2log (problem 3.3.6)

Percentage Accurate: 23.9% → 99.5%

Time: 11.1s

Alternatives: 12

Speedup: 68.3×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 23.9% 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: 99.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 0.001:\\ \;\;\;\;\frac{-1}{N \cdot \frac{-0.5 + \frac{0.08333333333333333 + \frac{-0.041666666666666664}{N}}{N}}{N} - N}\\ \mathbf{else}:\\ \;\;\;\;0 - \log \left(\frac{N}{N + 1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= (- (log (+ N 1.0)) (log N)) 0.001)
   (/
    -1.0
    (-
     (*
      N
      (/ (+ -0.5 (/ (+ 0.08333333333333333 (/ -0.041666666666666664 N)) N)) N))
     N))
   (- 0.0 (log (/ N (+ N 1.0))))))

C

double code(double N) {
	double tmp;
	if ((log((N + 1.0)) - log(N)) <= 0.001) {
		tmp = -1.0 / ((N * ((-0.5 + ((0.08333333333333333 + (-0.041666666666666664 / N)) / N)) / N)) - N);
	} else {
		tmp = 0.0 - 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)) <= 0.001d0) then
        tmp = (-1.0d0) / ((n * (((-0.5d0) + ((0.08333333333333333d0 + ((-0.041666666666666664d0) / n)) / n)) / n)) - n)
    else
        tmp = 0.0d0 - 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)) <= 0.001) {
		tmp = -1.0 / ((N * ((-0.5 + ((0.08333333333333333 + (-0.041666666666666664 / N)) / N)) / N)) - N);
	} else {
		tmp = 0.0 - Math.log((N / (N + 1.0)));
	}
	return tmp;
}

Python

def code(N):
	tmp = 0
	if (math.log((N + 1.0)) - math.log(N)) <= 0.001:
		tmp = -1.0 / ((N * ((-0.5 + ((0.08333333333333333 + (-0.041666666666666664 / N)) / N)) / N)) - N)
	else:
		tmp = 0.0 - math.log((N / (N + 1.0)))
	return tmp

Julia

function code(N)
	tmp = 0.0
	if (Float64(log(Float64(N + 1.0)) - log(N)) <= 0.001)
		tmp = Float64(-1.0 / Float64(Float64(N * Float64(Float64(-0.5 + Float64(Float64(0.08333333333333333 + Float64(-0.041666666666666664 / N)) / N)) / N)) - N));
	else
		tmp = Float64(0.0 - 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)) <= 0.001)
		tmp = -1.0 / ((N * ((-0.5 + ((0.08333333333333333 + (-0.041666666666666664 / N)) / N)) / N)) - N);
	else
		tmp = 0.0 - 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], 0.001], N[(-1.0 / N[(N[(N * N[(N[(-0.5 + N[(N[(0.08333333333333333 + N[(-0.041666666666666664 / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision]), $MachinePrecision] - N), $MachinePrecision]), $MachinePrecision], N[(0.0 - N[Log[N[(N / N[(N + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (log.f64 (+.f64 N #s(literal 1 binary64))) (log.f64 N)) < 1e-3

   1. Initial program 18.9%

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

   3. Taylor expanded in N around inf

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

   5. Simplified 99.8%

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

      1. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{N}{1 + \frac{\frac{-1}{2} + \frac{\frac{1}{3} + \frac{\frac{-1}{4}}{N}}{N}}{N}}}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{N}{1 + \frac{\frac{-1}{2} + \frac{\frac{1}{3} + \frac{\frac{-1}{4}}{N}}{N}}{N}}\right)}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \color{blue}{\left(1 + \frac{\frac{-1}{2} + \frac{\frac{1}{3} + \frac{\frac{-1}{4}}{N}}{N}}{N}\right)}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{\frac{-1}{2} + \frac{\frac{1}{3} + \frac{\frac{-1}{4}}{N}}{N}}{N}\right)}\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{-1}{2} + \frac{\frac{1}{3} + \frac{\frac{-1}{4}}{N}}{N}\right), \color{blue}{N}\right)\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{-1}{2}, \left(\frac{\frac{1}{3} + \frac{\frac{-1}{4}}{N}}{N}\right)\right), N\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\left(\frac{1}{3} + \frac{\frac{-1}{4}}{N}\right), N\right)\right), N\right)\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{3}, \left(\frac{\frac{-1}{4}}{N}\right)\right), N\right)\right), N\right)\right)\right)\right) \]

      9. /-lowering-/.f64 99.8%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\frac{-1}{4}, N\right)\right), N\right)\right), N\right)\right)\right)\right) \]

   7. Applied egg-rr 99.8%

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

   8. Taylor expanded in N around -inf

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(-1 \cdot \left(N \cdot \left(-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N} - 1\right)\right)\right)}\right) \]
   9. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\mathsf{neg}\left(N \cdot \left(-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N} - 1\right)\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\mathsf{neg}\left(\left(-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N} - 1\right) \cdot N\right)\right)\right) \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\left(-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N} - 1\right) \cdot \color{blue}{\left(\mathsf{neg}\left(N\right)\right)}\right)\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\left(-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N} - 1\right) \cdot \left(-1 \cdot \color{blue}{N}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N} - 1\right), \color{blue}{\left(-1 \cdot N\right)}\right)\right) \]

   10. Simplified 99.8%

       \[\leadsto \frac{1}{\color{blue}{\left(\frac{0.5 - \frac{0.08333333333333333 + \frac{-0.041666666666666664}{N}}{N}}{0 - N} + -1\right) \cdot \left(0 - N\right)}} \]
   11. Step-by-step derivation

       1. sub0-neg N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\left(\frac{\frac{1}{2} - \frac{\frac{1}{12} + \frac{\frac{-1}{24}}{N}}{N}}{0 - N} + -1\right) \cdot \left(\mathsf{neg}\left(N\right)\right)\right)\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\left(\mathsf{neg}\left(N\right)\right) \cdot \color{blue}{\left(\frac{\frac{1}{2} - \frac{\frac{1}{12} + \frac{\frac{-1}{24}}{N}}{N}}{0 - N} + -1\right)}\right)\right) \]

       3. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{\frac{1}{2} - \frac{\frac{1}{12} + \frac{\frac{-1}{24}}{N}}{N}}{0 - N} \cdot \left(\mathsf{neg}\left(N\right)\right) + \color{blue}{-1 \cdot \left(\mathsf{neg}\left(N\right)\right)}\right)\right) \]

       4. neg-mul-1 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{\frac{1}{2} - \frac{\frac{1}{12} + \frac{\frac{-1}{24}}{N}}{N}}{0 - N} \cdot \left(\mathsf{neg}\left(N\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(N\right)\right)\right)\right)\right)\right) \]

       5. remove-double-neg N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{\frac{1}{2} - \frac{\frac{1}{12} + \frac{\frac{-1}{24}}{N}}{N}}{0 - N} \cdot \left(\mathsf{neg}\left(N\right)\right) + N\right)\right) \]

       6. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\left(\frac{\frac{1}{2} - \frac{\frac{1}{12} + \frac{\frac{-1}{24}}{N}}{N}}{0 - N} \cdot \left(\mathsf{neg}\left(N\right)\right)\right), \color{blue}{N}\right)\right) \]

   12. Applied egg-rr 99.9%

       \[\leadsto \frac{1}{\color{blue}{\frac{-0.5 + \frac{0.08333333333333333 + \frac{-0.041666666666666664}{N}}{N}}{N} \cdot \left(0 - N\right) + N}} \]

   if 1e-3 < (-.f64 (log.f64 (+.f64 N #s(literal 1 binary64))) (log.f64 N))

   1. Initial program 93.0%

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

      1. diff-log N/A

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

      2. clear-num N/A

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

      3. neg-log N/A

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

      4. diff-log N/A

         \[\leadsto \mathsf{neg}\left(\left(\log N - \log \left(N + 1\right)\right)\right) \]

      5. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\log N - \log \left(N + 1\right)\right)\right) \]

      6. diff-log N/A

         \[\leadsto \mathsf{neg.f64}\left(\log \left(\frac{N}{N + 1}\right)\right) \]

      7. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\left(\frac{N}{N + 1}\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(N, \left(N + 1\right)\right)\right)\right) \]

      9. +-lowering-+.f64 95.5%

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(N, \mathsf{+.f64}\left(N, 1\right)\right)\right)\right) \]

   4. Applied egg-rr 95.5%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 0.001:\\ \;\;\;\;\frac{-1}{N \cdot \frac{-0.5 + \frac{0.08333333333333333 + \frac{-0.041666666666666664}{N}}{N}}{N} - N}\\ \mathbf{else}:\\ \;\;\;\;0 - \log \left(\frac{N}{N + 1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;N \leq 1020:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{N \cdot \frac{-0.5 + \frac{0.08333333333333333 + \frac{-0.041666666666666664}{N}}{N}}{N} - N}\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= N 1020.0)
   (log (/ (+ N 1.0) N))
   (/
    -1.0
    (-
     (*
      N
      (/ (+ -0.5 (/ (+ 0.08333333333333333 (/ -0.041666666666666664 N)) N)) N))
     N))))

C

double code(double N) {
	double tmp;
	if (N <= 1020.0) {
		tmp = log(((N + 1.0) / N));
	} else {
		tmp = -1.0 / ((N * ((-0.5 + ((0.08333333333333333 + (-0.041666666666666664 / N)) / N)) / N)) - N);
	}
	return tmp;
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= 1020.0d0) then
        tmp = log(((n + 1.0d0) / n))
    else
        tmp = (-1.0d0) / ((n * (((-0.5d0) + ((0.08333333333333333d0 + ((-0.041666666666666664d0) / n)) / n)) / n)) - n)
    end if
    code = tmp
end function

Java

public static double code(double N) {
	double tmp;
	if (N <= 1020.0) {
		tmp = Math.log(((N + 1.0) / N));
	} else {
		tmp = -1.0 / ((N * ((-0.5 + ((0.08333333333333333 + (-0.041666666666666664 / N)) / N)) / N)) - N);
	}
	return tmp;
}

Python

def code(N):
	tmp = 0
	if N <= 1020.0:
		tmp = math.log(((N + 1.0) / N))
	else:
		tmp = -1.0 / ((N * ((-0.5 + ((0.08333333333333333 + (-0.041666666666666664 / N)) / N)) / N)) - N)
	return tmp

Julia

function code(N)
	tmp = 0.0
	if (N <= 1020.0)
		tmp = log(Float64(Float64(N + 1.0) / N));
	else
		tmp = Float64(-1.0 / Float64(Float64(N * Float64(Float64(-0.5 + Float64(Float64(0.08333333333333333 + Float64(-0.041666666666666664 / N)) / N)) / N)) - N));
	end
	return tmp
end

MATLAB

function tmp_2 = code(N)
	tmp = 0.0;
	if (N <= 1020.0)
		tmp = log(((N + 1.0) / N));
	else
		tmp = -1.0 / ((N * ((-0.5 + ((0.08333333333333333 + (-0.041666666666666664 / N)) / N)) / N)) - N);
	end
	tmp_2 = tmp;
end

Wolfram

code[N_] := If[LessEqual[N, 1020.0], N[Log[N[(N[(N + 1.0), $MachinePrecision] / N), $MachinePrecision]], $MachinePrecision], N[(-1.0 / N[(N[(N * N[(N[(-0.5 + N[(N[(0.08333333333333333 + N[(-0.041666666666666664 / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision]), $MachinePrecision] - N), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if N < 1020

   1. Initial program 93.0%

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

      1. diff-log N/A

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

      2. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{log.f64}\left(\left(\frac{N + 1}{N}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{/.f64}\left(\left(N + 1\right), N\right)\right) \]

      4. +-lowering-+.f64 95.1%

         \[\leadsto \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(N, 1\right), N\right)\right) \]

   4. Applied egg-rr 95.1%

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

   if 1020 < N

   1. Initial program 18.9%

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

   3. Taylor expanded in N around inf

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

   5. Simplified 99.8%

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

      1. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{N}{1 + \frac{\frac{-1}{2} + \frac{\frac{1}{3} + \frac{\frac{-1}{4}}{N}}{N}}{N}}}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{N}{1 + \frac{\frac{-1}{2} + \frac{\frac{1}{3} + \frac{\frac{-1}{4}}{N}}{N}}{N}}\right)}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \color{blue}{\left(1 + \frac{\frac{-1}{2} + \frac{\frac{1}{3} + \frac{\frac{-1}{4}}{N}}{N}}{N}\right)}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{\frac{-1}{2} + \frac{\frac{1}{3} + \frac{\frac{-1}{4}}{N}}{N}}{N}\right)}\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{-1}{2} + \frac{\frac{1}{3} + \frac{\frac{-1}{4}}{N}}{N}\right), \color{blue}{N}\right)\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{-1}{2}, \left(\frac{\frac{1}{3} + \frac{\frac{-1}{4}}{N}}{N}\right)\right), N\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\left(\frac{1}{3} + \frac{\frac{-1}{4}}{N}\right), N\right)\right), N\right)\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{3}, \left(\frac{\frac{-1}{4}}{N}\right)\right), N\right)\right), N\right)\right)\right)\right) \]

      9. /-lowering-/.f64 99.8%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\frac{-1}{4}, N\right)\right), N\right)\right), N\right)\right)\right)\right) \]

   7. Applied egg-rr 99.8%

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

   8. Taylor expanded in N around -inf

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(-1 \cdot \left(N \cdot \left(-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N} - 1\right)\right)\right)}\right) \]
   9. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\mathsf{neg}\left(N \cdot \left(-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N} - 1\right)\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\mathsf{neg}\left(\left(-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N} - 1\right) \cdot N\right)\right)\right) \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\left(-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N} - 1\right) \cdot \color{blue}{\left(\mathsf{neg}\left(N\right)\right)}\right)\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\left(-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N} - 1\right) \cdot \left(-1 \cdot \color{blue}{N}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N} - 1\right), \color{blue}{\left(-1 \cdot N\right)}\right)\right) \]

   10. Simplified 99.8%

       \[\leadsto \frac{1}{\color{blue}{\left(\frac{0.5 - \frac{0.08333333333333333 + \frac{-0.041666666666666664}{N}}{N}}{0 - N} + -1\right) \cdot \left(0 - N\right)}} \]
   11. Step-by-step derivation

       1. sub0-neg N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\left(\frac{\frac{1}{2} - \frac{\frac{1}{12} + \frac{\frac{-1}{24}}{N}}{N}}{0 - N} + -1\right) \cdot \left(\mathsf{neg}\left(N\right)\right)\right)\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\left(\mathsf{neg}\left(N\right)\right) \cdot \color{blue}{\left(\frac{\frac{1}{2} - \frac{\frac{1}{12} + \frac{\frac{-1}{24}}{N}}{N}}{0 - N} + -1\right)}\right)\right) \]

       3. distribute-rgt-in N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{\frac{1}{2} - \frac{\frac{1}{12} + \frac{\frac{-1}{24}}{N}}{N}}{0 - N} \cdot \left(\mathsf{neg}\left(N\right)\right) + \color{blue}{-1 \cdot \left(\mathsf{neg}\left(N\right)\right)}\right)\right) \]

       4. neg-mul-1 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{\frac{1}{2} - \frac{\frac{1}{12} + \frac{\frac{-1}{24}}{N}}{N}}{0 - N} \cdot \left(\mathsf{neg}\left(N\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(N\right)\right)\right)\right)\right)\right) \]

       5. remove-double-neg N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{\frac{1}{2} - \frac{\frac{1}{12} + \frac{\frac{-1}{24}}{N}}{N}}{0 - N} \cdot \left(\mathsf{neg}\left(N\right)\right) + N\right)\right) \]

       6. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\left(\frac{\frac{1}{2} - \frac{\frac{1}{12} + \frac{\frac{-1}{24}}{N}}{N}}{0 - N} \cdot \left(\mathsf{neg}\left(N\right)\right)\right), \color{blue}{N}\right)\right) \]

   12. Applied egg-rr 99.9%

       \[\leadsto \frac{1}{\color{blue}{\frac{-0.5 + \frac{0.08333333333333333 + \frac{-0.041666666666666664}{N}}{N}}{N} \cdot \left(0 - N\right) + N}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;N \leq 1020:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{N \cdot \frac{-0.5 + \frac{0.08333333333333333 + \frac{-0.041666666666666664}{N}}{N}}{N} - N}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.0% accurate, 12.1× speedup

Math

\[\begin{array}{l} \\ \frac{-1}{N \cdot \frac{-0.5 + \frac{0.08333333333333333 + \frac{-0.041666666666666664}{N}}{N}}{N} - N} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (/
  -1.0
  (-
   (*
    N
    (/ (+ -0.5 (/ (+ 0.08333333333333333 (/ -0.041666666666666664 N)) N)) N))
   N)))

C

double code(double N) {
	return -1.0 / ((N * ((-0.5 + ((0.08333333333333333 + (-0.041666666666666664 / N)) / N)) / N)) - N);
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    code = (-1.0d0) / ((n * (((-0.5d0) + ((0.08333333333333333d0 + ((-0.041666666666666664d0) / n)) / n)) / n)) - n)
end function

Java

public static double code(double N) {
	return -1.0 / ((N * ((-0.5 + ((0.08333333333333333 + (-0.041666666666666664 / N)) / N)) / N)) - N);
}

Python

def code(N):
	return -1.0 / ((N * ((-0.5 + ((0.08333333333333333 + (-0.041666666666666664 / N)) / N)) / N)) - N)

Julia

function code(N)
	return Float64(-1.0 / Float64(Float64(N * Float64(Float64(-0.5 + Float64(Float64(0.08333333333333333 + Float64(-0.041666666666666664 / N)) / N)) / N)) - N))
end

MATLAB

function tmp = code(N)
	tmp = -1.0 / ((N * ((-0.5 + ((0.08333333333333333 + (-0.041666666666666664 / N)) / N)) / N)) - N);
end

Wolfram

code[N_] := N[(-1.0 / N[(N[(N * N[(N[(-0.5 + N[(N[(0.08333333333333333 + N[(-0.041666666666666664 / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision]), $MachinePrecision] - N), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 23.2%

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

3. Taylor expanded in N around inf

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

5. Simplified 97.0%

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

   1. clear-num N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{N}{1 + \frac{\frac{-1}{2} + \frac{\frac{1}{3} + \frac{\frac{-1}{4}}{N}}{N}}{N}}}} \]

   2. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{N}{1 + \frac{\frac{-1}{2} + \frac{\frac{1}{3} + \frac{\frac{-1}{4}}{N}}{N}}{N}}\right)}\right) \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \color{blue}{\left(1 + \frac{\frac{-1}{2} + \frac{\frac{1}{3} + \frac{\frac{-1}{4}}{N}}{N}}{N}\right)}\right)\right) \]

   4. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{\frac{-1}{2} + \frac{\frac{1}{3} + \frac{\frac{-1}{4}}{N}}{N}}{N}\right)}\right)\right)\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{-1}{2} + \frac{\frac{1}{3} + \frac{\frac{-1}{4}}{N}}{N}\right), \color{blue}{N}\right)\right)\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{-1}{2}, \left(\frac{\frac{1}{3} + \frac{\frac{-1}{4}}{N}}{N}\right)\right), N\right)\right)\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\left(\frac{1}{3} + \frac{\frac{-1}{4}}{N}\right), N\right)\right), N\right)\right)\right)\right) \]

   8. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{3}, \left(\frac{\frac{-1}{4}}{N}\right)\right), N\right)\right), N\right)\right)\right)\right) \]

   9. /-lowering-/.f64 97.0%

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\frac{-1}{4}, N\right)\right), N\right)\right), N\right)\right)\right)\right) \]

7. Applied egg-rr 97.0%

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

8. Taylor expanded in N around -inf

   \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(-1 \cdot \left(N \cdot \left(-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N} - 1\right)\right)\right)}\right) \]
9. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \mathsf{/.f64}\left(1, \left(\mathsf{neg}\left(N \cdot \left(-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N} - 1\right)\right)\right)\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(1, \left(\mathsf{neg}\left(\left(-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N} - 1\right) \cdot N\right)\right)\right) \]

   3. distribute-rgt-neg-in N/A

      \[\leadsto \mathsf{/.f64}\left(1, \left(\left(-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N} - 1\right) \cdot \color{blue}{\left(\mathsf{neg}\left(N\right)\right)}\right)\right) \]

   4. mul-1-neg N/A

      \[\leadsto \mathsf{/.f64}\left(1, \left(\left(-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N} - 1\right) \cdot \left(-1 \cdot \color{blue}{N}\right)\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N} - 1\right), \color{blue}{\left(-1 \cdot N\right)}\right)\right) \]

10. Simplified 97.3%

    \[\leadsto \frac{1}{\color{blue}{\left(\frac{0.5 - \frac{0.08333333333333333 + \frac{-0.041666666666666664}{N}}{N}}{0 - N} + -1\right) \cdot \left(0 - N\right)}} \]
11. Step-by-step derivation

    1. sub0-neg N/A

       \[\leadsto \mathsf{/.f64}\left(1, \left(\left(\frac{\frac{1}{2} - \frac{\frac{1}{12} + \frac{\frac{-1}{24}}{N}}{N}}{0 - N} + -1\right) \cdot \left(\mathsf{neg}\left(N\right)\right)\right)\right) \]

    2. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(1, \left(\left(\mathsf{neg}\left(N\right)\right) \cdot \color{blue}{\left(\frac{\frac{1}{2} - \frac{\frac{1}{12} + \frac{\frac{-1}{24}}{N}}{N}}{0 - N} + -1\right)}\right)\right) \]

    3. distribute-rgt-in N/A

       \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{\frac{1}{2} - \frac{\frac{1}{12} + \frac{\frac{-1}{24}}{N}}{N}}{0 - N} \cdot \left(\mathsf{neg}\left(N\right)\right) + \color{blue}{-1 \cdot \left(\mathsf{neg}\left(N\right)\right)}\right)\right) \]

    4. neg-mul-1 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{\frac{1}{2} - \frac{\frac{1}{12} + \frac{\frac{-1}{24}}{N}}{N}}{0 - N} \cdot \left(\mathsf{neg}\left(N\right)\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(N\right)\right)\right)\right)\right)\right) \]

    5. remove-double-neg N/A

       \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{\frac{1}{2} - \frac{\frac{1}{12} + \frac{\frac{-1}{24}}{N}}{N}}{0 - N} \cdot \left(\mathsf{neg}\left(N\right)\right) + N\right)\right) \]

    6. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\left(\frac{\frac{1}{2} - \frac{\frac{1}{12} + \frac{\frac{-1}{24}}{N}}{N}}{0 - N} \cdot \left(\mathsf{neg}\left(N\right)\right)\right), \color{blue}{N}\right)\right) \]

12. Applied egg-rr 97.4%

    \[\leadsto \frac{1}{\color{blue}{\frac{-0.5 + \frac{0.08333333333333333 + \frac{-0.041666666666666664}{N}}{N}}{N} \cdot \left(0 - N\right) + N}} \]

13. Final simplification 97.4%

    \[\leadsto \frac{-1}{N \cdot \frac{-0.5 + \frac{0.08333333333333333 + \frac{-0.041666666666666664}{N}}{N}}{N} - N} \]
14. Add Preprocessing

Alternative 4: 96.9% accurate, 12.1× speedup

Math

\[\begin{array}{l} \\ \frac{-1}{N \cdot \left(\frac{-0.5 + \frac{0.08333333333333333 + \frac{-0.041666666666666664}{N}}{N}}{N} + -1\right)} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (/
  -1.0
  (*
   N
   (+
    (/ (+ -0.5 (/ (+ 0.08333333333333333 (/ -0.041666666666666664 N)) N)) N)
    -1.0))))

C

double code(double N) {
	return -1.0 / (N * (((-0.5 + ((0.08333333333333333 + (-0.041666666666666664 / N)) / N)) / N) + -1.0));
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    code = (-1.0d0) / (n * ((((-0.5d0) + ((0.08333333333333333d0 + ((-0.041666666666666664d0) / n)) / n)) / n) + (-1.0d0)))
end function

Java

public static double code(double N) {
	return -1.0 / (N * (((-0.5 + ((0.08333333333333333 + (-0.041666666666666664 / N)) / N)) / N) + -1.0));
}

Python

def code(N):
	return -1.0 / (N * (((-0.5 + ((0.08333333333333333 + (-0.041666666666666664 / N)) / N)) / N) + -1.0))

Julia

function code(N)
	return Float64(-1.0 / Float64(N * Float64(Float64(Float64(-0.5 + Float64(Float64(0.08333333333333333 + Float64(-0.041666666666666664 / N)) / N)) / N) + -1.0)))
end

MATLAB

function tmp = code(N)
	tmp = -1.0 / (N * (((-0.5 + ((0.08333333333333333 + (-0.041666666666666664 / N)) / N)) / N) + -1.0));
end

Wolfram

code[N_] := N[(-1.0 / N[(N * N[(N[(N[(-0.5 + N[(N[(0.08333333333333333 + N[(-0.041666666666666664 / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 23.2%

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

3. Taylor expanded in N around inf

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

5. Simplified 97.0%

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

   1. clear-num N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{N}{1 + \frac{\frac{-1}{2} + \frac{\frac{1}{3} + \frac{\frac{-1}{4}}{N}}{N}}{N}}}} \]

   2. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{N}{1 + \frac{\frac{-1}{2} + \frac{\frac{1}{3} + \frac{\frac{-1}{4}}{N}}{N}}{N}}\right)}\right) \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \color{blue}{\left(1 + \frac{\frac{-1}{2} + \frac{\frac{1}{3} + \frac{\frac{-1}{4}}{N}}{N}}{N}\right)}\right)\right) \]

   4. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{\frac{-1}{2} + \frac{\frac{1}{3} + \frac{\frac{-1}{4}}{N}}{N}}{N}\right)}\right)\right)\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{-1}{2} + \frac{\frac{1}{3} + \frac{\frac{-1}{4}}{N}}{N}\right), \color{blue}{N}\right)\right)\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{-1}{2}, \left(\frac{\frac{1}{3} + \frac{\frac{-1}{4}}{N}}{N}\right)\right), N\right)\right)\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\left(\frac{1}{3} + \frac{\frac{-1}{4}}{N}\right), N\right)\right), N\right)\right)\right)\right) \]

   8. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{3}, \left(\frac{\frac{-1}{4}}{N}\right)\right), N\right)\right), N\right)\right)\right)\right) \]

   9. /-lowering-/.f64 97.0%

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\frac{-1}{4}, N\right)\right), N\right)\right), N\right)\right)\right)\right) \]

7. Applied egg-rr 97.0%

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

8. Taylor expanded in N around -inf

   \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(-1 \cdot \left(N \cdot \left(-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N} - 1\right)\right)\right)}\right) \]
9. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \mathsf{/.f64}\left(1, \left(\mathsf{neg}\left(N \cdot \left(-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N} - 1\right)\right)\right)\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(1, \left(\mathsf{neg}\left(\left(-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N} - 1\right) \cdot N\right)\right)\right) \]

   3. distribute-rgt-neg-in N/A

      \[\leadsto \mathsf{/.f64}\left(1, \left(\left(-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N} - 1\right) \cdot \color{blue}{\left(\mathsf{neg}\left(N\right)\right)}\right)\right) \]

   4. mul-1-neg N/A

      \[\leadsto \mathsf{/.f64}\left(1, \left(\left(-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N} - 1\right) \cdot \left(-1 \cdot \color{blue}{N}\right)\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N} - 1\right), \color{blue}{\left(-1 \cdot N\right)}\right)\right) \]

10. Simplified 97.3%

    \[\leadsto \frac{1}{\color{blue}{\left(\frac{0.5 - \frac{0.08333333333333333 + \frac{-0.041666666666666664}{N}}{N}}{0 - N} + -1\right) \cdot \left(0 - N\right)}} \]
11. Step-by-step derivation

    1. frac-2neg N/A

       \[\leadsto \frac{\mathsf{neg}\left(1\right)}{\color{blue}{\mathsf{neg}\left(\left(\frac{\frac{1}{2} - \frac{\frac{1}{12} + \frac{\frac{-1}{24}}{N}}{N}}{0 - N} + -1\right) \cdot \left(0 - N\right)\right)}} \]

    2. metadata-eval N/A

       \[\leadsto \frac{-1}{\mathsf{neg}\left(\color{blue}{\left(\frac{\frac{1}{2} - \frac{\frac{1}{12} + \frac{\frac{-1}{24}}{N}}{N}}{0 - N} + -1\right) \cdot \left(0 - N\right)}\right)} \]

    3. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(-1, \color{blue}{\left(\mathsf{neg}\left(\left(\frac{\frac{1}{2} - \frac{\frac{1}{12} + \frac{\frac{-1}{24}}{N}}{N}}{0 - N} + -1\right) \cdot \left(0 - N\right)\right)\right)}\right) \]

    4. sub0-neg N/A

       \[\leadsto \mathsf{/.f64}\left(-1, \left(\mathsf{neg}\left(\left(\frac{\frac{1}{2} - \frac{\frac{1}{12} + \frac{\frac{-1}{24}}{N}}{N}}{0 - N} + -1\right) \cdot \left(\mathsf{neg}\left(N\right)\right)\right)\right)\right) \]

    5. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(-1, \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(N\right)\right) \cdot \left(\frac{\frac{1}{2} - \frac{\frac{1}{12} + \frac{\frac{-1}{24}}{N}}{N}}{0 - N} + -1\right)\right)\right)\right) \]

    6. distribute-lft-neg-in N/A

       \[\leadsto \mathsf{/.f64}\left(-1, \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(N\right)\right)\right)\right) \cdot \color{blue}{\left(\frac{\frac{1}{2} - \frac{\frac{1}{12} + \frac{\frac{-1}{24}}{N}}{N}}{0 - N} + -1\right)}\right)\right) \]

    7. remove-double-neg N/A

       \[\leadsto \mathsf{/.f64}\left(-1, \left(N \cdot \left(\color{blue}{\frac{\frac{1}{2} - \frac{\frac{1}{12} + \frac{\frac{-1}{24}}{N}}{N}}{0 - N}} + -1\right)\right)\right) \]

    8. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(N, \color{blue}{\left(\frac{\frac{1}{2} - \frac{\frac{1}{12} + \frac{\frac{-1}{24}}{N}}{N}}{0 - N} + -1\right)}\right)\right) \]

    9. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{*.f64}\left(N, \mathsf{+.f64}\left(\left(\frac{\frac{1}{2} - \frac{\frac{1}{12} + \frac{\frac{-1}{24}}{N}}{N}}{0 - N}\right), \color{blue}{-1}\right)\right)\right) \]

12. Applied egg-rr 97.3%

    \[\leadsto \color{blue}{\frac{-1}{N \cdot \left(\frac{-0.5 + \frac{0.08333333333333333 + \frac{-0.041666666666666664}{N}}{N}}{N} + -1\right)}} \]
13. Add Preprocessing

Alternative 5: 96.5% 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 23.2%

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

3. Taylor expanded in N around inf

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

5. Simplified 97.0%

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

6. Final simplification 97.0%

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

Alternative 6: 95.6% accurate, 13.7× speedup

Math

\[\begin{array}{l} \\ \frac{-1}{N \cdot \left(\left(-1 - \frac{0.5}{N}\right) - \frac{-0.08333333333333333}{N \cdot N}\right)} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (/ -1.0 (* N (- (- -1.0 (/ 0.5 N)) (/ -0.08333333333333333 (* N N))))))

C

double code(double N) {
	return -1.0 / (N * ((-1.0 - (0.5 / N)) - (-0.08333333333333333 / (N * N))));
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    code = (-1.0d0) / (n * (((-1.0d0) - (0.5d0 / n)) - ((-0.08333333333333333d0) / (n * n))))
end function

Java

public static double code(double N) {
	return -1.0 / (N * ((-1.0 - (0.5 / N)) - (-0.08333333333333333 / (N * N))));
}

Python

def code(N):
	return -1.0 / (N * ((-1.0 - (0.5 / N)) - (-0.08333333333333333 / (N * N))))

Julia

function code(N)
	return Float64(-1.0 / Float64(N * Float64(Float64(-1.0 - Float64(0.5 / N)) - Float64(-0.08333333333333333 / Float64(N * N)))))
end

MATLAB

function tmp = code(N)
	tmp = -1.0 / (N * ((-1.0 - (0.5 / N)) - (-0.08333333333333333 / (N * N))));
end

Wolfram

code[N_] := N[(-1.0 / N[(N * N[(N[(-1.0 - N[(0.5 / N), $MachinePrecision]), $MachinePrecision] - N[(-0.08333333333333333 / N[(N * N), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 23.2%

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

3. Taylor expanded in N around inf

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

5. Simplified 97.0%

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

   1. clear-num N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{N}{1 + \frac{\frac{-1}{2} + \frac{\frac{1}{3} + \frac{\frac{-1}{4}}{N}}{N}}{N}}}} \]

   2. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{N}{1 + \frac{\frac{-1}{2} + \frac{\frac{1}{3} + \frac{\frac{-1}{4}}{N}}{N}}{N}}\right)}\right) \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \color{blue}{\left(1 + \frac{\frac{-1}{2} + \frac{\frac{1}{3} + \frac{\frac{-1}{4}}{N}}{N}}{N}\right)}\right)\right) \]

   4. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{\frac{-1}{2} + \frac{\frac{1}{3} + \frac{\frac{-1}{4}}{N}}{N}}{N}\right)}\right)\right)\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{-1}{2} + \frac{\frac{1}{3} + \frac{\frac{-1}{4}}{N}}{N}\right), \color{blue}{N}\right)\right)\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{-1}{2}, \left(\frac{\frac{1}{3} + \frac{\frac{-1}{4}}{N}}{N}\right)\right), N\right)\right)\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\left(\frac{1}{3} + \frac{\frac{-1}{4}}{N}\right), N\right)\right), N\right)\right)\right)\right) \]

   8. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{3}, \left(\frac{\frac{-1}{4}}{N}\right)\right), N\right)\right), N\right)\right)\right)\right) \]

   9. /-lowering-/.f64 97.0%

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\frac{-1}{4}, N\right)\right), N\right)\right), N\right)\right)\right)\right) \]

7. Applied egg-rr 97.0%

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

8. Taylor expanded in N around inf

   \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(N \cdot \left(\left(1 + \frac{1}{2} \cdot \frac{1}{N}\right) - \frac{\frac{1}{12}}{{N}^{2}}\right)\right)}\right) \]
9. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(N, \color{blue}{\left(\left(1 + \frac{1}{2} \cdot \frac{1}{N}\right) - \frac{\frac{1}{12}}{{N}^{2}}\right)}\right)\right) \]

   2. sub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(N, \left(\left(1 + \frac{1}{2} \cdot \frac{1}{N}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{\frac{1}{12}}{{N}^{2}}\right)\right)}\right)\right)\right) \]

   3. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(N, \mathsf{+.f64}\left(\left(1 + \frac{1}{2} \cdot \frac{1}{N}\right), \color{blue}{\left(\mathsf{neg}\left(\frac{\frac{1}{12}}{{N}^{2}}\right)\right)}\right)\right)\right) \]

   4. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(N, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot \frac{1}{N}\right)\right), \left(\mathsf{neg}\left(\color{blue}{\frac{\frac{1}{12}}{{N}^{2}}}\right)\right)\right)\right)\right) \]

   5. associate-*r/ N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(N, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\frac{1}{2} \cdot 1}{N}\right)\right), \left(\mathsf{neg}\left(\frac{\frac{1}{12}}{\color{blue}{{N}^{2}}}\right)\right)\right)\right)\right) \]

   6. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(N, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\frac{1}{2}}{N}\right)\right), \left(\mathsf{neg}\left(\frac{\frac{1}{12}}{{\color{blue}{N}}^{2}}\right)\right)\right)\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(N, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, N\right)\right), \left(\mathsf{neg}\left(\frac{\frac{1}{12}}{\color{blue}{{N}^{2}}}\right)\right)\right)\right)\right) \]

   8. distribute-neg-frac N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(N, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, N\right)\right), \left(\frac{\mathsf{neg}\left(\frac{1}{12}\right)}{\color{blue}{{N}^{2}}}\right)\right)\right)\right) \]

   9. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(N, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, N\right)\right), \left(\frac{\frac{-1}{12}}{{\color{blue}{N}}^{2}}\right)\right)\right)\right) \]

   10. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(N, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, N\right)\right), \mathsf{/.f64}\left(\frac{-1}{12}, \color{blue}{\left({N}^{2}\right)}\right)\right)\right)\right) \]

   11. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(N, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, N\right)\right), \mathsf{/.f64}\left(\frac{-1}{12}, \left(N \cdot \color{blue}{N}\right)\right)\right)\right)\right) \]

   12. *-lowering-*.f64 96.2%

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(N, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, N\right)\right), \mathsf{/.f64}\left(\frac{-1}{12}, \mathsf{*.f64}\left(N, \color{blue}{N}\right)\right)\right)\right)\right) \]

10. Simplified 96.2%

    \[\leadsto \frac{1}{\color{blue}{N \cdot \left(\left(1 + \frac{0.5}{N}\right) + \frac{-0.08333333333333333}{N \cdot N}\right)}} \]

11. Final simplification 96.2%

    \[\leadsto \frac{-1}{N \cdot \left(\left(-1 - \frac{0.5}{N}\right) - \frac{-0.08333333333333333}{N \cdot N}\right)} \]
12. Add Preprocessing

Alternative 7: 95.6% accurate, 15.8× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{-1}{-1 + \frac{-0.5 + \frac{0.08333333333333333}{N}}{N}}}{N} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (/ (/ -1.0 (+ -1.0 (/ (+ -0.5 (/ 0.08333333333333333 N)) N))) N))

C

double code(double N) {
	return (-1.0 / (-1.0 + ((-0.5 + (0.08333333333333333 / N)) / N))) / N;
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    code = ((-1.0d0) / ((-1.0d0) + (((-0.5d0) + (0.08333333333333333d0 / n)) / n))) / n
end function

Java

public static double code(double N) {
	return (-1.0 / (-1.0 + ((-0.5 + (0.08333333333333333 / N)) / N))) / N;
}

Python

def code(N):
	return (-1.0 / (-1.0 + ((-0.5 + (0.08333333333333333 / N)) / N))) / N

Julia

function code(N)
	return Float64(Float64(-1.0 / Float64(-1.0 + Float64(Float64(-0.5 + Float64(0.08333333333333333 / N)) / N))) / N)
end

MATLAB

function tmp = code(N)
	tmp = (-1.0 / (-1.0 + ((-0.5 + (0.08333333333333333 / N)) / N))) / N;
end

Wolfram

code[N_] := N[(N[(-1.0 / N[(-1.0 + N[(N[(-0.5 + N[(0.08333333333333333 / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision]

Derivation

1. Initial program 23.2%

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

3. Taylor expanded in N around inf

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

5. Simplified 97.0%

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

   1. flip3-+ N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{{1}^{3} + {\left(\frac{\frac{-1}{2} + \frac{\frac{1}{3} + \frac{\frac{-1}{4}}{N}}{N}}{N}\right)}^{3}}{1 \cdot 1 + \left(\frac{\frac{-1}{2} + \frac{\frac{1}{3} + \frac{\frac{-1}{4}}{N}}{N}}{N} \cdot \frac{\frac{-1}{2} + \frac{\frac{1}{3} + \frac{\frac{-1}{4}}{N}}{N}}{N} - 1 \cdot \frac{\frac{-1}{2} + \frac{\frac{1}{3} + \frac{\frac{-1}{4}}{N}}{N}}{N}\right)}\right), N\right) \]

   2. clear-num N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{\frac{1 \cdot 1 + \left(\frac{\frac{-1}{2} + \frac{\frac{1}{3} + \frac{\frac{-1}{4}}{N}}{N}}{N} \cdot \frac{\frac{-1}{2} + \frac{\frac{1}{3} + \frac{\frac{-1}{4}}{N}}{N}}{N} - 1 \cdot \frac{\frac{-1}{2} + \frac{\frac{1}{3} + \frac{\frac{-1}{4}}{N}}{N}}{N}\right)}{{1}^{3} + {\left(\frac{\frac{-1}{2} + \frac{\frac{1}{3} + \frac{\frac{-1}{4}}{N}}{N}}{N}\right)}^{3}}}\right), N\right) \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{1 \cdot 1 + \left(\frac{\frac{-1}{2} + \frac{\frac{1}{3} + \frac{\frac{-1}{4}}{N}}{N}}{N} \cdot \frac{\frac{-1}{2} + \frac{\frac{1}{3} + \frac{\frac{-1}{4}}{N}}{N}}{N} - 1 \cdot \frac{\frac{-1}{2} + \frac{\frac{1}{3} + \frac{\frac{-1}{4}}{N}}{N}}{N}\right)}{{1}^{3} + {\left(\frac{\frac{-1}{2} + \frac{\frac{1}{3} + \frac{\frac{-1}{4}}{N}}{N}}{N}\right)}^{3}}\right)\right), N\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(1 \cdot 1 + \left(\frac{\frac{-1}{2} + \frac{\frac{1}{3} + \frac{\frac{-1}{4}}{N}}{N}}{N} \cdot \frac{\frac{-1}{2} + \frac{\frac{1}{3} + \frac{\frac{-1}{4}}{N}}{N}}{N} - 1 \cdot \frac{\frac{-1}{2} + \frac{\frac{1}{3} + \frac{\frac{-1}{4}}{N}}{N}}{N}\right)\right), \left({1}^{3} + {\left(\frac{\frac{-1}{2} + \frac{\frac{1}{3} + \frac{\frac{-1}{4}}{N}}{N}}{N}\right)}^{3}\right)\right)\right), N\right) \]

7. Applied egg-rr 96.9%

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

8. Taylor expanded in N around -inf

   \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \color{blue}{\left(1 + -1 \cdot \frac{\frac{1}{12} \cdot \frac{1}{N} - \frac{1}{2}}{N}\right)}\right), N\right) \]
9. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(1 + \left(\mathsf{neg}\left(\frac{\frac{1}{12} \cdot \frac{1}{N} - \frac{1}{2}}{N}\right)\right)\right)\right), N\right) \]

   2. unsub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(1 - \frac{\frac{1}{12} \cdot \frac{1}{N} - \frac{1}{2}}{N}\right)\right), N\right) \]

   3. --lowering--.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \left(\frac{\frac{1}{12} \cdot \frac{1}{N} - \frac{1}{2}}{N}\right)\right)\right), N\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{1}{12} \cdot \frac{1}{N} - \frac{1}{2}\right), N\right)\right)\right), N\right) \]

   5. sub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{1}{12} \cdot \frac{1}{N} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), N\right)\right)\right), N\right) \]

   6. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{1}{12} \cdot \frac{1}{N} + \frac{-1}{2}\right), N\right)\right)\right), N\right) \]

   7. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{1}{12} \cdot \frac{1}{N}\right), \frac{-1}{2}\right), N\right)\right)\right), N\right) \]

   8. associate-*r/ N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{\frac{1}{12} \cdot 1}{N}\right), \frac{-1}{2}\right), N\right)\right)\right), N\right) \]

   9. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{\frac{1}{12}}{N}\right), \frac{-1}{2}\right), N\right)\right)\right), N\right) \]

   10. /-lowering-/.f64 96.2%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{12}, N\right), \frac{-1}{2}\right), N\right)\right)\right), N\right) \]

10. Simplified 96.2%

    \[\leadsto \frac{\frac{1}{\color{blue}{1 - \frac{\frac{0.08333333333333333}{N} + -0.5}{N}}}}{N} \]

11. Final simplification 96.2%

    \[\leadsto \frac{\frac{-1}{-1 + \frac{-0.5 + \frac{0.08333333333333333}{N}}{N}}}{N} \]
12. Add Preprocessing

Alternative 8: 95.2% accurate, 15.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 23.2%

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

3. Taylor expanded in N around inf

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

5. Simplified 97.0%

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

   1. clear-num N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{N}{1 + \frac{\frac{-1}{2} + \frac{\frac{1}{3} + \frac{\frac{-1}{4}}{N}}{N}}{N}}}} \]

   2. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{N}{1 + \frac{\frac{-1}{2} + \frac{\frac{1}{3} + \frac{\frac{-1}{4}}{N}}{N}}{N}}\right)}\right) \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \color{blue}{\left(1 + \frac{\frac{-1}{2} + \frac{\frac{1}{3} + \frac{\frac{-1}{4}}{N}}{N}}{N}\right)}\right)\right) \]

   4. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{\frac{-1}{2} + \frac{\frac{1}{3} + \frac{\frac{-1}{4}}{N}}{N}}{N}\right)}\right)\right)\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{-1}{2} + \frac{\frac{1}{3} + \frac{\frac{-1}{4}}{N}}{N}\right), \color{blue}{N}\right)\right)\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{-1}{2}, \left(\frac{\frac{1}{3} + \frac{\frac{-1}{4}}{N}}{N}\right)\right), N\right)\right)\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\left(\frac{1}{3} + \frac{\frac{-1}{4}}{N}\right), N\right)\right), N\right)\right)\right)\right) \]

   8. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{3}, \left(\frac{\frac{-1}{4}}{N}\right)\right), N\right)\right), N\right)\right)\right)\right) \]

   9. /-lowering-/.f64 97.0%

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\frac{-1}{4}, N\right)\right), N\right)\right), N\right)\right)\right)\right) \]

7. Applied egg-rr 97.0%

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

8. Taylor expanded in N around -inf

   \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \color{blue}{\left(1 + -1 \cdot \frac{\frac{1}{2} - \frac{1}{3} \cdot \frac{1}{N}}{N}\right)}\right)\right) \]
9. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \left(1 + \left(\mathsf{neg}\left(\frac{\frac{1}{2} - \frac{1}{3} \cdot \frac{1}{N}}{N}\right)\right)\right)\right)\right) \]

   2. unsub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \left(1 - \color{blue}{\frac{\frac{1}{2} - \frac{1}{3} \cdot \frac{1}{N}}{N}}\right)\right)\right) \]

   3. --lowering--.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{\_.f64}\left(1, \color{blue}{\left(\frac{\frac{1}{2} - \frac{1}{3} \cdot \frac{1}{N}}{N}\right)}\right)\right)\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{1}{2} - \frac{1}{3} \cdot \frac{1}{N}\right), \color{blue}{N}\right)\right)\right)\right) \]

   5. --lowering--.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{1}{2}, \left(\frac{1}{3} \cdot \frac{1}{N}\right)\right), N\right)\right)\right)\right) \]

   6. associate-*r/ N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{3} \cdot 1}{N}\right)\right), N\right)\right)\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{1}{2}, \left(\frac{\frac{1}{3}}{N}\right)\right), N\right)\right)\right)\right) \]

   8. /-lowering-/.f64 95.9%

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{1}{3}, N\right)\right), N\right)\right)\right)\right) \]

10. Simplified 95.9%

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

11. Final simplification 95.9%

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

Alternative 9: 95.2% accurate, 18.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 23.2%

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

3. Taylor expanded in N around inf

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + \frac{\frac{1}{3}}{{N}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{N}\right), \color{blue}{N}\right) \]

5. Simplified 95.8%

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

6. Final simplification 95.8%

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

Alternative 10: 93.1% accurate, 22.8× speedup

Math

\[\begin{array}{l} \\ \frac{-1}{N \cdot \left(-1 - \frac{0.5}{N}\right)} \end{array} \]

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 23.2%

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

3. Taylor expanded in N around inf

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

5. Simplified 97.0%

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

   1. clear-num N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{N}{1 + \frac{\frac{-1}{2} + \frac{\frac{1}{3} + \frac{\frac{-1}{4}}{N}}{N}}{N}}}} \]

   2. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{N}{1 + \frac{\frac{-1}{2} + \frac{\frac{1}{3} + \frac{\frac{-1}{4}}{N}}{N}}{N}}\right)}\right) \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \color{blue}{\left(1 + \frac{\frac{-1}{2} + \frac{\frac{1}{3} + \frac{\frac{-1}{4}}{N}}{N}}{N}\right)}\right)\right) \]

   4. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{\frac{-1}{2} + \frac{\frac{1}{3} + \frac{\frac{-1}{4}}{N}}{N}}{N}\right)}\right)\right)\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{-1}{2} + \frac{\frac{1}{3} + \frac{\frac{-1}{4}}{N}}{N}\right), \color{blue}{N}\right)\right)\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{-1}{2}, \left(\frac{\frac{1}{3} + \frac{\frac{-1}{4}}{N}}{N}\right)\right), N\right)\right)\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\left(\frac{1}{3} + \frac{\frac{-1}{4}}{N}\right), N\right)\right), N\right)\right)\right)\right) \]

   8. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{3}, \left(\frac{\frac{-1}{4}}{N}\right)\right), N\right)\right), N\right)\right)\right)\right) \]

   9. /-lowering-/.f64 97.0%

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{3}, \mathsf{/.f64}\left(\frac{-1}{4}, N\right)\right), N\right)\right), N\right)\right)\right)\right) \]

7. Applied egg-rr 97.0%

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

8. Taylor expanded in N around inf

   \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(N \cdot \left(1 + \frac{1}{2} \cdot \frac{1}{N}\right)\right)}\right) \]
9. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(N, \color{blue}{\left(1 + \frac{1}{2} \cdot \frac{1}{N}\right)}\right)\right) \]

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(N, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{N}\right)}\right)\right)\right) \]

   3. associate-*r/ N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(N, \mathsf{+.f64}\left(1, \left(\frac{\frac{1}{2} \cdot 1}{\color{blue}{N}}\right)\right)\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(N, \mathsf{+.f64}\left(1, \left(\frac{\frac{1}{2}}{N}\right)\right)\right)\right) \]

   5. /-lowering-/.f64 93.8%

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{N}\right)\right)\right)\right) \]

10. Simplified 93.8%

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

11. Final simplification 93.8%

    \[\leadsto \frac{-1}{N \cdot \left(-1 - \frac{0.5}{N}\right)} \]
12. Add Preprocessing

Alternative 11: 92.5% accurate, 29.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 23.2%

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

3. Taylor expanded in N around inf

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(1 - \frac{1}{2} \cdot \frac{1}{N}\right), \color{blue}{N}\right) \]

   2. sub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\left(1 + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{N}\right)\right)\right), N\right) \]

   3. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{N}\right)\right)\right), N\right) \]

   4. associate-*r/ N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{2} \cdot 1}{N}\right)\right)\right), N\right) \]

   5. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\mathsf{neg}\left(\frac{\frac{1}{2}}{N}\right)\right)\right), N\right) \]

   6. distribute-neg-frac N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(\frac{1}{2}\right)}{N}\right)\right), N\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\frac{-1}{2}}{N}\right)\right), N\right) \]

   8. /-lowering-/.f64 93.2%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{2}, N\right)\right), N\right) \]

5. Simplified 93.2%

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

6. Final simplification 93.2%

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

Alternative 12: 84.4% 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 23.2%

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

3. Taylor expanded in N around inf

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

   1. /-lowering-/.f64 85.1%

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{N}\right) \]

5. Simplified 85.1%

   \[\leadsto \color{blue}{\frac{1}{N}} \]
6. Add Preprocessing

Developer Target 1: 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 2024140 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  :pre (and (> N 1.0) (< N 1e+40))

  :alt
  (! :herbie-platform default (log1p (/ 1 N)))

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