2log (problem 3.3.6)

Percentage Accurate: 23.4% → 99.4%

Time: 10.6s

Alternatives: 8

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: 23.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 0.6× speedup

Math

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

FPCore

(FPCore (N)
 :precision binary64
 (if (<= (- (log (+ N 1.0)) (log N)) 0.0005)
   (-
    (/ (+ (/ (+ (/ 0.3333333333333333 N) -0.5) N) 1.0) 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)) <= 0.0005) {
		tmp = (((((0.3333333333333333 / N) + -0.5) / N) + 1.0) / 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)) <= 0.0005d0) then
        tmp = (((((0.3333333333333333d0 / n) + (-0.5d0)) / n) + 1.0d0) / 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)) <= 0.0005) {
		tmp = (((((0.3333333333333333 / N) + -0.5) / N) + 1.0) / 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)) <= 0.0005:
		tmp = (((((0.3333333333333333 / N) + -0.5) / N) + 1.0) / 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)) <= 0.0005)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(0.3333333333333333 / N) + -0.5) / N) + 1.0) / 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)) <= 0.0005)
		tmp = (((((0.3333333333333333 / N) + -0.5) / N) + 1.0) / 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], 0.0005], N[(N[(N[(N[(N[(N[(0.3333333333333333 / N), $MachinePrecision] + -0.5), $MachinePrecision] / N), $MachinePrecision] + 1.0), $MachinePrecision] / N), $MachinePrecision] - N[(0.25 / N[Power[N, 4.0], $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 #s(literal 1 binary64))) (log.f64 N)) < 5.0000000000000001e-4

   1. Initial program 20.2%

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

      1. +-commutative 20.2%

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

      2. log1p-define 20.2%

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

   3. Simplified 20.2%

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

   5. Taylor expanded in N around -inf 99.8%

      \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{N} - 0.3333333333333333}{N} - 0.5}{N} - 1}{N}} \]
   6. Step-by-step derivation

      1. mul-1-neg 99.8%

         \[\leadsto \color{blue}{-\frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{N} - 0.3333333333333333}{N} - 0.5}{N} - 1}{N}} \]

      2. distribute-neg-frac2 99.8%

         \[\leadsto \color{blue}{\frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{N} - 0.3333333333333333}{N} - 0.5}{N} - 1}{-N}} \]

   7. Simplified 99.8%

      \[\leadsto \color{blue}{\frac{-1 - \frac{-0.5 + \frac{0.3333333333333333 + \frac{-0.25}{N}}{N}}{N}}{-N}} \]

   8. Taylor expanded in N around 0 99.3%

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

   9. Taylor expanded in N around inf 99.7%

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

   11. Simplified 99.8%

       \[\leadsto \color{blue}{\frac{1 + \frac{\frac{0.3333333333333333}{N} + -0.5}{N}}{N} - \frac{0.25}{{N}^{4}}} \]

   if 5.0000000000000001e-4 < (-.f64 (log.f64 (+.f64 N #s(literal 1 binary64))) (log.f64 N))

   1. Initial program 89.4%

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

      1. +-commutative 89.4%

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

      2. log1p-define 89.4%

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

   3. Simplified 89.4%

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

      1. add-log-exp 89.1%

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

      2. add-cube-cbrt 88.6%

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

      3. log-prod 88.7%

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

      4. pow2 88.7%

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

      5. exp-diff 88.7%

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

      6. log1p-undefine 88.7%

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

      7. rem-exp-log 88.9%

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

      8. add-exp-log 89.2%

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

      9. +-commutative 89.2%

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

      10. exp-diff 89.4%

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

      11. log1p-undefine 89.4%

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

      12. rem-exp-log 89.5%

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

      13. add-exp-log 89.6%

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

   6. Applied egg-rr 89.6%

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

      1. sum-log 89.4%

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

      2. unpow2 89.4%

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

      3. add-cube-cbrt 92.3%

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

      4. clear-num 92.2%

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

      5. log-div 93.4%

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

      6. metadata-eval 93.4%

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

   8. Applied egg-rr 93.4%

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

      1. neg-sub0 93.4%

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

   10. Simplified 93.4%

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

4. Final simplification 99.2%

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

Alternative 2: 99.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;N \leq 1950:\\ \;\;\;\;-\log \left(\frac{N}{N + 1}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-0.5 + \frac{0.3333333333333333 + \frac{-0.25}{N}}{N}}{N} + 1}{N}\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= N 1950.0)
   (- (log (/ N (+ N 1.0))))
   (/ (+ (/ (+ -0.5 (/ (+ 0.3333333333333333 (/ -0.25 N)) N)) N) 1.0) N)))

C

double code(double N) {
	double tmp;
	if (N <= 1950.0) {
		tmp = -log((N / (N + 1.0)));
	} else {
		tmp = (((-0.5 + ((0.3333333333333333 + (-0.25 / N)) / N)) / N) + 1.0) / N;
	}
	return tmp;
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= 1950.0d0) then
        tmp = -log((n / (n + 1.0d0)))
    else
        tmp = ((((-0.5d0) + ((0.3333333333333333d0 + ((-0.25d0) / n)) / n)) / n) + 1.0d0) / n
    end if
    code = tmp
end function

Java

public static double code(double N) {
	double tmp;
	if (N <= 1950.0) {
		tmp = -Math.log((N / (N + 1.0)));
	} else {
		tmp = (((-0.5 + ((0.3333333333333333 + (-0.25 / N)) / N)) / N) + 1.0) / N;
	}
	return tmp;
}

Python

def code(N):
	tmp = 0
	if N <= 1950.0:
		tmp = -math.log((N / (N + 1.0)))
	else:
		tmp = (((-0.5 + ((0.3333333333333333 + (-0.25 / N)) / N)) / N) + 1.0) / N
	return tmp

Julia

function code(N)
	tmp = 0.0
	if (N <= 1950.0)
		tmp = Float64(-log(Float64(N / Float64(N + 1.0))));
	else
		tmp = Float64(Float64(Float64(Float64(-0.5 + Float64(Float64(0.3333333333333333 + Float64(-0.25 / N)) / N)) / N) + 1.0) / N);
	end
	return tmp
end

MATLAB

function tmp_2 = code(N)
	tmp = 0.0;
	if (N <= 1950.0)
		tmp = -log((N / (N + 1.0)));
	else
		tmp = (((-0.5 + ((0.3333333333333333 + (-0.25 / N)) / N)) / N) + 1.0) / N;
	end
	tmp_2 = tmp;
end

Wolfram

code[N_] := If[LessEqual[N, 1950.0], (-N[Log[N[(N / N[(N + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), N[(N[(N[(N[(-0.5 + N[(N[(0.3333333333333333 + N[(-0.25 / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision] + 1.0), $MachinePrecision] / N), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if N < 1950

   1. Initial program 89.4%

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

      1. +-commutative 89.4%

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

      2. log1p-define 89.4%

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

   3. Simplified 89.4%

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

      1. add-log-exp 89.1%

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

      2. add-cube-cbrt 88.6%

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

      3. log-prod 88.7%

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

      4. pow2 88.7%

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

      5. exp-diff 88.7%

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

      6. log1p-undefine 88.7%

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

      7. rem-exp-log 88.9%

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

      8. add-exp-log 89.2%

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

      9. +-commutative 89.2%

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

      10. exp-diff 89.4%

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

      11. log1p-undefine 89.4%

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

      12. rem-exp-log 89.5%

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

      13. add-exp-log 89.6%

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

   6. Applied egg-rr 89.6%

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

      1. sum-log 89.4%

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

      2. unpow2 89.4%

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

      3. add-cube-cbrt 92.3%

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

      4. clear-num 92.2%

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

      5. log-div 93.4%

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

      6. metadata-eval 93.4%

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

   8. Applied egg-rr 93.4%

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

      1. neg-sub0 93.4%

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

   10. Simplified 93.4%

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

   if 1950 < N

   1. Initial program 20.2%

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

      1. +-commutative 20.2%

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

      2. log1p-define 20.2%

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

   3. Simplified 20.2%

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

   5. Taylor expanded in N around -inf 99.8%

      \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{N} - 0.3333333333333333}{N} - 0.5}{N} - 1}{N}} \]
   6. Step-by-step derivation

      1. mul-1-neg 99.8%

         \[\leadsto \color{blue}{-\frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{N} - 0.3333333333333333}{N} - 0.5}{N} - 1}{N}} \]

      2. distribute-neg-frac2 99.8%

         \[\leadsto \color{blue}{\frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{N} - 0.3333333333333333}{N} - 0.5}{N} - 1}{-N}} \]

   7. Simplified 99.8%

      \[\leadsto \color{blue}{\frac{-1 - \frac{-0.5 + \frac{0.3333333333333333 + \frac{-0.25}{N}}{N}}{N}}{-N}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;N \leq 1950:\\ \;\;\;\;-\log \left(\frac{N}{N + 1}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-0.5 + \frac{0.3333333333333333 + \frac{-0.25}{N}}{N}}{N} + 1}{N}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;N \leq 1350:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-0.5 + \frac{0.3333333333333333 + \frac{-0.25}{N}}{N}}{N} + 1}{N}\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= N 1350.0)
   (log (/ (+ N 1.0) N))
   (/ (+ (/ (+ -0.5 (/ (+ 0.3333333333333333 (/ -0.25 N)) N)) N) 1.0) N)))

C

double code(double N) {
	double tmp;
	if (N <= 1350.0) {
		tmp = log(((N + 1.0) / N));
	} else {
		tmp = (((-0.5 + ((0.3333333333333333 + (-0.25 / N)) / N)) / N) + 1.0) / N;
	}
	return tmp;
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= 1350.0d0) then
        tmp = log(((n + 1.0d0) / n))
    else
        tmp = ((((-0.5d0) + ((0.3333333333333333d0 + ((-0.25d0) / n)) / n)) / n) + 1.0d0) / n
    end if
    code = tmp
end function

Java

public static double code(double N) {
	double tmp;
	if (N <= 1350.0) {
		tmp = Math.log(((N + 1.0) / N));
	} else {
		tmp = (((-0.5 + ((0.3333333333333333 + (-0.25 / N)) / N)) / N) + 1.0) / N;
	}
	return tmp;
}

Python

def code(N):
	tmp = 0
	if N <= 1350.0:
		tmp = math.log(((N + 1.0) / N))
	else:
		tmp = (((-0.5 + ((0.3333333333333333 + (-0.25 / N)) / N)) / N) + 1.0) / N
	return tmp

Julia

function code(N)
	tmp = 0.0
	if (N <= 1350.0)
		tmp = log(Float64(Float64(N + 1.0) / N));
	else
		tmp = Float64(Float64(Float64(Float64(-0.5 + Float64(Float64(0.3333333333333333 + Float64(-0.25 / N)) / N)) / N) + 1.0) / N);
	end
	return tmp
end

MATLAB

function tmp_2 = code(N)
	tmp = 0.0;
	if (N <= 1350.0)
		tmp = log(((N + 1.0) / N));
	else
		tmp = (((-0.5 + ((0.3333333333333333 + (-0.25 / N)) / N)) / N) + 1.0) / N;
	end
	tmp_2 = tmp;
end

Wolfram

code[N_] := If[LessEqual[N, 1350.0], N[Log[N[(N[(N + 1.0), $MachinePrecision] / N), $MachinePrecision]], $MachinePrecision], N[(N[(N[(N[(-0.5 + N[(N[(0.3333333333333333 + N[(-0.25 / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision] + 1.0), $MachinePrecision] / N), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if N < 1350

   1. Initial program 89.4%

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

      1. +-commutative 89.4%

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

      2. log1p-define 89.4%

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

   3. Simplified 89.4%

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

      1. add-log-exp 89.4%

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

      2. log1p-expm1-u 89.4%

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

      3. log1p-undefine 89.4%

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

      4. diff-log 89.2%

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

      5. log1p-undefine 89.2%

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

      6. rem-exp-log 89.7%

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

      7. +-commutative 89.7%

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

      8. add-exp-log 89.5%

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

      9. log1p-undefine 89.5%

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

      10. log1p-expm1-u 89.5%

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

      11. add-exp-log 92.3%

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

   6. Applied egg-rr 92.3%

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

   if 1350 < N

   1. Initial program 20.2%

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

      1. +-commutative 20.2%

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

      2. log1p-define 20.2%

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

   3. Simplified 20.2%

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

   5. Taylor expanded in N around -inf 99.8%

      \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{N} - 0.3333333333333333}{N} - 0.5}{N} - 1}{N}} \]
   6. Step-by-step derivation

      1. mul-1-neg 99.8%

         \[\leadsto \color{blue}{-\frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{N} - 0.3333333333333333}{N} - 0.5}{N} - 1}{N}} \]

      2. distribute-neg-frac2 99.8%

         \[\leadsto \color{blue}{\frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{N} - 0.3333333333333333}{N} - 0.5}{N} - 1}{-N}} \]

   7. Simplified 99.8%

      \[\leadsto \color{blue}{\frac{-1 - \frac{-0.5 + \frac{0.3333333333333333 + \frac{-0.25}{N}}{N}}{N}}{-N}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;N \leq 1350:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-0.5 + \frac{0.3333333333333333 + \frac{-0.25}{N}}{N}}{N} + 1}{N}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.4% accurate, 13.7× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{-0.5 + \frac{0.3333333333333333 + \frac{-0.25}{N}}{N}}{N} + 1}{N} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (/ (+ (/ (+ -0.5 (/ (+ 0.3333333333333333 (/ -0.25 N)) N)) N) 1.0) N))

C

double code(double N) {
	return (((-0.5 + ((0.3333333333333333 + (-0.25 / N)) / N)) / N) + 1.0) / N;
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    code = ((((-0.5d0) + ((0.3333333333333333d0 + ((-0.25d0) / n)) / n)) / n) + 1.0d0) / n
end function

Java

public static double code(double N) {
	return (((-0.5 + ((0.3333333333333333 + (-0.25 / N)) / N)) / N) + 1.0) / N;
}

Python

def code(N):
	return (((-0.5 + ((0.3333333333333333 + (-0.25 / N)) / N)) / N) + 1.0) / N

Julia

function code(N)
	return Float64(Float64(Float64(Float64(-0.5 + Float64(Float64(0.3333333333333333 + Float64(-0.25 / N)) / N)) / N) + 1.0) / N)
end

MATLAB

function tmp = code(N)
	tmp = (((-0.5 + ((0.3333333333333333 + (-0.25 / N)) / N)) / N) + 1.0) / N;
end

Wolfram

code[N_] := N[(N[(N[(N[(-0.5 + N[(N[(0.3333333333333333 + N[(-0.25 / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision] + 1.0), $MachinePrecision] / N), $MachinePrecision]

Derivation

1. Initial program 26.4%

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

   1. +-commutative 26.4%

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

   2. log1p-define 26.4%

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

3. Simplified 26.4%

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

5. Taylor expanded in N around -inf 96.4%

   \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{N} - 0.3333333333333333}{N} - 0.5}{N} - 1}{N}} \]
6. Step-by-step derivation

   1. mul-1-neg 96.4%

      \[\leadsto \color{blue}{-\frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{N} - 0.3333333333333333}{N} - 0.5}{N} - 1}{N}} \]

   2. distribute-neg-frac2 96.4%

      \[\leadsto \color{blue}{\frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{N} - 0.3333333333333333}{N} - 0.5}{N} - 1}{-N}} \]

7. Simplified 96.4%

   \[\leadsto \color{blue}{\frac{-1 - \frac{-0.5 + \frac{0.3333333333333333 + \frac{-0.25}{N}}{N}}{N}}{-N}} \]

8. Final simplification 96.4%

   \[\leadsto \frac{\frac{-0.5 + \frac{0.3333333333333333 + \frac{-0.25}{N}}{N}}{N} + 1}{N} \]
9. Add Preprocessing

Alternative 5: 95.2% accurate, 18.6× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\frac{0.3333333333333333}{N} + -0.5}{N} + 1}{N} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (/ (+ (/ (+ (/ 0.3333333333333333 N) -0.5) N) 1.0) N))

C

double code(double N) {
	return ((((0.3333333333333333 / N) + -0.5) / N) + 1.0) / N;
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    code = ((((0.3333333333333333d0 / n) + (-0.5d0)) / n) + 1.0d0) / n
end function

Java

public static double code(double N) {
	return ((((0.3333333333333333 / N) + -0.5) / N) + 1.0) / N;
}

Python

def code(N):
	return ((((0.3333333333333333 / N) + -0.5) / N) + 1.0) / N

Julia

function code(N)
	return Float64(Float64(Float64(Float64(Float64(0.3333333333333333 / N) + -0.5) / N) + 1.0) / N)
end

MATLAB

function tmp = code(N)
	tmp = ((((0.3333333333333333 / N) + -0.5) / N) + 1.0) / N;
end

Wolfram

code[N_] := N[(N[(N[(N[(N[(0.3333333333333333 / N), $MachinePrecision] + -0.5), $MachinePrecision] / N), $MachinePrecision] + 1.0), $MachinePrecision] / N), $MachinePrecision]

Derivation

1. Initial program 26.4%

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

   1. +-commutative 26.4%

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

   2. log1p-define 26.4%

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

3. Simplified 26.4%

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

5. Taylor expanded in N around inf 94.8%

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

   1. associate--l+ 94.8%

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

   2. unpow2 94.8%

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

   3. associate-/r* 94.8%

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

   4. metadata-eval 94.8%

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

   5. associate-*r/ 94.8%

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

   6. associate-*r/ 94.8%

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

   7. metadata-eval 94.8%

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

   8. div-sub 94.8%

      \[\leadsto \frac{1 + \color{blue}{\frac{0.3333333333333333 \cdot \frac{1}{N} - 0.5}{N}}}{N} \]

   9. sub-neg 94.8%

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

   10. metadata-eval 94.8%

       \[\leadsto \frac{1 + \frac{0.3333333333333333 \cdot \frac{1}{N} + \color{blue}{-0.5}}{N}}{N} \]

   11. +-commutative 94.8%

       \[\leadsto \frac{1 + \frac{\color{blue}{-0.5 + 0.3333333333333333 \cdot \frac{1}{N}}}{N}}{N} \]

   12. associate-*r/ 94.8%

       \[\leadsto \frac{1 + \frac{-0.5 + \color{blue}{\frac{0.3333333333333333 \cdot 1}{N}}}{N}}{N} \]

   13. metadata-eval 94.8%

       \[\leadsto \frac{1 + \frac{-0.5 + \frac{\color{blue}{0.3333333333333333}}{N}}{N}}{N} \]

7. Simplified 94.8%

   \[\leadsto \color{blue}{\frac{1 + \frac{-0.5 + \frac{0.3333333333333333}{N}}{N}}{N}} \]

8. Final simplification 94.8%

   \[\leadsto \frac{\frac{\frac{0.3333333333333333}{N} + -0.5}{N} + 1}{N} \]
9. Add Preprocessing

Alternative 6: 92.6% accurate, 22.8× speedup

Math

\[\begin{array}{l} \\ \frac{-1}{\frac{N}{-1 - \frac{-0.5}{N}}} \end{array} \]

FPCore

(FPCore (N) :precision binary64 (/ -1.0 (/ N (- -1.0 (/ -0.5 N)))))

C

double code(double N) {
	return -1.0 / (N / (-1.0 - (-0.5 / N)));
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    code = (-1.0d0) / (n / ((-1.0d0) - ((-0.5d0) / n)))
end function

Java

public static double code(double N) {
	return -1.0 / (N / (-1.0 - (-0.5 / N)));
}

Python

def code(N):
	return -1.0 / (N / (-1.0 - (-0.5 / N)))

Julia

function code(N)
	return Float64(-1.0 / Float64(N / Float64(-1.0 - Float64(-0.5 / N))))
end

MATLAB

function tmp = code(N)
	tmp = -1.0 / (N / (-1.0 - (-0.5 / N)));
end

Wolfram

code[N_] := N[(-1.0 / N[(N / N[(-1.0 - N[(-0.5 / N), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 26.4%

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

   1. +-commutative 26.4%

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

   2. log1p-define 26.4%

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

3. Simplified 26.4%

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

5. Taylor expanded in N around inf 91.9%

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

   1. associate-*r/ 91.9%

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

   2. metadata-eval 91.9%

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

7. Simplified 91.9%

   \[\leadsto \color{blue}{\frac{1 - \frac{0.5}{N}}{N}} \]
8. Step-by-step derivation

   1. clear-num 92.0%

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

   2. inv-pow 92.0%

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

9. Applied egg-rr 92.0%

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

    1. unpow-1 92.0%

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

    2. sub-neg 92.0%

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

    3. distribute-neg-frac 92.0%

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

    4. metadata-eval 92.0%

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

11. Simplified 92.0%

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

12. Final simplification 92.0%

    \[\leadsto \frac{-1}{\frac{N}{-1 - \frac{-0.5}{N}}} \]
13. Add Preprocessing

Alternative 7: 92.6% accurate, 29.3× speedup

Math

\[\begin{array}{l} \\ \frac{1 - \frac{0.5}{N}}{N} \end{array} \]

FPCore

(FPCore (N) :precision binary64 (/ (- 1.0 (/ 0.5 N)) N))

C

double code(double N) {
	return (1.0 - (0.5 / N)) / N;
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    code = (1.0d0 - (0.5d0 / n)) / n
end function

Java

public static double code(double N) {
	return (1.0 - (0.5 / N)) / N;
}

Python

def code(N):
	return (1.0 - (0.5 / N)) / N

Julia

function code(N)
	return Float64(Float64(1.0 - Float64(0.5 / N)) / N)
end

MATLAB

function tmp = code(N)
	tmp = (1.0 - (0.5 / N)) / N;
end

Wolfram

code[N_] := N[(N[(1.0 - N[(0.5 / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision]

Derivation

1. Initial program 26.4%

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

   1. +-commutative 26.4%

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

   2. log1p-define 26.4%

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

3. Simplified 26.4%

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

5. Taylor expanded in N around inf 91.9%

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

   1. associate-*r/ 91.9%

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

   2. metadata-eval 91.9%

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

7. Simplified 91.9%

   \[\leadsto \color{blue}{\frac{1 - \frac{0.5}{N}}{N}} \]
8. Add Preprocessing

Alternative 8: 84.8% 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 26.4%

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

   1. +-commutative 26.4%

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

   2. log1p-define 26.4%

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

3. Simplified 26.4%

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

5. Taylor expanded in N around inf 82.4%

   \[\leadsto \color{blue}{\frac{1}{N}} \]
6. 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 2024108 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  :pre (and (> N 1.0) (< N 1e+40))

  :alt
  (log1p (/ 1.0 N))

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