2log (problem 3.3.6)

Percentage Accurate: 24.3% → 99.4%

Time: 9.6s

Alternatives: 12

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: 24.3% 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

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 0.0006:\\ \;\;\;\;\frac{0.3333333333333333}{{N}^{3}} + \left(\frac{1}{0.5 + \left(N + \left(\frac{0.25}{N} + \frac{0.125}{{N}^{2}}\right)\right)} - \frac{0.25}{{N}^{4}}\right)\\ \mathbf{else}:\\ \;\;\;\;-\log \left(\frac{N}{N + 1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= (- (log (+ N 1.0)) (log N)) 0.0006)
   (+
    (/ 0.3333333333333333 (pow N 3.0))
    (-
     (/ 1.0 (+ 0.5 (+ N (+ (/ 0.25 N) (/ 0.125 (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 / (0.5 + (N + ((0.25 / N) + (0.125 / 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 / (0.5d0 + (n + ((0.25d0 / n) + (0.125d0 / (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 / (0.5 + (N + ((0.25 / N) + (0.125 / 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 / (0.5 + (N + ((0.25 / N) + (0.125 / 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 / Float64(0.5 + Float64(N + Float64(Float64(0.25 / N) + Float64(0.125 / (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 / (0.5 + (N + ((0.25 / N) + (0.125 / (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[(0.5 + N[(N + N[(N[(0.25 / N), $MachinePrecision] + N[(0.125 / N[Power[N, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.25 / N[Power[N, 4.0], $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 19.9%

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

      1. +-commutative 19.9%

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

      2. log1p-def 20.0%

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

   3. Simplified 20.0%

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

   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%

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

   7. Taylor expanded in N around 0 99.7%

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

      1. associate--r+ 99.6%

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

      2. associate-*r/ 99.6%

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

      3. metadata-eval 99.6%

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

      4. associate-*r/ 99.6%

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

      5. metadata-eval 99.6%

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

   9. Simplified 99.6%

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

       1. frac-sub 99.4%

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

       2. unpow2 99.4%

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

       3. cube-mult 99.3%

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

       4. clear-num 99.4%

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

       5. *-un-lft-identity 99.4%

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

       6. unpow2 99.4%

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

       7. distribute-lft-out-- 99.4%

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

   11. Applied egg-rr 99.4%

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

   12. Taylor expanded in N around inf 99.7%

       \[\leadsto \frac{0.3333333333333333}{{N}^{3}} + \left(\frac{1}{\color{blue}{0.5 + \left(N + \left(0.125 \cdot \frac{1}{{N}^{2}} + 0.25 \cdot \frac{1}{N}\right)\right)}} - \frac{0.25}{{N}^{4}}\right) \]
   13. Step-by-step derivation

       1. +-commutative 99.7%

          \[\leadsto \frac{0.3333333333333333}{{N}^{3}} + \left(\frac{1}{0.5 + \left(N + \color{blue}{\left(0.25 \cdot \frac{1}{N} + 0.125 \cdot \frac{1}{{N}^{2}}\right)}\right)} - \frac{0.25}{{N}^{4}}\right) \]

       2. associate-*r/ 99.7%

          \[\leadsto \frac{0.3333333333333333}{{N}^{3}} + \left(\frac{1}{0.5 + \left(N + \left(\color{blue}{\frac{0.25 \cdot 1}{N}} + 0.125 \cdot \frac{1}{{N}^{2}}\right)\right)} - \frac{0.25}{{N}^{4}}\right) \]

       3. metadata-eval 99.7%

          \[\leadsto \frac{0.3333333333333333}{{N}^{3}} + \left(\frac{1}{0.5 + \left(N + \left(\frac{\color{blue}{0.25}}{N} + 0.125 \cdot \frac{1}{{N}^{2}}\right)\right)} - \frac{0.25}{{N}^{4}}\right) \]

       4. associate-*r/ 99.7%

          \[\leadsto \frac{0.3333333333333333}{{N}^{3}} + \left(\frac{1}{0.5 + \left(N + \left(\frac{0.25}{N} + \color{blue}{\frac{0.125 \cdot 1}{{N}^{2}}}\right)\right)} - \frac{0.25}{{N}^{4}}\right) \]

       5. metadata-eval 99.7%

          \[\leadsto \frac{0.3333333333333333}{{N}^{3}} + \left(\frac{1}{0.5 + \left(N + \left(\frac{0.25}{N} + \frac{\color{blue}{0.125}}{{N}^{2}}\right)\right)} - \frac{0.25}{{N}^{4}}\right) \]

   14. Simplified 99.7%

       \[\leadsto \frac{0.3333333333333333}{{N}^{3}} + \left(\frac{1}{\color{blue}{0.5 + \left(N + \left(\frac{0.25}{N} + \frac{0.125}{{N}^{2}}\right)\right)}} - \frac{0.25}{{N}^{4}}\right) \]

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

   1. Initial program 93.3%

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

      1. +-commutative 93.3%

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

      2. log1p-def 93.3%

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

   3. Simplified 93.3%

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

      1. add-log-exp 92.6%

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

      2. add-cube-cbrt 91.7%

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

      3. log-prod 91.5%

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

      4. pow2 91.5%

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

      5. exp-diff 91.2%

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

      6. log1p-udef 91.2%

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

      7. rem-exp-log 91.2%

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

      8. add-exp-log 91.5%

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

      9. +-commutative 91.5%

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

      10. exp-diff 91.7%

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

      11. log1p-udef 91.7%

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

      12. rem-exp-log 91.3%

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

      13. add-exp-log 91.5%

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

   5. Applied egg-rr 91.5%

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

      1. log-pow 91.1%

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

      2. distribute-lft1-in 91.1%

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

      3. metadata-eval 91.1%

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

   7. Simplified 91.1%

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

      1. add-log-exp 91.2%

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

      2. *-commutative 91.2%

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

      3. exp-to-pow 91.5%

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

      4. pow3 91.7%

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

      5. add-cube-cbrt 94.2%

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

      6. clear-num 94.1%

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

      7. log-div 96.3%

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

      8. metadata-eval 96.3%

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

   9. Applied egg-rr 96.3%

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

       1. neg-sub0 96.3%

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

   11. Simplified 96.3%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 0.0006:\\ \;\;\;\;\frac{0.3333333333333333}{{N}^{3}} + \left(\frac{1}{0.5 + \left(N + \left(\frac{0.25}{N} + \frac{0.125}{{N}^{2}}\right)\right)} - \frac{0.25}{{N}^{4}}\right)\\ \mathbf{else}:\\ \;\;\;\;-\log \left(\frac{N}{N + 1}\right)\\ \end{array} \]

Alternative 2: 99.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 0.0005:\\ \;\;\;\;\frac{0.3333333333333333}{{N}^{3}} + \left(N \cdot \left({N}^{-3} \cdot \left(N + -0.5\right)\right) - \frac{0.25}{{N}^{4}}\right)\\ \mathbf{else}:\\ \;\;\;\;-\log \left(\frac{N}{N + 1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= (- (log (+ N 1.0)) (log N)) 0.0005)
   (+
    (/ 0.3333333333333333 (pow N 3.0))
    (- (* N (* (pow N -3.0) (+ N -0.5))) (/ 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)) <= 0.0005) {
		tmp = (0.3333333333333333 / pow(N, 3.0)) + ((N * (pow(N, -3.0) * (N + -0.5))) - (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)) <= 0.0005d0) then
        tmp = (0.3333333333333333d0 / (n ** 3.0d0)) + ((n * ((n ** (-3.0d0)) * (n + (-0.5d0)))) - (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)) <= 0.0005) {
		tmp = (0.3333333333333333 / Math.pow(N, 3.0)) + ((N * (Math.pow(N, -3.0) * (N + -0.5))) - (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)) <= 0.0005:
		tmp = (0.3333333333333333 / math.pow(N, 3.0)) + ((N * (math.pow(N, -3.0) * (N + -0.5))) - (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)) <= 0.0005)
		tmp = Float64(Float64(0.3333333333333333 / (N ^ 3.0)) + Float64(Float64(N * Float64((N ^ -3.0) * Float64(N + -0.5))) - 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)) <= 0.0005)
		tmp = (0.3333333333333333 / (N ^ 3.0)) + ((N * ((N ^ -3.0) * (N + -0.5))) - (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], 0.0005], N[(N[(0.3333333333333333 / N[Power[N, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(N * N[(N[Power[N, -3.0], $MachinePrecision] * N[(N + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.25 / N[Power[N, 4.0], $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)) < 5.0000000000000001e-4

   1. Initial program 19.7%

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

      1. +-commutative 19.7%

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

      2. log1p-def 19.7%

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

   3. Simplified 19.7%

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

   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%

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

   7. Taylor expanded in N around 0 99.7%

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

      1. associate--r+ 99.7%

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

      2. associate-*r/ 99.7%

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

      3. metadata-eval 99.7%

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

      4. associate-*r/ 99.7%

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

      5. metadata-eval 99.7%

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

   9. Simplified 99.7%

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

       1. frac-sub 99.4%

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

       2. unpow2 99.4%

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

       3. cube-mult 99.3%

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

       4. div-inv 99.2%

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

       5. *-un-lft-identity 99.2%

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

       6. unpow2 99.2%

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

       7. distribute-lft-out-- 99.3%

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

       8. pow-flip 99.4%

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

       9. metadata-eval 99.4%

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

   11. Applied egg-rr 99.4%

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

       1. associate-*l* 99.3%

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

       2. sub-neg 99.3%

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

       3. metadata-eval 99.3%

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

   13. Simplified 99.3%

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

   if 5.0000000000000001e-4 < (-.f64 (log.f64 (+.f64 N 1)) (log.f64 N))

   1. Initial program 92.4%

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

      1. +-commutative 92.4%

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

      2. log1p-def 92.4%

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

   3. Simplified 92.4%

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

      1. add-log-exp 91.8%

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

      2. add-cube-cbrt 90.9%

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

      3. log-prod 90.7%

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

      4. pow2 90.7%

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

      5. exp-diff 90.4%

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

      6. log1p-udef 90.4%

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

      7. rem-exp-log 90.4%

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

      8. add-exp-log 91.0%

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

      9. +-commutative 91.0%

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

      10. exp-diff 91.2%

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

      11. log1p-udef 91.2%

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

      12. rem-exp-log 90.8%

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

      13. add-exp-log 91.0%

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

   5. Applied egg-rr 91.0%

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

      1. log-pow 90.8%

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

      2. distribute-lft1-in 90.8%

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

      3. metadata-eval 90.8%

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

   7. Simplified 90.8%

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

      1. add-log-exp 90.7%

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

      2. *-commutative 90.7%

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

      3. exp-to-pow 90.9%

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

      4. pow3 91.1%

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

      5. add-cube-cbrt 93.5%

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

      6. clear-num 93.4%

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

      7. log-div 95.8%

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

      8. metadata-eval 95.8%

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

   9. Applied egg-rr 95.8%

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

       1. neg-sub0 95.8%

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

   11. Simplified 95.8%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 0.0005:\\ \;\;\;\;\frac{0.3333333333333333}{{N}^{3}} + \left(N \cdot \left({N}^{-3} \cdot \left(N + -0.5\right)\right) - \frac{0.25}{{N}^{4}}\right)\\ \mathbf{else}:\\ \;\;\;\;-\log \left(\frac{N}{N + 1}\right)\\ \end{array} \]

Alternative 3: 99.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 0.0005:\\ \;\;\;\;\frac{0.3333333333333333}{{N}^{3}} + \left(\left(N \cdot \left(N - 0.5\right)\right) \cdot {N}^{-3} - \frac{0.25}{{N}^{4}}\right)\\ \mathbf{else}:\\ \;\;\;\;-\log \left(\frac{N}{N + 1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= (- (log (+ N 1.0)) (log N)) 0.0005)
   (+
    (/ 0.3333333333333333 (pow N 3.0))
    (- (* (* N (- N 0.5)) (pow N -3.0)) (/ 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)) <= 0.0005) {
		tmp = (0.3333333333333333 / pow(N, 3.0)) + (((N * (N - 0.5)) * pow(N, -3.0)) - (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)) <= 0.0005d0) then
        tmp = (0.3333333333333333d0 / (n ** 3.0d0)) + (((n * (n - 0.5d0)) * (n ** (-3.0d0))) - (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)) <= 0.0005) {
		tmp = (0.3333333333333333 / Math.pow(N, 3.0)) + (((N * (N - 0.5)) * Math.pow(N, -3.0)) - (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)) <= 0.0005:
		tmp = (0.3333333333333333 / math.pow(N, 3.0)) + (((N * (N - 0.5)) * math.pow(N, -3.0)) - (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)) <= 0.0005)
		tmp = Float64(Float64(0.3333333333333333 / (N ^ 3.0)) + Float64(Float64(Float64(N * Float64(N - 0.5)) * (N ^ -3.0)) - 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)) <= 0.0005)
		tmp = (0.3333333333333333 / (N ^ 3.0)) + (((N * (N - 0.5)) * (N ^ -3.0)) - (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], 0.0005], N[(N[(0.3333333333333333 / N[Power[N, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N * N[(N - 0.5), $MachinePrecision]), $MachinePrecision] * N[Power[N, -3.0], $MachinePrecision]), $MachinePrecision] - N[(0.25 / N[Power[N, 4.0], $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)) < 5.0000000000000001e-4

   1. Initial program 19.7%

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

      1. +-commutative 19.7%

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

      2. log1p-def 19.7%

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

   3. Simplified 19.7%

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

   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%

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

   7. Taylor expanded in N around 0 99.7%

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

      1. associate--r+ 99.7%

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

      2. associate-*r/ 99.7%

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

      3. metadata-eval 99.7%

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

      4. associate-*r/ 99.7%

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

      5. metadata-eval 99.7%

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

   9. Simplified 99.7%

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

       1. frac-sub 99.4%

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

       2. unpow2 99.4%

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

       3. cube-mult 99.3%

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

       4. div-inv 99.2%

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

       5. *-un-lft-identity 99.2%

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

       6. unpow2 99.2%

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

       7. distribute-lft-out-- 99.3%

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

       8. pow-flip 99.4%

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

       9. metadata-eval 99.4%

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

   11. Applied egg-rr 99.4%

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

   if 5.0000000000000001e-4 < (-.f64 (log.f64 (+.f64 N 1)) (log.f64 N))

   1. Initial program 92.4%

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

      1. +-commutative 92.4%

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

      2. log1p-def 92.4%

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

   3. Simplified 92.4%

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

      1. add-log-exp 91.8%

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

      2. add-cube-cbrt 90.9%

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

      3. log-prod 90.7%

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

      4. pow2 90.7%

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

      5. exp-diff 90.4%

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

      6. log1p-udef 90.4%

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

      7. rem-exp-log 90.4%

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

      8. add-exp-log 91.0%

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

      9. +-commutative 91.0%

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

      10. exp-diff 91.2%

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

      11. log1p-udef 91.2%

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

      12. rem-exp-log 90.8%

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

      13. add-exp-log 91.0%

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

   5. Applied egg-rr 91.0%

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

      1. log-pow 90.8%

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

      2. distribute-lft1-in 90.8%

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

      3. metadata-eval 90.8%

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

   7. Simplified 90.8%

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

      1. add-log-exp 90.7%

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

      2. *-commutative 90.7%

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

      3. exp-to-pow 90.9%

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

      4. pow3 91.1%

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

      5. add-cube-cbrt 93.5%

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

      6. clear-num 93.4%

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

      7. log-div 95.8%

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

      8. metadata-eval 95.8%

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

   9. Applied egg-rr 95.8%

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

       1. neg-sub0 95.8%

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

   11. Simplified 95.8%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 0.0005:\\ \;\;\;\;\frac{0.3333333333333333}{{N}^{3}} + \left(\left(N \cdot \left(N - 0.5\right)\right) \cdot {N}^{-3} - \frac{0.25}{{N}^{4}}\right)\\ \mathbf{else}:\\ \;\;\;\;-\log \left(\frac{N}{N + 1}\right)\\ \end{array} \]

Alternative 4: 99.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 0.0005:\\ \;\;\;\;\frac{0.3333333333333333}{{N}^{3}} + \left(\left(\frac{1}{N} - \frac{0.5}{{N}^{2}}\right) - \frac{0.25}{{N}^{4}}\right)\\ \mathbf{else}:\\ \;\;\;\;-\log \left(\frac{N}{N + 1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= (- (log (+ N 1.0)) (log N)) 0.0005)
   (+
    (/ 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)) <= 0.0005) {
		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)) <= 0.0005d0) 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)) <= 0.0005) {
		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)) <= 0.0005:
		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)) <= 0.0005)
		tmp = Float64(Float64(0.3333333333333333 / (N ^ 3.0)) + Float64(Float64(Float64(1.0 / N) - 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)) <= 0.0005)
		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], 0.0005], N[(N[(0.3333333333333333 / N[Power[N, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(1.0 / N), $MachinePrecision] - N[(0.5 / N[Power[N, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.25 / N[Power[N, 4.0], $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)) < 5.0000000000000001e-4

   1. Initial program 19.7%

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

      1. +-commutative 19.7%

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

      2. log1p-def 19.7%

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

   3. Simplified 19.7%

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

   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%

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

   7. Taylor expanded in N around 0 99.7%

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

      1. associate--r+ 99.7%

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

      2. associate-*r/ 99.7%

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

      3. metadata-eval 99.7%

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

      4. associate-*r/ 99.7%

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

      5. metadata-eval 99.7%

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

   9. Simplified 99.7%

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

   if 5.0000000000000001e-4 < (-.f64 (log.f64 (+.f64 N 1)) (log.f64 N))

   1. Initial program 92.4%

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

      1. +-commutative 92.4%

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

      2. log1p-def 92.4%

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

   3. Simplified 92.4%

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

      1. add-log-exp 91.8%

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

      2. add-cube-cbrt 90.9%

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

      3. log-prod 90.7%

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

      4. pow2 90.7%

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

      5. exp-diff 90.4%

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

      6. log1p-udef 90.4%

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

      7. rem-exp-log 90.4%

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

      8. add-exp-log 91.0%

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

      9. +-commutative 91.0%

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

      10. exp-diff 91.2%

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

      11. log1p-udef 91.2%

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

      12. rem-exp-log 90.8%

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

      13. add-exp-log 91.0%

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

   5. Applied egg-rr 91.0%

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

      1. log-pow 90.8%

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

      2. distribute-lft1-in 90.8%

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

      3. metadata-eval 90.8%

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

   7. Simplified 90.8%

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

      1. add-log-exp 90.7%

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

      2. *-commutative 90.7%

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

      3. exp-to-pow 90.9%

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

      4. pow3 91.1%

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

      5. add-cube-cbrt 93.5%

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

      6. clear-num 93.4%

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

      7. log-div 95.8%

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

      8. metadata-eval 95.8%

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

   9. Applied egg-rr 95.8%

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

       1. neg-sub0 95.8%

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

   11. Simplified 95.8%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 0.0005:\\ \;\;\;\;\frac{0.3333333333333333}{{N}^{3}} + \left(\left(\frac{1}{N} - \frac{0.5}{{N}^{2}}\right) - \frac{0.25}{{N}^{4}}\right)\\ \mathbf{else}:\\ \;\;\;\;-\log \left(\frac{N}{N + 1}\right)\\ \end{array} \]

Alternative 5: 99.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 0.0005:\\ \;\;\;\;\frac{0.3333333333333333}{{N}^{3}} + \left(\frac{1}{N} - \left(\frac{0.25}{{N}^{4}} + \frac{0.5}{{N}^{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-\log \left(\frac{N}{N + 1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= (- (log (+ N 1.0)) (log N)) 0.0005)
   (+
    (/ 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)) <= 0.0005) {
		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)) <= 0.0005d0) 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)) <= 0.0005) {
		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)) <= 0.0005:
		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)) <= 0.0005)
		tmp = Float64(Float64(0.3333333333333333 / (N ^ 3.0)) + Float64(Float64(1.0 / N) - Float64(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)) <= 0.0005)
		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], 0.0005], N[(N[(0.3333333333333333 / N[Power[N, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / N), $MachinePrecision] - N[(N[(0.25 / N[Power[N, 4.0], $MachinePrecision]), $MachinePrecision] + N[(0.5 / N[Power[N, 2.0], $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)) < 5.0000000000000001e-4

   1. Initial program 19.7%

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

      1. +-commutative 19.7%

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

      2. log1p-def 19.7%

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

   3. Simplified 19.7%

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

   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 5.0000000000000001e-4 < (-.f64 (log.f64 (+.f64 N 1)) (log.f64 N))

   1. Initial program 92.4%

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

      1. +-commutative 92.4%

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

      2. log1p-def 92.4%

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

   3. Simplified 92.4%

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

      1. add-log-exp 91.8%

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

      2. add-cube-cbrt 90.9%

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

      3. log-prod 90.7%

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

      4. pow2 90.7%

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

      5. exp-diff 90.4%

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

      6. log1p-udef 90.4%

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

      7. rem-exp-log 90.4%

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

      8. add-exp-log 91.0%

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

      9. +-commutative 91.0%

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

      10. exp-diff 91.2%

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

      11. log1p-udef 91.2%

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

      12. rem-exp-log 90.8%

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

      13. add-exp-log 91.0%

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

   5. Applied egg-rr 91.0%

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

      1. log-pow 90.8%

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

      2. distribute-lft1-in 90.8%

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

      3. metadata-eval 90.8%

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

   7. Simplified 90.8%

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

      1. add-log-exp 90.7%

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

      2. *-commutative 90.7%

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

      3. exp-to-pow 90.9%

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

      4. pow3 91.1%

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

      5. add-cube-cbrt 93.5%

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

      6. clear-num 93.4%

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

      7. log-div 95.8%

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

      8. metadata-eval 95.8%

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

   9. Applied egg-rr 95.8%

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

       1. neg-sub0 95.8%

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

   11. Simplified 95.8%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 0.0005:\\ \;\;\;\;\frac{0.3333333333333333}{{N}^{3}} + \left(\frac{1}{N} - \left(\frac{0.25}{{N}^{4}} + \frac{0.5}{{N}^{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-\log \left(\frac{N}{N + 1}\right)\\ \end{array} \]

Alternative 6: 99.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 0.00015:\\ \;\;\;\;\frac{0.3333333333333333}{{N}^{3}} + \left(\frac{1}{N + \left(0.5 + \frac{0.25}{N}\right)} - \frac{0.25}{{N}^{4}}\right)\\ \mathbf{else}:\\ \;\;\;\;-\log \left(\frac{N}{N + 1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= (- (log (+ N 1.0)) (log N)) 0.00015)
   (+
    (/ 0.3333333333333333 (pow N 3.0))
    (- (/ 1.0 (+ N (+ 0.5 (/ 0.25 N)))) (/ 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)) <= 0.00015) {
		tmp = (0.3333333333333333 / pow(N, 3.0)) + ((1.0 / (N + (0.5 + (0.25 / N)))) - (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)) <= 0.00015d0) then
        tmp = (0.3333333333333333d0 / (n ** 3.0d0)) + ((1.0d0 / (n + (0.5d0 + (0.25d0 / n)))) - (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)) <= 0.00015) {
		tmp = (0.3333333333333333 / Math.pow(N, 3.0)) + ((1.0 / (N + (0.5 + (0.25 / N)))) - (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)) <= 0.00015:
		tmp = (0.3333333333333333 / math.pow(N, 3.0)) + ((1.0 / (N + (0.5 + (0.25 / N)))) - (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)) <= 0.00015)
		tmp = Float64(Float64(0.3333333333333333 / (N ^ 3.0)) + Float64(Float64(1.0 / Float64(N + Float64(0.5 + Float64(0.25 / N)))) - 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)) <= 0.00015)
		tmp = (0.3333333333333333 / (N ^ 3.0)) + ((1.0 / (N + (0.5 + (0.25 / N)))) - (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], 0.00015], N[(N[(0.3333333333333333 / N[Power[N, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / N[(N + N[(0.5 + N[(0.25 / N), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.25 / N[Power[N, 4.0], $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)) < 1.49999999999999987e-4

   1. Initial program 18.4%

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

      1. +-commutative 18.4%

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

      2. log1p-def 18.4%

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

   3. Simplified 18.4%

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

   4. Taylor expanded in N around inf 99.8%

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

      1. associate--l+ 99.8%

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

      2. associate-*r/ 99.8%

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

      3. metadata-eval 99.8%

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

      4. +-commutative 99.8%

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

      5. associate-*r/ 99.8%

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

      6. metadata-eval 99.8%

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

      7. associate-*r/ 99.8%

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

      8. metadata-eval 99.8%

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

   6. Simplified 99.8%

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

   7. Taylor expanded in N around 0 99.8%

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

      1. associate--r+ 99.8%

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

      2. associate-*r/ 99.8%

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

      3. metadata-eval 99.8%

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

      4. associate-*r/ 99.8%

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

      5. metadata-eval 99.8%

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

   9. Simplified 99.8%

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

       1. frac-sub 99.5%

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

       2. unpow2 99.5%

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

       3. cube-mult 99.5%

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

       4. clear-num 99.6%

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

       5. *-un-lft-identity 99.6%

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

       6. unpow2 99.6%

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

       7. distribute-lft-out-- 99.6%

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

   11. Applied egg-rr 99.6%

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

   12. Taylor expanded in N around inf 99.5%

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

       1. associate-+r+ 99.5%

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

       2. +-commutative 99.5%

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

       3. associate-+l+ 99.5%

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

       4. associate-*r/ 99.5%

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

       5. metadata-eval 99.5%

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

   14. Simplified 99.5%

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

   if 1.49999999999999987e-4 < (-.f64 (log.f64 (+.f64 N 1)) (log.f64 N))

   1. Initial program 89.1%

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

      1. +-commutative 89.1%

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

      2. log1p-def 89.1%

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

   3. Simplified 89.1%

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

      1. add-log-exp 88.5%

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

      2. add-cube-cbrt 87.9%

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

      3. log-prod 87.6%

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

      4. pow2 87.6%

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

      5. exp-diff 87.4%

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

      6. log1p-udef 87.4%

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

      7. rem-exp-log 87.8%

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

      8. add-exp-log 87.8%

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

      9. +-commutative 87.8%

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

      10. exp-diff 87.8%

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

      11. log1p-udef 87.8%

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

      12. rem-exp-log 87.9%

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

      13. add-exp-log 87.6%

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

   5. Applied egg-rr 87.6%

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

      1. log-pow 87.5%

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

      2. distribute-lft1-in 87.5%

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

      3. metadata-eval 87.5%

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

   7. Simplified 87.5%

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

      1. add-log-exp 87.4%

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

      2. *-commutative 87.4%

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

      3. exp-to-pow 87.6%

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

      4. pow3 87.7%

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

      5. add-cube-cbrt 90.7%

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

      6. clear-num 90.7%

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

      7. log-div 93.0%

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

      8. metadata-eval 93.0%

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

   9. Applied egg-rr 93.0%

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

       1. neg-sub0 93.0%

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

   11. Simplified 93.0%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 0.00015:\\ \;\;\;\;\frac{0.3333333333333333}{{N}^{3}} + \left(\frac{1}{N + \left(0.5 + \frac{0.25}{N}\right)} - \frac{0.25}{{N}^{4}}\right)\\ \mathbf{else}:\\ \;\;\;\;-\log \left(\frac{N}{N + 1}\right)\\ \end{array} \]

Alternative 7: 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 18.2%

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

      1. +-commutative 18.2%

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

      2. log1p-def 18.2%

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

   3. Simplified 18.2%

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

   4. Taylor expanded in N around inf 99.4%

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

      1. associate--l+ 99.4%

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

      2. associate-*r/ 99.4%

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

      3. metadata-eval 99.4%

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

      4. associate-*r/ 99.4%

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

      5. metadata-eval 99.4%

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

   6. Simplified 99.4%

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

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

   1. Initial program 88.5%

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

      1. +-commutative 88.5%

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

      2. log1p-def 88.5%

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

   3. Simplified 88.5%

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

      1. add-log-exp 87.9%

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

      2. add-cube-cbrt 87.3%

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

      3. log-prod 87.0%

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

      4. pow2 87.0%

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

      5. exp-diff 86.9%

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

      6. log1p-udef 86.9%

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

      7. rem-exp-log 87.6%

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

      8. add-exp-log 87.2%

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

      9. +-commutative 87.2%

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

      10. exp-diff 87.3%

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

      11. log1p-udef 87.3%

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

      12. rem-exp-log 87.6%

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

      13. add-exp-log 87.2%

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

   5. Applied egg-rr 87.2%

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

      1. log-pow 87.1%

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

      2. distribute-lft1-in 87.1%

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

      3. metadata-eval 87.1%

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

   7. Simplified 87.1%

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

      1. add-log-exp 87.0%

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

      2. *-commutative 87.0%

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

      3. exp-to-pow 87.2%

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

      4. pow3 87.2%

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

      5. add-cube-cbrt 90.3%

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

      6. clear-num 90.3%

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

      7. log-div 92.5%

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

      8. metadata-eval 92.5%

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

   9. Applied egg-rr 92.5%

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

       1. neg-sub0 92.5%

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

   11. Simplified 92.5%

       \[\leadsto \color{blue}{-\log \left(\frac{N}{N + 1}\right)} \]
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 8: 97.8% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (N)
 :precision binary64
 (if (<= (- (log (+ N 1.0)) (log N)) 2e-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)) <= 2e-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)) <= 2d-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)) <= 2e-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)) <= 2e-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)) <= 2e-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)) <= 2e-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], 2e-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)) < 1.99999999999999991e-6

   1. Initial program 14.9%

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

      1. +-commutative 14.9%

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

      2. log1p-def 14.9%

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

   3. Simplified 14.9%

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

   4. Taylor expanded in N around inf 99.1%

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

      1. associate-*r/ 99.1%

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

      2. metadata-eval 99.1%

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

   6. Simplified 99.1%

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

   if 1.99999999999999991e-6 < (-.f64 (log.f64 (+.f64 N 1)) (log.f64 N))

   1. Initial program 83.0%

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

      1. +-commutative 83.0%

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

      2. log1p-def 83.0%

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

   3. Simplified 83.0%

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

      1. add-log-exp 82.7%

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

      2. add-cube-cbrt 82.1%

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

      3. log-prod 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 82.3%

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

      6. log1p-udef 82.3%

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

      7. rem-exp-log 82.5%

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

      8. add-exp-log 82.8%

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

      9. +-commutative 82.8%

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

      10. exp-diff 82.8%

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

      11. log1p-udef 82.8%

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

      12. rem-exp-log 83.1%

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

      13. add-exp-log 82.3%

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

   5. Applied egg-rr 82.3%

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

      1. log-pow 82.7%

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

      2. distribute-lft1-in 82.7%

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

      3. metadata-eval 82.7%

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

   7. Simplified 82.7%

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

      1. add-log-exp 82.4%

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

      2. *-commutative 82.4%

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

      3. exp-to-pow 82.5%

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

      4. pow3 82.3%

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

      5. add-cube-cbrt 86.1%

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

      6. clear-num 86.0%

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

      7. log-div 87.2%

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

      8. metadata-eval 87.2%

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

   9. Applied egg-rr 87.2%

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

       1. neg-sub0 87.2%

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

   11. Simplified 87.2%

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

4. Final simplification 97.6%

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

Alternative 9: 93.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;N \leq 130000000:\\ \;\;\;\;-\log \left(\frac{N}{N + 1}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N}\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= N 130000000.0) (- (log (/ N (+ N 1.0)))) (/ 1.0 N)))

C

double code(double N) {
	double tmp;
	if (N <= 130000000.0) {
		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 <= 130000000.0d0) 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 <= 130000000.0) {
		tmp = -Math.log((N / (N + 1.0)));
	} else {
		tmp = 1.0 / N;
	}
	return tmp;
}

Python

def code(N):
	tmp = 0
	if N <= 130000000.0:
		tmp = -math.log((N / (N + 1.0)))
	else:
		tmp = 1.0 / N
	return tmp

Julia

function code(N)
	tmp = 0.0
	if (N <= 130000000.0)
		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 <= 130000000.0)
		tmp = -log((N / (N + 1.0)));
	else
		tmp = 1.0 / N;
	end
	tmp_2 = tmp;
end

Wolfram

code[N_] := If[LessEqual[N, 130000000.0], (-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 < 1.3e8

   1. Initial program 75.6%

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

      1. +-commutative 75.6%

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

      2. log1p-def 75.6%

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

   3. Simplified 75.6%

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

      1. add-log-exp 75.5%

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

      2. add-cube-cbrt 75.1%

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

      3. log-prod 75.4%

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

      4. pow2 75.4%

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

      5. exp-diff 75.2%

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

      6. log1p-udef 75.2%

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

      7. rem-exp-log 75.2%

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

      8. add-exp-log 75.9%

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

      9. +-commutative 75.9%

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

      10. exp-diff 75.9%

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

      11. log1p-udef 75.9%

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

      12. rem-exp-log 75.9%

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

      13. add-exp-log 75.9%

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

   5. Applied egg-rr 75.9%

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

      1. log-pow 76.1%

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

      2. distribute-lft1-in 76.1%

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

      3. metadata-eval 76.1%

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

   7. Simplified 76.1%

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

      1. add-log-exp 75.9%

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

      2. *-commutative 75.9%

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

      3. exp-to-pow 75.9%

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

      4. pow3 75.8%

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

      5. add-cube-cbrt 79.1%

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

      6. clear-num 79.0%

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

      7. log-div 80.2%

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

      8. metadata-eval 80.2%

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

   9. Applied egg-rr 80.2%

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

       1. neg-sub0 80.2%

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

   11. Simplified 80.2%

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

   if 1.3e8 < N

   1. Initial program 10.4%

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

      1. +-commutative 10.4%

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

      2. log1p-def 10.4%

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

   3. Simplified 10.4%

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

   4. Taylor expanded in N around inf 95.6%

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

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;N \leq 130000000:\\ \;\;\;\;-\log \left(\frac{N}{N + 1}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N}\\ \end{array} \]

Alternative 10: 97.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;N \leq 200000:\\ \;\;\;\;-\log \left(\frac{N}{N + 1}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(N + -0.5\right) \cdot {N}^{-2}\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= N 200000.0) (- (log (/ N (+ N 1.0)))) (* (+ N -0.5) (pow N -2.0))))

C

double code(double N) {
	double tmp;
	if (N <= 200000.0) {
		tmp = -log((N / (N + 1.0)));
	} else {
		tmp = (N + -0.5) * pow(N, -2.0);
	}
	return tmp;
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= 200000.0d0) then
        tmp = -log((n / (n + 1.0d0)))
    else
        tmp = (n + (-0.5d0)) * (n ** (-2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double N) {
	double tmp;
	if (N <= 200000.0) {
		tmp = -Math.log((N / (N + 1.0)));
	} else {
		tmp = (N + -0.5) * Math.pow(N, -2.0);
	}
	return tmp;
}

Python

def code(N):
	tmp = 0
	if N <= 200000.0:
		tmp = -math.log((N / (N + 1.0)))
	else:
		tmp = (N + -0.5) * math.pow(N, -2.0)
	return tmp

Julia

function code(N)
	tmp = 0.0
	if (N <= 200000.0)
		tmp = Float64(-log(Float64(N / Float64(N + 1.0))));
	else
		tmp = Float64(Float64(N + -0.5) * (N ^ -2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(N)
	tmp = 0.0;
	if (N <= 200000.0)
		tmp = -log((N / (N + 1.0)));
	else
		tmp = (N + -0.5) * (N ^ -2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[N_] := If[LessEqual[N, 200000.0], (-N[Log[N[(N / N[(N + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), N[(N[(N + -0.5), $MachinePrecision] * N[Power[N, -2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if N < 2e5

   1. Initial program 83.0%

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

      1. +-commutative 83.0%

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

      2. log1p-def 83.0%

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

   3. Simplified 83.0%

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

      1. add-log-exp 82.7%

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

      2. add-cube-cbrt 82.1%

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

      3. log-prod 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 82.3%

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

      6. log1p-udef 82.3%

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

      7. rem-exp-log 82.5%

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

      8. add-exp-log 82.8%

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

      9. +-commutative 82.8%

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

      10. exp-diff 82.8%

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

      11. log1p-udef 82.8%

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

      12. rem-exp-log 83.1%

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

      13. add-exp-log 82.3%

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

   5. Applied egg-rr 82.3%

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

      1. log-pow 82.7%

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

      2. distribute-lft1-in 82.7%

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

      3. metadata-eval 82.7%

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

   7. Simplified 82.7%

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

      1. add-log-exp 82.4%

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

      2. *-commutative 82.4%

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

      3. exp-to-pow 82.5%

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

      4. pow3 82.3%

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

      5. add-cube-cbrt 86.1%

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

      6. clear-num 86.0%

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

      7. log-div 87.2%

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

      8. metadata-eval 87.2%

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

   9. Applied egg-rr 87.2%

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

       1. neg-sub0 87.2%

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

   11. Simplified 87.2%

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

   if 2e5 < N

   1. Initial program 14.9%

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

      1. +-commutative 14.9%

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

      2. log1p-def 14.9%

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

   3. Simplified 14.9%

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

      1. add-log-exp 14.9%

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

      2. add-cube-cbrt 14.9%

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

      3. log-prod 14.9%

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

      4. pow2 14.9%

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

      5. exp-diff 14.9%

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

      6. log1p-udef 14.9%

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

      7. rem-exp-log 16.2%

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

      8. add-exp-log 15.5%

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

      9. +-commutative 15.5%

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

      10. exp-diff 15.5%

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

      11. log1p-udef 15.4%

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

      12. rem-exp-log 16.8%

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

      13. add-exp-log 15.7%

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

   5. Applied egg-rr 15.7%

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

      1. log-pow 15.7%

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

      2. distribute-lft1-in 15.7%

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

      3. metadata-eval 15.7%

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

   7. Simplified 15.7%

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

   8. Taylor expanded in N around inf 98.7%

      \[\leadsto 3 \cdot \color{blue}{\left(0.3333333333333333 \cdot \frac{1}{N} - 0.16666666666666666 \cdot \frac{1}{{N}^{2}}\right)} \]
   9. Step-by-step derivation

      1. associate-*r/ 98.6%

         \[\leadsto 3 \cdot \left(\color{blue}{\frac{0.3333333333333333 \cdot 1}{N}} - 0.16666666666666666 \cdot \frac{1}{{N}^{2}}\right) \]

      2. metadata-eval 98.6%

         \[\leadsto 3 \cdot \left(\frac{\color{blue}{0.3333333333333333}}{N} - 0.16666666666666666 \cdot \frac{1}{{N}^{2}}\right) \]

      3. associate-*r/ 98.6%

         \[\leadsto 3 \cdot \left(\frac{0.3333333333333333}{N} - \color{blue}{\frac{0.16666666666666666 \cdot 1}{{N}^{2}}}\right) \]

      4. metadata-eval 98.6%

         \[\leadsto 3 \cdot \left(\frac{0.3333333333333333}{N} - \frac{\color{blue}{0.16666666666666666}}{{N}^{2}}\right) \]

   10. Simplified 98.6%

       \[\leadsto 3 \cdot \color{blue}{\left(\frac{0.3333333333333333}{N} - \frac{0.16666666666666666}{{N}^{2}}\right)} \]
   11. Step-by-step derivation

       1. clear-num 98.5%

          \[\leadsto 3 \cdot \left(\color{blue}{\frac{1}{\frac{N}{0.3333333333333333}}} - \frac{0.16666666666666666}{{N}^{2}}\right) \]

       2. frac-sub 98.5%

          \[\leadsto 3 \cdot \color{blue}{\frac{1 \cdot {N}^{2} - \frac{N}{0.3333333333333333} \cdot 0.16666666666666666}{\frac{N}{0.3333333333333333} \cdot {N}^{2}}} \]

       3. *-un-lft-identity 98.5%

          \[\leadsto 3 \cdot \frac{\color{blue}{{N}^{2}} - \frac{N}{0.3333333333333333} \cdot 0.16666666666666666}{\frac{N}{0.3333333333333333} \cdot {N}^{2}} \]

       4. div-inv 98.5%

          \[\leadsto 3 \cdot \frac{{N}^{2} - \color{blue}{\left(N \cdot \frac{1}{0.3333333333333333}\right)} \cdot 0.16666666666666666}{\frac{N}{0.3333333333333333} \cdot {N}^{2}} \]

       5. metadata-eval 98.5%

          \[\leadsto 3 \cdot \frac{{N}^{2} - \left(N \cdot \color{blue}{3}\right) \cdot 0.16666666666666666}{\frac{N}{0.3333333333333333} \cdot {N}^{2}} \]

       6. div-inv 98.6%

          \[\leadsto 3 \cdot \frac{{N}^{2} - \left(N \cdot 3\right) \cdot 0.16666666666666666}{\color{blue}{\left(N \cdot \frac{1}{0.3333333333333333}\right)} \cdot {N}^{2}} \]

       7. metadata-eval 98.6%

          \[\leadsto 3 \cdot \frac{{N}^{2} - \left(N \cdot 3\right) \cdot 0.16666666666666666}{\left(N \cdot \color{blue}{3}\right) \cdot {N}^{2}} \]

   12. Applied egg-rr 98.6%

       \[\leadsto 3 \cdot \color{blue}{\frac{{N}^{2} - \left(N \cdot 3\right) \cdot 0.16666666666666666}{\left(N \cdot 3\right) \cdot {N}^{2}}} \]
   13. Step-by-step derivation

       1. unpow2 98.6%

          \[\leadsto 3 \cdot \frac{\color{blue}{N \cdot N} - \left(N \cdot 3\right) \cdot 0.16666666666666666}{\left(N \cdot 3\right) \cdot {N}^{2}} \]

       2. associate-*l* 98.6%

          \[\leadsto 3 \cdot \frac{N \cdot N - \color{blue}{N \cdot \left(3 \cdot 0.16666666666666666\right)}}{\left(N \cdot 3\right) \cdot {N}^{2}} \]

       3. metadata-eval 98.6%

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

       4. distribute-lft-out-- 98.6%

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

       5. sub-neg 98.6%

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

       6. metadata-eval 98.6%

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

       7. *-commutative 98.6%

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

       8. associate-*r* 98.5%

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

       9. unpow2 98.5%

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

       10. cube-mult 98.5%

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

   14. Simplified 98.5%

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

       1. expm1-log1p-u 98.5%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(3 \cdot \frac{N \cdot \left(N + -0.5\right)}{3 \cdot {N}^{3}}\right)\right)} \]

       2. expm1-udef 17.1%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(3 \cdot \frac{N \cdot \left(N + -0.5\right)}{3 \cdot {N}^{3}}\right)} - 1} \]

       3. associate-*r/ 17.1%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{3 \cdot \left(N \cdot \left(N + -0.5\right)\right)}{3 \cdot {N}^{3}}}\right)} - 1 \]

       4. times-frac 17.1%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{3}{3} \cdot \frac{N \cdot \left(N + -0.5\right)}{{N}^{3}}}\right)} - 1 \]

       5. metadata-eval 17.1%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{1} \cdot \frac{N \cdot \left(N + -0.5\right)}{{N}^{3}}\right)} - 1 \]

       6. metadata-eval 17.1%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{1}{1}} \cdot \frac{N \cdot \left(N + -0.5\right)}{{N}^{3}}\right)} - 1 \]

       7. times-frac 17.1%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{1 \cdot \left(N \cdot \left(N + -0.5\right)\right)}{1 \cdot {N}^{3}}}\right)} - 1 \]

       8. *-un-lft-identity 17.1%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{N \cdot \left(N + -0.5\right)}}{1 \cdot {N}^{3}}\right)} - 1 \]

       9. *-un-lft-identity 17.1%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{N \cdot \left(N + -0.5\right)}{\color{blue}{{N}^{3}}}\right)} - 1 \]

   16. Applied egg-rr 17.1%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{N \cdot \left(N + -0.5\right)}{{N}^{3}}\right)} - 1} \]
   17. Step-by-step derivation

       1. expm1-def 98.7%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{N \cdot \left(N + -0.5\right)}{{N}^{3}}\right)\right)} \]

       2. expm1-log1p 98.7%

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

       3. *-commutative 98.7%

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

       4. associate-*r/ 98.7%

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

       5. cube-mult 98.6%

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

       6. unpow2 98.6%

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

       7. associate-/r* 98.7%

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

       8. *-inverses 98.7%

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

       9. unpow2 98.7%

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

       10. associate-/r* 99.0%

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

       11. *-rgt-identity 99.0%

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

       12. associate-*r/ 98.7%

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

       13. unpow-1 98.7%

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

       14. metadata-eval 98.7%

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

       15. unpow-1 98.7%

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

       16. metadata-eval 98.7%

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

       17. sqr-pow 98.9%

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

   18. Simplified 98.9%

       \[\leadsto \color{blue}{\left(N + -0.5\right) \cdot {N}^{-2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;N \leq 200000:\\ \;\;\;\;-\log \left(\frac{N}{N + 1}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(N + -0.5\right) \cdot {N}^{-2}\\ \end{array} \]

Alternative 11: 92.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;N \leq 98000000:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N}\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= N 98000000.0) (log (/ (+ N 1.0) N)) (/ 1.0 N)))

C

double code(double N) {
	double tmp;
	if (N <= 98000000.0) {
		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 <= 98000000.0d0) 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 <= 98000000.0) {
		tmp = Math.log(((N + 1.0) / N));
	} else {
		tmp = 1.0 / N;
	}
	return tmp;
}

Python

def code(N):
	tmp = 0
	if N <= 98000000.0:
		tmp = math.log(((N + 1.0) / N))
	else:
		tmp = 1.0 / N
	return tmp

Julia

function code(N)
	tmp = 0.0
	if (N <= 98000000.0)
		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 <= 98000000.0)
		tmp = log(((N + 1.0) / N));
	else
		tmp = 1.0 / N;
	end
	tmp_2 = tmp;
end

Wolfram

code[N_] := If[LessEqual[N, 98000000.0], 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 < 9.8e7

   1. Initial program 76.0%

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

      1. +-commutative 76.0%

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

      2. log1p-def 76.0%

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

   3. Simplified 76.0%

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

      1. add-log-exp 76.0%

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

      2. log1p-expm1-u 76.0%

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

      3. log1p-udef 75.9%

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

      4. diff-log 75.8%

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

      5. log1p-udef 75.8%

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

      6. rem-exp-log 75.2%

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

      7. +-commutative 75.2%

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

      8. add-exp-log 75.3%

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

      9. log1p-udef 75.4%

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

      10. log1p-expm1-u 75.4%

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

      11. add-exp-log 79.5%

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

   5. Applied egg-rr 79.5%

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

   if 9.8e7 < N

   1. Initial program 10.6%

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

      1. +-commutative 10.6%

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

      2. log1p-def 10.7%

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

   3. Simplified 10.7%

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

   4. Taylor expanded in N around inf 95.4%

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

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;N \leq 98000000:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N}\\ \end{array} \]

Alternative 12: 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 23.4%

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

   1. +-commutative 23.4%

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

   2. log1p-def 23.4%

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

3. Simplified 23.4%

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

4. Taylor expanded in N around inf 85.1%

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

5. Final simplification 85.1%

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

Developer target: 99.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{log1p}\left(\frac{1}{N}\right) \end{array} \]

FPCore

(FPCore (N) :precision binary64 (log1p (/ 1.0 N)))

C

double code(double N) {
	return log1p((1.0 / N));
}

Java

public static double code(double N) {
	return Math.log1p((1.0 / N));
}

Python

def code(N):
	return math.log1p((1.0 / N))

Julia

function code(N)
	return log1p(Float64(1.0 / N))
end

Wolfram

code[N_] := N[Log[1 + N[(1.0 / N), $MachinePrecision]], $MachinePrecision]

Reproduce

herbie shell --seed 2024024 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  :pre (and (> N 1.0) (< N 1e+40))

  :herbie-target
  (log1p (/ 1.0 N))

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