333333333333 (pow N 3.0))))
   (- (log (/ N (+ N 1.0))))))

C

double code(double N) {
	double tmp;
	if ((log((N + 1.0)) - log(N)) <= 0.0001) {
		tmp = (1.0 / N) + ((-0.5 / pow(N, 2.0)) + (0.3333333333333333 / pow(N, 3.0)));
	} else {
		tmp = -log((N / (N + 1.0)));
	}
	return tmp;
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((log((n + 1.0d0)) - log(n)) <= 0.0001d0) then
        tmp = (1.0d0 / n) + (((-0.5d0) / (n ** 2.0d0)) + (0.3333333333333333d0 / (n ** 3.0d0)))
    else
        tmp = -log((n / (n + 1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double N) {
	double tmp;
	if ((Math.log((N + 1.0)) - Math.log(N)) <= 0.0001) {
		tmp = (1.0 / N) + ((-0.5 / Math.pow(N, 2.0)) + (0.3333333333333333 / Math.pow(N, 3.0)));
	} else {
		tmp = -Math.log((N / (N + 1.0)));
	}
	return tmp;
}

Python

def code(N):
	tmp = 0
	if (math.log((N + 1.0)) - math.log(N)) <= 0.0001:
		tmp = (1.0 / N) + ((-0.5 / math.pow(N, 2.0)) + (0.3333333333333333 / math.pow(N, 3.0)))
	else:
		tmp = -math.log((N / (N + 1.0)))
	return tmp

Julia

function code(N)
	tmp = 0.0
	if (Float64(log(Float64(N + 1.0)) - log(N)) <= 0.0001)
		tmp = Float64(Float64(1.0 / N) + Float64(Float64(-0.5 / (N ^ 2.0)) + Float64(0.3333333333333333 / (N ^ 3.0))));
	else
		tmp = Float64(-log(Float64(N / Float64(N + 1.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(N)
	tmp = 0.0;
	if ((log((N + 1.0)) - log(N)) <= 0.0001)
		tmp = (1.0 / N) + ((-0.5 / (N ^ 2.0)) + (0.3333333333333333 / (N ^ 3.0)));
	else
		tmp = -log((N / (N + 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[N_] := If[LessEqual[N[(N[Log[N[(N + 1.0), $MachinePrecision]], $MachinePrecision] - N[Log[N], $MachinePrecision]), $MachinePrecision], 0.0001], N[(N[(1.0 / N), $MachinePrecision] + N[(N[(-0.5 / N[Power[N, 2.0], $MachinePrecision]), $MachinePrecision] + N[(0.3333333333333333 / N[Power[N, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-N[Log[N[(N / N[(N + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (log.f64 (+.f64 N 1)) (log.f64 N)) < 1.00000000000000005e-4

   1. Initial program 18.0%

      \[\log \left(N + 1\right) - \log N \]
   2. Step-by-step derivation

      1. +-commutative 18.0%

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

      2. log1p-def 18.0%

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

   3. Simplified 18.0%

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

   4. Taylor expanded in N around inf 99.5%

      \[\leadsto \color{blue}{\left(0.3333333333333333 \cdot \frac{1}{{N}^{3}} + \frac{1}{N}\right) - 0.5 \cdot \frac{1}{{N}^{2}}} \]
   5. Step-by-step derivation

      1. sub-neg 99.5%

         \[\leadsto \color{blue}{\left(0.3333333333333333 \cdot \frac{1}{{N}^{3}} + \frac{1}{N}\right) + \left(-0.5 \cdot \frac{1}{{N}^{2}}\right)} \]

      2. +-commutative 99.5%

         \[\leadsto \color{blue}{\left(\frac{1}{N} + 0.3333333333333333 \cdot \frac{1}{{N}^{3}}\right)} + \left(-0.5 \cdot \frac{1}{{N}^{2}}\right) \]

      3. associate-+l+ 99.6%

         \[\leadsto \color{blue}{\frac{1}{N} + \left(0.3333333333333333 \cdot \frac{1}{{N}^{3}} + \left(-0.5 \cdot \frac{1}{{N}^{2}}\right)\right)} \]

      4. associate-*r/ 99.6%

         \[\leadsto \frac{1}{N} + \left(\color{blue}{\frac{0.3333333333333333 \cdot 1}{{N}^{3}}} + \left(-0.5 \cdot \frac{1}{{N}^{2}}\right)\right) \]

      5. metadata-eval 99.6%

         \[\leadsto \frac{1}{N} + \left(\frac{\color{blue}{0.3333333333333333}}{{N}^{3}} + \left(-0.5 \cdot \frac{1}{{N}^{2}}\right)\right) \]

      6. associate-*r/ 99.6%

         \[\leadsto \frac{1}{N} + \left(\frac{0.3333333333333333}{{N}^{3}} + \left(-\color{blue}{\frac{0.5 \cdot 1}{{N}^{2}}}\right)\right) \]

      7. metadata-eval 99.6%

         \[\leadsto \frac{1}{N} + \left(\frac{0.3333333333333333}{{N}^{3}} + \left(-\frac{\color{blue}{0.5}}{{N}^{2}}\right)\right) \]

      8. distribute-neg-frac 99.6%

         \[\leadsto \frac{1}{N} + \left(\frac{0.3333333333333333}{{N}^{3}} + \color{blue}{\frac{-0.5}{{N}^{2}}}\right) \]

      9. metadata-eval 99.6%

         \[\leadsto \frac{1}{N} + \left(\frac{0.3333333333333333}{{N}^{3}} + \frac{\color{blue}{-0.5}}{{N}^{2}}\right) \]

   6. Simplified 99.6%

      \[\leadsto \color{blue}{\frac{1}{N} + \left(\frac{0.3333333333333333}{{N}^{3}} + \frac{-0.5}{{N}^{2}}\right)} \]

   if 1.00000000000000005e-4 < (-.f64 (log.f64 (+.f64 N 1)) (log.f64 N))

   1. Initial program 86.8%

      \[\log \left(N + 1\right) - \log N \]
   2. Step-by-step derivation

      1. +-commutative 86.8%

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

      2. log1p-def 86.8%

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

   3. Simplified 86.8%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(N\right) - \log N} \]
   4. Step-by-step derivation

      1. add-log-exp 86.8%

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

      2. add-cube-cbrt 86.6%

         \[\leadsto \log \color{blue}{\left(\left(\sqrt[3]{e^{\mathsf{log1p}\left(N\right) - \log N}} \cdot \sqrt[3]{e^{\mathsf{log1p}\left(N\right) - \log N}}\right) \cdot \sqrt[3]{e^{\mathsf{log1p}\left(N\right) - \log N}}\right)} \]

      3. log-prod 86.4%

         \[\leadsto \color{blue}{\log \left(\sqrt[3]{e^{\mathsf{log1p}\left(N\right) - \log N}} \cdot \sqrt[3]{e^{\mathsf{log1p}\left(N\right) - \log N}}\right) + \log \left(\sqrt[3]{e^{\mathsf{log1p}\left(N\right) - \log N}}\right)} \]

      4. pow2 86.4%

         \[\leadsto \log \color{blue}{\left({\left(\sqrt[3]{e^{\mathsf{log1p}\left(N\right) - \log N}}\right)}^{2}\right)} + \log \left(\sqrt[3]{e^{\mathsf{log1p}\left(N\right) - \log N}}\right) \]

      5. exp-diff 86.4%

         \[\leadsto \log \left({\left(\sqrt[3]{\color{blue}{\frac{e^{\mathsf{log1p}\left(N\right)}}{e^{\log N}}}}\right)}^{2}\right) + \log \left(\sqrt[3]{e^{\mathsf{log1p}\left(N\right) - \log N}}\right) \]

      6. log1p-udef 86.4%

         \[\leadsto \log \left({\left(\sqrt[3]{\frac{e^{\color{blue}{\log \left(1 + N\right)}}}{e^{\log N}}}\right)}^{2}\right) + \log \left(\sqrt[3]{e^{\mathsf{log1p}\left(N\right) - \log N}}\right) \]

      7. rem-exp-log 86.8%

         \[\leadsto \log \left({\left(\sqrt[3]{\frac{\color{blue}{1 + N}}{e^{\log N}}}\right)}^{2}\right) + \log \left(\sqrt[3]{e^{\mathsf{log1p}\left(N\right) - \log N}}\right) \]

      8. add-exp-log 87.5%

         \[\leadsto \log \left({\left(\sqrt[3]{\frac{1 + N}{\color{blue}{N}}}\right)}^{2}\right) + \log \left(\sqrt[3]{e^{\mathsf{log1p}\left(N\right) - \log N}}\right) \]

      9. +-commutative 87.5%

         \[\leadsto \log \left({\left(\sqrt[3]{\frac{\color{blue}{N + 1}}{N}}\right)}^{2}\right) + \log \left(\sqrt[3]{e^{\mathsf{log1p}\left(N\right) - \log N}}\right) \]

      10. exp-diff 87.5%

          \[\leadsto \log \left({\left(\sqrt[3]{\frac{N + 1}{N}}\right)}^{2}\right) + \log \left(\sqrt[3]{\color{blue}{\frac{e^{\mathsf{log1p}\left(N\right)}}{e^{\log N}}}}\right) \]

      11. log1p-udef 87.5%

          \[\leadsto \log \left({\left(\sqrt[3]{\frac{N + 1}{N}}\right)}^{2}\right) + \log \left(\sqrt[3]{\frac{e^{\color{blue}{\log \left(1 + N\right)}}}{e^{\log N}}}\right) \]

      12. rem-exp-log 88.1%

          \[\leadsto \log \left({\left(\sqrt[3]{\frac{N + 1}{N}}\right)}^{2}\right) + \log \left(\sqrt[3]{\frac{\color{blue}{1 + N}}{e^{\log N}}}\right) \]

      13. add-exp-log 87.7%

          \[\leadsto \log \left({\left(\sqrt[3]{\frac{N + 1}{N}}\right)}^{2}\right) + \log \left(\sqrt[3]{\frac{1 + N}{\color{blue}{N}}}\right) \]

   5. Applied egg-rr 87.7%

      \[\leadsto \color{blue}{\log \left({\left(\sqrt[3]{\frac{N + 1}{N}}\right)}^{2}\right) + \log \left(\sqrt[3]{\frac{N + 1}{N}}\right)} \]
   6. Step-by-step derivation

      1. log-pow 87.7%

         \[\leadsto \color{blue}{2 \cdot \log \left(\sqrt[3]{\frac{N + 1}{N}}\right)} + \log \left(\sqrt[3]{\frac{N + 1}{N}}\right) \]

      2. distribute-lft1-in 87.7%

         \[\leadsto \color{blue}{\left(2 + 1\right) \cdot \log \left(\sqrt[3]{\frac{N + 1}{N}}\right)} \]

      3. metadata-eval 87.7%

         \[\leadsto \color{blue}{3} \cdot \log \left(\sqrt[3]{\frac{N + 1}{N}}\right) \]

   7. Simplified 87.7%

      \[\leadsto \color{blue}{3 \cdot \log \left(\sqrt[3]{\frac{N + 1}{N}}\right)} \]
   8. Step-by-step derivation

      1. *-commutative 87.7%

         \[\leadsto \color{blue}{\log \left(\sqrt[3]{\frac{N + 1}{N}}\right) \cdot 3} \]

      2. add-log-exp 87.9%

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

      3. exp-to-pow 87.9%

         \[\leadsto \log \color{blue}{\left({\left(\sqrt[3]{\frac{N + 1}{N}}\right)}^{3}\right)} \]

      4. pow3 87.8%

         \[\leadsto \log \color{blue}{\left(\left(\sqrt[3]{\frac{N + 1}{N}} \cdot \sqrt[3]{\frac{N + 1}{N}}\right) \cdot \sqrt[3]{\frac{N + 1}{N}}\right)} \]

      5. add-cube-cbrt 90.4%

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

      6. clear-num 90.4%

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

      7. log-div 92.3%

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

      8. metadata-eval 92.3%

         \[\leadsto \color{blue}{0} - \log \left(\frac{N}{N + 1}\right) \]

   9. Applied egg-rr 92.3%

      \[\leadsto \color{blue}{0 - \log \left(\frac{N}{N + 1}\right)} \]
   10. Step-by-step derivation

       1. neg-sub0 92.3%

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

   11. Simplified 92.3%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 0.0001:\\ \;\;\;\;\frac{1}{N} + \left(\frac{-0.5}{{N}^{2}} + \frac{0.3333333333333333}{{N}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;-\log \left(\frac{N}{N + 1}\right)\\ \end{array} \]

Alternative 6: 97.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\frac{1}{N} - \frac{0.5}{{N}^{2}}\\ \mathbf{else}:\\ \;\;\;\;-\log \left(\frac{N}{N + 1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= (- (log (+ N 1.0)) (log N)) 5e-6)
   (- (/ 1.0 N) (/ 0.5 (pow N 2.0)))
   (- (log (/ N (+ N 1.0))))))

C

double code(double N) {
	double tmp;
	if ((log((N + 1.0)) - log(N)) <= 5e-6) {
		tmp = (1.0 / N) - (0.5 / pow(N, 2.0));
	} else {
		tmp = -log((N / (N + 1.0)));
	}
	return tmp;
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((log((n + 1.0d0)) - log(n)) <= 5d-6) then
        tmp = (1.0d0 / n) - (0.5d0 / (n ** 2.0d0))
    else
        tmp = -log((n / (n + 1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double N) {
	double tmp;
	if ((Math.log((N + 1.0)) - Math.log(N)) <= 5e-6) {
		tmp = (1.0 / N) - (0.5 / Math.pow(N, 2.0));
	} else {
		tmp = -Math.log((N / (N + 1.0)));
	}
	return tmp;
}

Python

def code(N):
	tmp = 0
	if (math.log((N + 1.0)) - math.log(N)) <= 5e-6:
		tmp = (1.0 / N) - (0.5 / math.pow(N, 2.0))
	else:
		tmp = -math.log((N / (N + 1.0)))
	return tmp

Julia

function code(N)
	tmp = 0.0
	if (Float64(log(Float64(N + 1.0)) - log(N)) <= 5e-6)
		tmp = Float64(Float64(1.0 / N) - Float64(0.5 / (N ^ 2.0)));
	else
		tmp = Float64(-log(Float64(N / Float64(N + 1.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(N)
	tmp = 0.0;
	if ((log((N + 1.0)) - log(N)) <= 5e-6)
		tmp = (1.0 / N) - (0.5 / (N ^ 2.0));
	else
		tmp = -log((N / (N + 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[N_] := If[LessEqual[N[(N[Log[N[(N + 1.0), $MachinePrecision]], $MachinePrecision] - N[Log[N], $MachinePrecision]), $MachinePrecision], 5e-6], N[(N[(1.0 / N), $MachinePrecision] - N[(0.5 / N[Power[N, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-N[Log[N[(N / N[(N + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (log.f64 (+.f64 N 1)) (log.f64 N)) < 5.00000000000000041e-6

   1. Initial program 15.2%

      \[\log \left(N + 1\right) - \log N \]
   2. Step-by-step derivation

      1. +-commutative 15.2%

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

      2. log1p-def 15.2%

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

   3. Simplified 15.2%

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

   4. Taylor expanded in N around inf 99.0%

      \[\leadsto \color{blue}{\frac{1}{N} - 0.5 \cdot \frac{1}{{N}^{2}}} \]
   5. Step-by-step derivation

      1. associate-*r/ 99.0%

         \[\leadsto \frac{1}{N} - \color{blue}{\frac{0.5 \cdot 1}{{N}^{2}}} \]

      2. metadata-eval 99.0%

         \[\leadsto \frac{1}{N} - \frac{\color{blue}{0.5}}{{N}^{2}} \]

   6. Simplified 99.0%

      \[\leadsto \color{blue}{\frac{1}{N} - \frac{0.5}{{N}^{2}}} \]

   if 5.00000000000000041e-6 < (-.f64 (log.f64 (+.f64 N 1)) (log.f64 N))

   1. Initial program 82.8%

      \[\log \left(N + 1\right) - \log N \]
   2. Step-by-step derivation

      1. +-commutative 82.8%

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

      2. log1p-def 82.8%

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

   3. Simplified 82.8%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(N\right) - \log N} \]
   4. Step-by-step derivation

      1. add-log-exp 82.9%

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

      2. add-cube-cbrt 82.6%

         \[\leadsto \log \color{blue}{\left(\left(\sqrt[3]{e^{\mathsf{log1p}\left(N\right) - \log N}} \cdot \sqrt[3]{e^{\mathsf{log1p}\left(N\right) - \log N}}\right) \cdot \sqrt[3]{e^{\mathsf{log1p}\left(N\right) - \log N}}\right)} \]

      3. log-prod 82.5%

         \[\leadsto \color{blue}{\log \left(\sqrt[3]{e^{\mathsf{log1p}\left(N\right) - \log N}} \cdot \sqrt[3]{e^{\mathsf{log1p}\left(N\right) - \log N}}\right) + \log \left(\sqrt[3]{e^{\mathsf{log1p}\left(N\right) - \log N}}\right)} \]

      4. pow2 82.5%

         \[\leadsto \log \color{blue}{\left({\left(\sqrt[3]{e^{\mathsf{log1p}\left(N\right) - \log N}}\right)}^{2}\right)} + \log \left(\sqrt[3]{e^{\mathsf{log1p}\left(N\right) - \log N}}\right) \]

      5. exp-diff 82.7%

         \[\leadsto \log \left({\left(\sqrt[3]{\color{blue}{\frac{e^{\mathsf{log1p}\left(N\right)}}{e^{\log N}}}}\right)}^{2}\right) + \log \left(\sqrt[3]{e^{\mathsf{log1p}\left(N\right) - \log N}}\right) \]

      6. log1p-udef 82.7%

         \[\leadsto \log \left({\left(\sqrt[3]{\frac{e^{\color{blue}{\log \left(1 + N\right)}}}{e^{\log N}}}\right)}^{2}\right) + \log \left(\sqrt[3]{e^{\mathsf{log1p}\left(N\right) - \log N}}\right) \]

      7. rem-exp-log 82.8%

         \[\leadsto \log \left({\left(\sqrt[3]{\frac{\color{blue}{1 + N}}{e^{\log N}}}\right)}^{2}\right) + \log \left(\sqrt[3]{e^{\mathsf{log1p}\left(N\right) - \log N}}\right) \]

      8. add-exp-log 83.4%

         \[\leadsto \log \left({\left(\sqrt[3]{\frac{1 + N}{\color{blue}{N}}}\right)}^{2}\right) + \log \left(\sqrt[3]{e^{\mathsf{log1p}\left(N\right) - \log N}}\right) \]

      9. +-commutative 83.4%

         \[\leadsto \log \left({\left(\sqrt[3]{\frac{\color{blue}{N + 1}}{N}}\right)}^{2}\right) + \log \left(\sqrt[3]{e^{\mathsf{log1p}\left(N\right) - \log N}}\right) \]

      10. exp-diff 83.5%

          \[\leadsto \log \left({\left(\sqrt[3]{\frac{N + 1}{N}}\right)}^{2}\right) + \log \left(\sqrt[3]{\color{blue}{\frac{e^{\mathsf{log1p}\left(N\right)}}{e^{\log N}}}}\right) \]

      11. log1p-udef 83.5%

          \[\leadsto \log \left({\left(\sqrt[3]{\frac{N + 1}{N}}\right)}^{2}\right) + \log \left(\sqrt[3]{\frac{e^{\color{blue}{\log \left(1 + N\right)}}}{e^{\log N}}}\right) \]

      12. rem-exp-log 84.1%

          \[\leadsto \log \left({\left(\sqrt[3]{\frac{N + 1}{N}}\right)}^{2}\right) + \log \left(\sqrt[3]{\frac{\color{blue}{1 + N}}{e^{\log N}}}\right) \]

      13. add-exp-log 83.5%

          \[\leadsto \log \left({\left(\sqrt[3]{\frac{N + 1}{N}}\right)}^{2}\right) + \log \left(\sqrt[3]{\frac{1 + N}{\color{blue}{N}}}\right) \]

   5. Applied egg-rr 83.5%

      \[\leadsto \color{blue}{\log \left({\left(\sqrt[3]{\frac{N + 1}{N}}\right)}^{2}\right) + \log \left(\sqrt[3]{\frac{N + 1}{N}}\right)} \]
   6. Step-by-step derivation

      1. log-pow 83.6%

         \[\leadsto \color{blue}{2 \cdot \log \left(\sqrt[3]{\frac{N + 1}{N}}\right)} + \log \left(\sqrt[3]{\frac{N + 1}{N}}\right) \]

      2. distribute-lft1-in 83.6%

         \[\leadsto \color{blue}{\left(2 + 1\right) \cdot \log \left(\sqrt[3]{\frac{N + 1}{N}}\right)} \]

      3. metadata-eval 83.6%

         \[\leadsto \color{blue}{3} \cdot \log \left(\sqrt[3]{\frac{N + 1}{N}}\right) \]

   7. Simplified 83.6%

      \[\leadsto \color{blue}{3 \cdot \log \left(\sqrt[3]{\frac{N + 1}{N}}\right)} \]
   8. Step-by-step derivation

      1. *-commutative 83.6%

         \[\leadsto \color{blue}{\log \left(\sqrt[3]{\frac{N + 1}{N}}\right) \cdot 3} \]

      2. add-log-exp 83.7%

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

      3. exp-to-pow 83.7%

         \[\leadsto \log \color{blue}{\left({\left(\sqrt[3]{\frac{N + 1}{N}}\right)}^{3}\right)} \]

      4. pow3 83.5%

         \[\leadsto \log \color{blue}{\left(\left(\sqrt[3]{\frac{N + 1}{N}} \cdot \sqrt[3]{\frac{N + 1}{N}}\right) \cdot \sqrt[3]{\frac{N + 1}{N}}\right)} \]

      5. add-cube-cbrt 86.4%

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

      6. clear-num 86.4%

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

      7. log-div 88.6%

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

      8. metadata-eval 88.6%

         \[\leadsto \color{blue}{0} - \log \left(\frac{N}{N + 1}\right) \]

   9. Applied egg-rr 88.6%

      \[\leadsto \color{blue}{0 - \log \left(\frac{N}{N + 1}\right)} \]
   10. Step-by-step derivation

       1. neg-sub0 88.6%

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

   11. Simplified 88.6%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\frac{1}{N} - \frac{0.5}{{N}^{2}}\\ \mathbf{else}:\\ \;\;\;\;-\log \left(\frac{N}{N + 1}\right)\\ \end{array} \]

Alternative 7: 92.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;N \leq 105000000:\\ \;\;\;\;-\log \left(\frac{N}{N + 1}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N}\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= N 105000000.0) (- (log (/ N (+ N 1.0)))) (/ 1.0 N)))

C

double code(double N) {
	double tmp;
	if (N <= 105000000.0) {
		tmp = -log((N / (N + 1.0)));
	} else {
		tmp = 1.0 / N;
	}
	return tmp;
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= 105000000.0d0) then
        tmp = -log((n / (n + 1.0d0)))
    else
        tmp = 1.0d0 / n
    end if
    code = tmp
end function

Java

public static double code(double N) {
	double tmp;
	if (N <= 105000000.0) {
		tmp = -Math.log((N / (N + 1.0)));
	} else {
		tmp = 1.0 / N;
	}
	return tmp;
}

Python

def code(N):
	tmp = 0
	if N <= 105000000.0:
		tmp = -math.log((N / (N + 1.0)))
	else:
		tmp = 1.0 / N
	return tmp

Julia

function code(N)
	tmp = 0.0
	if (N <= 105000000.0)
		tmp = Float64(-log(Float64(N / Float64(N + 1.0))));
	else
		tmp = Float64(1.0 / N);
	end
	return tmp
end

MATLAB

function tmp_2 = code(N)
	tmp = 0.0;
	if (N <= 105000000.0)
		tmp = -log((N / (N + 1.0)));
	else
		tmp = 1.0 / N;
	end
	tmp_2 = tmp;
end

Wolfram

code[N_] := If[LessEqual[N, 105000000.0], (-N[Log[N[(N / N[(N + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), N[(1.0 / N), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if N < 1.05e8

   1. Initial program 73.6%

      \[\log \left(N + 1\right) - \log N \]
   2. Step-by-step derivation

      1. +-commutative 73.6%

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

      2. log1p-def 73.6%

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

   3. Simplified 73.6%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(N\right) - \log N} \]
   4. Step-by-step derivation

      1. add-log-exp 73.5%

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

      2. add-cube-cbrt 73.4%

         \[\leadsto \log \color{blue}{\left(\left(\sqrt[3]{e^{\mathsf{log1p}\left(N\right) - \log N}} \cdot \sqrt[3]{e^{\mathsf{log1p}\left(N\right) - \log N}}\right) \cdot \sqrt[3]{e^{\mathsf{log1p}\left(N\right) - \log N}}\right)} \]

      3. log-prod 73.2%

         \[\leadsto \color{blue}{\log \left(\sqrt[3]{e^{\mathsf{log1p}\left(N\right) - \log N}} \cdot \sqrt[3]{e^{\mathsf{log1p}\left(N\right) - \log N}}\right) + \log \left(\sqrt[3]{e^{\mathsf{log1p}\left(N\right) - \log N}}\right)} \]

      4. pow2 73.2%

         \[\leadsto \log \color{blue}{\left({\left(\sqrt[3]{e^{\mathsf{log1p}\left(N\right) - \log N}}\right)}^{2}\right)} + \log \left(\sqrt[3]{e^{\mathsf{log1p}\left(N\right) - \log N}}\right) \]

      5. exp-diff 73.2%

         \[\leadsto \log \left({\left(\sqrt[3]{\color{blue}{\frac{e^{\mathsf{log1p}\left(N\right)}}{e^{\log N}}}}\right)}^{2}\right) + \log \left(\sqrt[3]{e^{\mathsf{log1p}\left(N\right) - \log N}}\right) \]

      6. log1p-udef 73.2%

         \[\leadsto \log \left({\left(\sqrt[3]{\frac{e^{\color{blue}{\log \left(1 + N\right)}}}{e^{\log N}}}\right)}^{2}\right) + \log \left(\sqrt[3]{e^{\mathsf{log1p}\left(N\right) - \log N}}\right) \]

      7. rem-exp-log 73.7%

         \[\leadsto \log \left({\left(\sqrt[3]{\frac{\color{blue}{1 + N}}{e^{\log N}}}\right)}^{2}\right) + \log \left(\sqrt[3]{e^{\mathsf{log1p}\left(N\right) - \log N}}\right) \]

      8. add-exp-log 74.7%

         \[\leadsto \log \left({\left(\sqrt[3]{\frac{1 + N}{\color{blue}{N}}}\right)}^{2}\right) + \log \left(\sqrt[3]{e^{\mathsf{log1p}\left(N\right) - \log N}}\right) \]

      9. +-commutative 74.7%

         \[\leadsto \log \left({\left(\sqrt[3]{\frac{\color{blue}{N + 1}}{N}}\right)}^{2}\right) + \log \left(\sqrt[3]{e^{\mathsf{log1p}\left(N\right) - \log N}}\right) \]

      10. exp-diff 74.6%

          \[\leadsto \log \left({\left(\sqrt[3]{\frac{N + 1}{N}}\right)}^{2}\right) + \log \left(\sqrt[3]{\color{blue}{\frac{e^{\mathsf{log1p}\left(N\right)}}{e^{\log N}}}}\right) \]

      11. log1p-udef 74.6%

          \[\leadsto \log \left({\left(\sqrt[3]{\frac{N + 1}{N}}\right)}^{2}\right) + \log \left(\sqrt[3]{\frac{e^{\color{blue}{\log \left(1 + N\right)}}}{e^{\log N}}}\right) \]

      12. rem-exp-log 75.1%

          \[\leadsto \log \left({\left(\sqrt[3]{\frac{N + 1}{N}}\right)}^{2}\right) + \log \left(\sqrt[3]{\frac{\color{blue}{1 + N}}{e^{\log N}}}\right) \]

      13. add-exp-log 74.7%

          \[\leadsto \log \left({\left(\sqrt[3]{\frac{N + 1}{N}}\right)}^{2}\right) + \log \left(\sqrt[3]{\frac{1 + N}{\color{blue}{N}}}\right) \]

   5. Applied egg-rr 74.7%

      \[\leadsto \color{blue}{\log \left({\left(\sqrt[3]{\frac{N + 1}{N}}\right)}^{2}\right) + \log \left(\sqrt[3]{\frac{N + 1}{N}}\right)} \]
   6. Step-by-step derivation

      1. log-pow 75.0%

         \[\leadsto \color{blue}{2 \cdot \log \left(\sqrt[3]{\frac{N + 1}{N}}\right)} + \log \left(\sqrt[3]{\frac{N + 1}{N}}\right) \]

      2. distribute-lft1-in 75.0%

         \[\leadsto \color{blue}{\left(2 + 1\right) \cdot \log \left(\sqrt[3]{\frac{N + 1}{N}}\right)} \]

      3. metadata-eval 75.0%

         \[\leadsto \color{blue}{3} \cdot \log \left(\sqrt[3]{\frac{N + 1}{N}}\right) \]

   7. Simplified 75.0%

      \[\leadsto \color{blue}{3 \cdot \log \left(\sqrt[3]{\frac{N + 1}{N}}\right)} \]
   8. Step-by-step derivation

      1. *-commutative 75.0%

         \[\leadsto \color{blue}{\log \left(\sqrt[3]{\frac{N + 1}{N}}\right) \cdot 3} \]

      2. add-log-exp 74.8%

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

      3. exp-to-pow 74.8%

         \[\leadsto \log \color{blue}{\left({\left(\sqrt[3]{\frac{N + 1}{N}}\right)}^{3}\right)} \]

      4. pow3 74.6%

         \[\leadsto \log \color{blue}{\left(\left(\sqrt[3]{\frac{N + 1}{N}} \cdot \sqrt[3]{\frac{N + 1}{N}}\right) \cdot \sqrt[3]{\frac{N + 1}{N}}\right)} \]

      5. add-cube-cbrt 78.1%

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

      6. clear-num 78.1%

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

      7. log-div 79.9%

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

      8. metadata-eval 79.9%

         \[\leadsto \color{blue}{0} - \log \left(\frac{N}{N + 1}\right) \]

   9. Applied egg-rr 79.9%

      \[\leadsto \color{blue}{0 - \log \left(\frac{N}{N + 1}\right)} \]
   10. Step-by-step derivation

       1. neg-sub0 79.9%

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

   11. Simplified 79.9%

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

   if 1.05e8 < N

   1. Initial program 9.4%

      \[\log \left(N + 1\right) - \log N \]
   2. Step-by-step derivation

      1. +-commutative 9.4%

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

      2. log1p-def 9.4%

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

   3. Simplified 9.4%

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

   4. Taylor expanded in N around inf 95.9%

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

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;N \leq 105000000:\\ \;\;\;\;-\log \left(\frac{N}{N + 1}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N}\\ \end{array} \]

Alternative 8: 92.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;N \leq 106000000:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N}\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= N 106000000.0) (log (/ (+ N 1.0) N)) (/ 1.0 N)))

C

double code(double N) {
	double tmp;
	if (N <= 106000000.0) {
		tmp = log(((N + 1.0) / N));
	} else {
		tmp = 1.0 / N;
	}
	return tmp;
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= 106000000.0d0) then
        tmp = log(((n + 1.0d0) / n))
    else
        tmp = 1.0d0 / n
    end if
    code = tmp
end function

Java

public static double code(double N) {
	double tmp;
	if (N <= 106000000.0) {
		tmp = Math.log(((N + 1.0) / N));
	} else {
		tmp = 1.0 / N;
	}
	return tmp;
}

Python

def code(N):
	tmp = 0
	if N <= 106000000.0:
		tmp = math.log(((N + 1.0) / N))
	else:
		tmp = 1.0 / N
	return tmp

Julia

function code(N)
	tmp = 0.0
	if (N <= 106000000.0)
		tmp = log(Float64(Float64(N + 1.0) / N));
	else
		tmp = Float64(1.0 / N);
	end
	return tmp
end

MATLAB

function tmp_2 = code(N)
	tmp = 0.0;
	if (N <= 106000000.0)
		tmp = log(((N + 1.0) / N));
	else
		tmp = 1.0 / N;
	end
	tmp_2 = tmp;
end

Wolfram

code[N_] := If[LessEqual[N, 106000000.0], N[Log[N[(N[(N + 1.0), $MachinePrecision] / N), $MachinePrecision]], $MachinePrecision], N[(1.0 / N), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if N < 1.06e8

   1. Initial program 73.2%

      \[\log \left(N + 1\right) - \log N \]
   2. Step-by-step derivation

      1. +-commutative 73.2%

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

      2. log1p-def 73.2%

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

   3. Simplified 73.2%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(N\right) - \log N} \]
   4. Step-by-step derivation

      1. add-log-exp 73.2%

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

      2. log1p-expm1-u 73.2%

         \[\leadsto \log \left(e^{\mathsf{log1p}\left(N\right)}\right) - \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log N\right)\right)} \]

      3. log1p-udef 73.2%

         \[\leadsto \log \left(e^{\mathsf{log1p}\left(N\right)}\right) - \color{blue}{\log \left(1 + \mathsf{expm1}\left(\log N\right)\right)} \]

      4. diff-log 73.1%

         \[\leadsto \color{blue}{\log \left(\frac{e^{\mathsf{log1p}\left(N\right)}}{1 + \mathsf{expm1}\left(\log N\right)}\right)} \]

      5. log1p-udef 73.1%

         \[\leadsto \log \left(\frac{e^{\color{blue}{\log \left(1 + N\right)}}}{1 + \mathsf{expm1}\left(\log N\right)}\right) \]

      6. rem-exp-log 73.8%

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

      7. +-commutative 73.8%

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

      8. add-exp-log 73.8%

         \[\leadsto \log \left(\frac{N + 1}{\color{blue}{e^{\log \left(1 + \mathsf{expm1}\left(\log N\right)\right)}}}\right) \]

      9. log1p-udef 73.7%

         \[\leadsto \log \left(\frac{N + 1}{e^{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log N\right)\right)}}}\right) \]

      10. log1p-expm1-u 73.8%

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

      11. add-exp-log 77.8%

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

   5. Applied egg-rr 77.8%

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

   if 1.06e8 < N

   1. Initial program 9.1%

      \[\log \left(N + 1\right) - \log N \]
   2. Step-by-step derivation

      1. +-commutative 9.1%

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

      2. log1p-def 9.2%

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

   3. Simplified 9.2%

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

   4. Taylor expanded in N around inf 96.1%

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

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;N \leq 106000000:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N}\\ \end{array} \]

Alternative 9: 84.0% accurate, 68.3× speedup

Math

\[\begin{array}{l} \\ \frac{1}{N} \end{array} \]

FPCore

(FPCore (N) :precision binary64 (/ 1.0 N))

C

double code(double N) {
	return 1.0 / N;
}

Fortran

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

Java

public static double code(double N) {
	return 1.0 / N;
}

Python

def code(N):
	return 1.0 / N

Julia

function code(N)
	return Float64(1.0 / N)
end

MATLAB

function tmp = code(N)
	tmp = 1.0 / N;
end

Wolfram

code[N_] := N[(1.0 / N), $MachinePrecision]

Derivation

1. Initial program 23.9%

   \[\log \left(N + 1\right) - \log N \]
2. Step-by-step derivation

   1. +-commutative 23.9%

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

   2. log1p-def 23.9%

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

3. Simplified 23.9%

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

4. Taylor expanded in N around inf 84.2%

   \[\leadsto \color{blue}{\frac{1}{N}} \]

5. Final simplification 84.2%

   \[\leadsto \frac{1}{N} \]

Developer target: 99.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;N \geq 1000:\\ \;\;\;\;\left(\left(\frac{1}{N} + \frac{-1}{2 \cdot {N}^{2}}\right) + \frac{1}{3 \cdot {N}^{3}}\right) + \frac{-1}{4 \cdot {N}^{4}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(1 + \frac{1}{N}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (>= N 1000.0)
   (+
    (+ (+ (/ 1.0 N) (/ -1.0 (* 2.0 (pow N 2.0)))) (/ 1.0 (* 3.0 (pow N 3.0))))
    (/ -1.0 (* 4.0 (pow N 4.0))))
   (log (+ 1.0 (/ 1.0 N)))))

C

double code(double N) {
	double tmp;
	if (N >= 1000.0) {
		tmp = (((1.0 / N) + (-1.0 / (2.0 * pow(N, 2.0)))) + (1.0 / (3.0 * pow(N, 3.0)))) + (-1.0 / (4.0 * pow(N, 4.0)));
	} else {
		tmp = log((1.0 + (1.0 / N)));
	}
	return tmp;
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n >= 1000.0d0) then
        tmp = (((1.0d0 / n) + ((-1.0d0) / (2.0d0 * (n ** 2.0d0)))) + (1.0d0 / (3.0d0 * (n ** 3.0d0)))) + ((-1.0d0) / (4.0d0 * (n ** 4.0d0)))
    else
        tmp = log((1.0d0 + (1.0d0 / n)))
    end if
    code = tmp
end function

Java

public static double code(double N) {
	double tmp;
	if (N >= 1000.0) {
		tmp = (((1.0 / N) + (-1.0 / (2.0 * Math.pow(N, 2.0)))) + (1.0 / (3.0 * Math.pow(N, 3.0)))) + (-1.0 / (4.0 * Math.pow(N, 4.0)));
	} else {
		tmp = Math.log((1.0 + (1.0 / N)));
	}
	return tmp;
}

Python

def code(N):
	tmp = 0
	if N >= 1000.0:
		tmp = (((1.0 / N) + (-1.0 / (2.0 * math.pow(N, 2.0)))) + (1.0 / (3.0 * math.pow(N, 3.0)))) + (-1.0 / (4.0 * math.pow(N, 4.0)))
	else:
		tmp = math.log((1.0 + (1.0 / N)))
	return tmp

Julia

function code(N)
	tmp = 0.0
	if (N >= 1000.0)
		tmp = Float64(Float64(Float64(Float64(1.0 / N) + Float64(-1.0 / Float64(2.0 * (N ^ 2.0)))) + Float64(1.0 / Float64(3.0 * (N ^ 3.0)))) + Float64(-1.0 / Float64(4.0 * (N ^ 4.0))));
	else
		tmp = log(Float64(1.0 + Float64(1.0 / N)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(N)
	tmp = 0.0;
	if (N >= 1000.0)
		tmp = (((1.0 / N) + (-1.0 / (2.0 * (N ^ 2.0)))) + (1.0 / (3.0 * (N ^ 3.0)))) + (-1.0 / (4.0 * (N ^ 4.0)));
	else
		tmp = log((1.0 + (1.0 / N)));
	end
	tmp_2 = tmp;
end

Wolfram

code[N_] := If[GreaterEqual[N, 1000.0], N[(N[(N[(N[(1.0 / N), $MachinePrecision] + N[(-1.0 / N[(2.0 * N[Power[N, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(3.0 * N[Power[N, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[(4.0 * N[Power[N, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[N[(1.0 + N[(1.0 / N), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Reproduce

herbie shell --seed 2023333 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  :pre (and (> N 1.0) (< N 1e+40))

  :herbie-target
  (if (>= N 1000.0) (+ (+ (+ (/ 1.0 N) (/ -1.0 (* 2.0 (pow N 2.0)))) (/ 1.0 (* 3.0 (pow N 3.0)))) (/ -1.0 (* 4.0 (pow N 4.0)))) (log (+ 1.0 (/ 1.0 N))))

  (- (log (+ N 1.0)) (log N)))