2log (problem 3.3.6)

Percentage Accurate: 24.3% → 99.3%

Time: 10.4s

Alternatives: 9

Speedup: 68.3×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 24.3% 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: 99.3% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (N)
 :precision binary64
 (if (<= (- (log (+ N 1.0)) (log N)) 0.0005)
   (+
    (/ 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(1.0 / N) - Float64(Float64(0.5 / (N ^ 2.0)) + Float64(0.25 / (N ^ 4.0)))));
	else
		tmp = Float64(-log(Float64(N / Float64(N + 1.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(N)
	tmp = 0.0;
	if ((log((N + 1.0)) - log(N)) <= 0.0005)
		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[(1.0 / N), $MachinePrecision] - N[(N[(0.5 / N[Power[N, 2.0], $MachinePrecision]), $MachinePrecision] + N[(0.25 / N[Power[N, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-N[Log[N[(N / N[(N + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (log.f64 (+.f64 N 1)) (log.f64 N)) < 5.0000000000000001e-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. Add Preprocessing

   5. Taylor expanded in N around inf 99.7%

      \[\leadsto \color{blue}{\left(0.3333333333333333 \cdot \frac{1}{{N}^{3}} + \frac{1}{N}\right) - \left(0.25 \cdot \frac{1}{{N}^{4}} + 0.5 \cdot \frac{1}{{N}^{2}}\right)} \]
   6. 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%

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

   7. 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. Add Preprocessing
   5. 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%

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

   6. Applied egg-rr 91.0%

      \[\leadsto \color{blue}{\log \left({\left(\sqrt[3]{\frac{N + 1}{N}}\right)}^{2}\right) + \log \left(\sqrt[3]{\frac{N + 1}{N}}\right)} \]
   7. 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%

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

   8. Simplified 90.8%

      \[\leadsto \color{blue}{3 \cdot \log \left(\sqrt[3]{\frac{N + 1}{N}}\right)} \]
   9. 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%

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

   10. Applied egg-rr 95.8%

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

       1. neg-sub0 95.8%

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

   12. 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%

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

Alternative 2: 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. Add Preprocessing

   5. Taylor expanded in N around inf 99.7%

      \[\leadsto \color{blue}{\left(0.3333333333333333 \cdot \frac{1}{{N}^{3}} + \frac{1}{N}\right) - \left(0.25 \cdot \frac{1}{{N}^{4}} + 0.5 \cdot \frac{1}{{N}^{2}}\right)} \]
   6. 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%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in N around 0 99.7%

      \[\leadsto \frac{0.3333333333333333}{{N}^{3}} + \color{blue}{\left(\frac{1}{N} - \left(0.5 \cdot \frac{1}{{N}^{2}} + 0.25 \cdot \frac{1}{{N}^{4}}\right)\right)} \]
   9. 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. sub-neg 99.7%

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

      3. associate-*r/ 99.7%

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

      4. metadata-eval 99.7%

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

      5. associate-*r/ 99.7%

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

      6. metadata-eval 99.7%

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

      7. distribute-neg-frac 99.7%

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

      8. metadata-eval 99.7%

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

   10. 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. Add Preprocessing
   5. 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%

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

   6. Applied egg-rr 91.0%

      \[\leadsto \color{blue}{\log \left({\left(\sqrt[3]{\frac{N + 1}{N}}\right)}^{2}\right) + \log \left(\sqrt[3]{\frac{N + 1}{N}}\right)} \]
   7. 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%

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

   8. Simplified 90.8%

      \[\leadsto \color{blue}{3 \cdot \log \left(\sqrt[3]{\frac{N + 1}{N}}\right)} \]
   9. 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%

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

   10. Applied egg-rr 95.8%

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

       1. neg-sub0 95.8%

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

   12. 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%

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

Alternative 3: 98.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 0.00015:\\ \;\;\;\;\frac{e^{\frac{0.20833333333333334}{{N}^{2}}}}{N \cdot e^{\frac{0.5}{N}}}\\ \mathbf{else}:\\ \;\;\;\;-\log \left(\frac{N}{N + 1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= (- (log (+ N 1.0)) (log N)) 0.00015)
   (/ (exp (/ 0.20833333333333334 (pow N 2.0))) (* N (exp (/ 0.5 N))))
   (- (log (/ N (+ N 1.0))))))

C

double code(double N) {
	double tmp;
	if ((log((N + 1.0)) - log(N)) <= 0.00015) {
		tmp = exp((0.20833333333333334 / pow(N, 2.0))) / (N * exp((0.5 / N)));
	} 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 = exp((0.20833333333333334d0 / (n ** 2.0d0))) / (n * exp((0.5d0 / n)))
    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 = Math.exp((0.20833333333333334 / Math.pow(N, 2.0))) / (N * Math.exp((0.5 / N)));
	} 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 = math.exp((0.20833333333333334 / math.pow(N, 2.0))) / (N * math.exp((0.5 / N)))
	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(exp(Float64(0.20833333333333334 / (N ^ 2.0))) / Float64(N * exp(Float64(0.5 / N))));
	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 = exp((0.20833333333333334 / (N ^ 2.0))) / (N * exp((0.5 / N)));
	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[Exp[N[(0.20833333333333334 / N[Power[N, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N * N[Exp[N[(0.5 / N), $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. Add Preprocessing
   5. Step-by-step derivation

      1. add-exp-log 18.4%

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

   6. Applied egg-rr 18.4%

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

   7. Taylor expanded in N around inf 94.6%

      \[\leadsto e^{\color{blue}{\left(\log \left(\frac{1}{N}\right) + 0.20833333333333334 \cdot \frac{1}{{N}^{2}}\right) - 0.5 \cdot \frac{1}{N}}} \]
   8. Step-by-step derivation

      1. sub-neg 94.6%

         \[\leadsto e^{\color{blue}{\left(\log \left(\frac{1}{N}\right) + 0.20833333333333334 \cdot \frac{1}{{N}^{2}}\right) + \left(-0.5 \cdot \frac{1}{N}\right)}} \]

      2. log-rec 94.6%

         \[\leadsto e^{\left(\color{blue}{\left(-\log N\right)} + 0.20833333333333334 \cdot \frac{1}{{N}^{2}}\right) + \left(-0.5 \cdot \frac{1}{N}\right)} \]

      3. +-commutative 94.6%

         \[\leadsto e^{\color{blue}{\left(0.20833333333333334 \cdot \frac{1}{{N}^{2}} + \left(-\log N\right)\right)} + \left(-0.5 \cdot \frac{1}{N}\right)} \]

      4. unsub-neg 94.6%

         \[\leadsto e^{\color{blue}{\left(0.20833333333333334 \cdot \frac{1}{{N}^{2}} - \log N\right)} + \left(-0.5 \cdot \frac{1}{N}\right)} \]

      5. associate-*r/ 94.6%

         \[\leadsto e^{\left(\color{blue}{\frac{0.20833333333333334 \cdot 1}{{N}^{2}}} - \log N\right) + \left(-0.5 \cdot \frac{1}{N}\right)} \]

      6. metadata-eval 94.6%

         \[\leadsto e^{\left(\frac{\color{blue}{0.20833333333333334}}{{N}^{2}} - \log N\right) + \left(-0.5 \cdot \frac{1}{N}\right)} \]

      7. associate-*r/ 94.6%

         \[\leadsto e^{\left(\frac{0.20833333333333334}{{N}^{2}} - \log N\right) + \left(-\color{blue}{\frac{0.5 \cdot 1}{N}}\right)} \]

      8. metadata-eval 94.6%

         \[\leadsto e^{\left(\frac{0.20833333333333334}{{N}^{2}} - \log N\right) + \left(-\frac{\color{blue}{0.5}}{N}\right)} \]

      9. distribute-neg-frac 94.6%

         \[\leadsto e^{\left(\frac{0.20833333333333334}{{N}^{2}} - \log N\right) + \color{blue}{\frac{-0.5}{N}}} \]

      10. metadata-eval 94.6%

          \[\leadsto e^{\left(\frac{0.20833333333333334}{{N}^{2}} - \log N\right) + \frac{\color{blue}{-0.5}}{N}} \]

   9. Simplified 94.6%

      \[\leadsto e^{\color{blue}{\left(\frac{0.20833333333333334}{{N}^{2}} - \log N\right) + \frac{-0.5}{N}}} \]

   10. Taylor expanded in N around 0 94.6%

       \[\leadsto \color{blue}{e^{0.20833333333333334 \cdot \frac{1}{{N}^{2}} - \left(\log N + 0.5 \cdot \frac{1}{N}\right)}} \]
   11. Step-by-step derivation

       1. exp-diff 94.6%

          \[\leadsto \color{blue}{\frac{e^{0.20833333333333334 \cdot \frac{1}{{N}^{2}}}}{e^{\log N + 0.5 \cdot \frac{1}{N}}}} \]

       2. associate-*r/ 94.6%

          \[\leadsto \frac{e^{\color{blue}{\frac{0.20833333333333334 \cdot 1}{{N}^{2}}}}}{e^{\log N + 0.5 \cdot \frac{1}{N}}} \]

       3. metadata-eval 94.6%

          \[\leadsto \frac{e^{\frac{\color{blue}{0.20833333333333334}}{{N}^{2}}}}{e^{\log N + 0.5 \cdot \frac{1}{N}}} \]

       4. exp-sum 94.7%

          \[\leadsto \frac{e^{\frac{0.20833333333333334}{{N}^{2}}}}{\color{blue}{e^{\log N} \cdot e^{0.5 \cdot \frac{1}{N}}}} \]

       5. rem-exp-log 99.4%

          \[\leadsto \frac{e^{\frac{0.20833333333333334}{{N}^{2}}}}{\color{blue}{N} \cdot e^{0.5 \cdot \frac{1}{N}}} \]

       6. associate-*r/ 99.4%

          \[\leadsto \frac{e^{\frac{0.20833333333333334}{{N}^{2}}}}{N \cdot e^{\color{blue}{\frac{0.5 \cdot 1}{N}}}} \]

       7. metadata-eval 99.4%

          \[\leadsto \frac{e^{\frac{0.20833333333333334}{{N}^{2}}}}{N \cdot e^{\frac{\color{blue}{0.5}}{N}}} \]

   12. Simplified 99.4%

       \[\leadsto \color{blue}{\frac{e^{\frac{0.20833333333333334}{{N}^{2}}}}{N \cdot e^{\frac{0.5}{N}}}} \]

   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. Add Preprocessing
   5. 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%

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

   6. Applied egg-rr 87.6%

      \[\leadsto \color{blue}{\log \left({\left(\sqrt[3]{\frac{N + 1}{N}}\right)}^{2}\right) + \log \left(\sqrt[3]{\frac{N + 1}{N}}\right)} \]
   7. 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%

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

   8. Simplified 87.5%

      \[\leadsto \color{blue}{3 \cdot \log \left(\sqrt[3]{\frac{N + 1}{N}}\right)} \]
   9. 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%

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

   10. Applied egg-rr 93.0%

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

       1. neg-sub0 93.0%

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

   12. Simplified 93.0%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 0.00015:\\ \;\;\;\;\frac{e^{\frac{0.20833333333333334}{{N}^{2}}}}{N \cdot e^{\frac{0.5}{N}}}\\ \mathbf{else}:\\ \;\;\;\;-\log \left(\frac{N}{N + 1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 18.2%

      \[\log \left(N + 1\right) - \log N \]
   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. Add Preprocessing

   5. Taylor expanded in N around inf 99.4%

      \[\leadsto \color{blue}{\left(0.3333333333333333 \cdot \frac{1}{{N}^{3}} + \frac{1}{N}\right) - 0.5 \cdot \frac{1}{{N}^{2}}} \]
   6. 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%

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

   7. 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. Add Preprocessing
   5. 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%

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

   6. Applied egg-rr 87.2%

      \[\leadsto \color{blue}{\log \left({\left(\sqrt[3]{\frac{N + 1}{N}}\right)}^{2}\right) + \log \left(\sqrt[3]{\frac{N + 1}{N}}\right)} \]
   7. 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%

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

   8. Simplified 87.1%

      \[\leadsto \color{blue}{3 \cdot \log \left(\sqrt[3]{\frac{N + 1}{N}}\right)} \]
   9. 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%

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

   10. Applied egg-rr 92.5%

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

       1. neg-sub0 92.5%

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

   12. 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%

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

Alternative 5: 97.8% accurate, 0.6× 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. Add Preprocessing

   5. Taylor expanded in N around inf 99.1%

      \[\leadsto \color{blue}{\frac{1}{N} - 0.5 \cdot \frac{1}{{N}^{2}}} \]
   6. 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%

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

   7. 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. Add Preprocessing
   5. 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%

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

   6. Applied egg-rr 82.3%

      \[\leadsto \color{blue}{\log \left({\left(\sqrt[3]{\frac{N + 1}{N}}\right)}^{2}\right) + \log \left(\sqrt[3]{\frac{N + 1}{N}}\right)} \]
   7. 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%

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

   8. Simplified 82.7%

      \[\leadsto \color{blue}{3 \cdot \log \left(\sqrt[3]{\frac{N + 1}{N}}\right)} \]
   9. 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%

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

   10. Applied egg-rr 87.2%

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

       1. neg-sub0 87.2%

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

   12. 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%

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

Alternative 6: 93.1% accurate, 1.8× 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. Add Preprocessing
   5. 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%

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

   6. Applied egg-rr 75.9%

      \[\leadsto \color{blue}{\log \left({\left(\sqrt[3]{\frac{N + 1}{N}}\right)}^{2}\right) + \log \left(\sqrt[3]{\frac{N + 1}{N}}\right)} \]
   7. 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%

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

   8. Simplified 76.1%

      \[\leadsto \color{blue}{3 \cdot \log \left(\sqrt[3]{\frac{N + 1}{N}}\right)} \]
   9. 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%

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

   10. Applied egg-rr 80.2%

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

       1. neg-sub0 80.2%

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

   12. 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. Add Preprocessing

   5. 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%

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

Alternative 7: 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. Add Preprocessing
   5. 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%

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

   6. 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. Add Preprocessing

   5. 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%

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

Alternative 8: 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. Add Preprocessing

5. Taylor expanded in N around inf 85.1%

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

6. Final simplification 85.1%

   \[\leadsto \frac{1}{N} \]
7. Add Preprocessing

Alternative 9: 8.4% accurate, 205.0× speedup

Math

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

FPCore

(FPCore (N) :precision binary64 N)

C

double code(double N) {
	return N;
}

Fortran

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

Java

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

Python

def code(N):
	return N

Julia

function code(N)
	return N
end

MATLAB

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

Wolfram

code[N_] := N

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. Add Preprocessing
5. Step-by-step derivation

   1. add-log-exp 23.4%

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

   2. add-cube-cbrt 23.3%

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

   3. log-prod 23.4%

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

   4. pow2 23.4%

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

   5. exp-diff 23.4%

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

   6. log1p-udef 23.4%

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

   7. rem-exp-log 24.5%

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

   8. add-exp-log 23.9%

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

   9. +-commutative 23.9%

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

   10. exp-diff 23.9%

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

   11. log1p-udef 23.8%

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

   12. rem-exp-log 25.1%

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

   13. add-exp-log 24.0%

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

6. Applied egg-rr 24.0%

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

   1. log-pow 24.1%

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

   2. distribute-lft1-in 24.1%

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

   3. metadata-eval 24.1%

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

8. Simplified 24.1%

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

9. Taylor expanded in N around inf 84.8%

   \[\leadsto 3 \cdot \color{blue}{\frac{0.3333333333333333}{N}} \]
10. Step-by-step derivation

    1. associate-*r/ 85.1%

       \[\leadsto \color{blue}{\frac{3 \cdot 0.3333333333333333}{N}} \]

    2. metadata-eval 85.1%

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

    3. inv-pow 85.1%

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

    4. metadata-eval 85.1%

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

    5. sqrt-pow1 85.0%

       \[\leadsto \color{blue}{\sqrt{{N}^{-2}}} \]

    6. pow-to-exp 81.5%

       \[\leadsto \sqrt{\color{blue}{e^{\log N \cdot -2}}} \]

    7. *-commutative 81.5%

       \[\leadsto \sqrt{e^{\color{blue}{-2 \cdot \log N}}} \]

    8. add-sqr-sqrt 0.0%

       \[\leadsto \sqrt{e^{\color{blue}{\sqrt{-2 \cdot \log N} \cdot \sqrt{-2 \cdot \log N}}}} \]

    9. sqrt-unprod 8.3%

       \[\leadsto \sqrt{e^{\color{blue}{\sqrt{\left(-2 \cdot \log N\right) \cdot \left(-2 \cdot \log N\right)}}}} \]

    10. sqr-neg 8.3%

        \[\leadsto \sqrt{e^{\sqrt{\color{blue}{\left(--2 \cdot \log N\right) \cdot \left(--2 \cdot \log N\right)}}}} \]

    11. sqrt-unprod 8.3%

        \[\leadsto \sqrt{e^{\color{blue}{\sqrt{--2 \cdot \log N} \cdot \sqrt{--2 \cdot \log N}}}} \]

    12. add-sqr-sqrt 8.3%

        \[\leadsto \sqrt{e^{\color{blue}{--2 \cdot \log N}}} \]

    13. distribute-lft-neg-in 8.3%

        \[\leadsto \sqrt{e^{\color{blue}{\left(--2\right) \cdot \log N}}} \]

    14. metadata-eval 8.3%

        \[\leadsto \sqrt{e^{\color{blue}{2} \cdot \log N}} \]

    15. log-pow 8.3%

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

    16. add-exp-log 8.3%

        \[\leadsto \sqrt{\color{blue}{{N}^{2}}} \]

    17. sqrt-pow1 8.3%

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

    18. metadata-eval 8.3%

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

11. Applied egg-rr 8.3%

    \[\leadsto \color{blue}{{N}^{1}} \]
12. Step-by-step derivation

    1. unpow1 8.3%

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

13. Simplified 8.3%

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

14. Final simplification 8.3%

    \[\leadsto N \]
15. Add Preprocessing

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)))