2log (problem 3.3.6)

Percentage Accurate: 23.7% → 99.5%

Time: 16.8s

Alternatives: 18

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: 23.7% 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, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := N \cdot \left(N + 1\right)\\ \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 0.001:\\ \;\;\;\;\frac{1}{N - N \cdot \frac{-0.5 + \frac{0.08333333333333333 + \frac{-0.041666666666666664}{N}}{N}}{N}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log t\_0 \cdot \log \left(\frac{N}{N + 1}\right)}{\log \left(\frac{1}{t\_0}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (let* ((t_0 (* N (+ N 1.0))))
   (if (<= (- (log (+ N 1.0)) (log N)) 0.001)
     (/
      1.0
      (-
       N
       (*
        N
        (/
         (+ -0.5 (/ (+ 0.08333333333333333 (/ -0.041666666666666664 N)) N))
         N))))
     (/ (* (log t_0) (log (/ N (+ N 1.0)))) (log (/ 1.0 t_0))))))

C

double code(double N) {
	double t_0 = N * (N + 1.0);
	double tmp;
	if ((log((N + 1.0)) - log(N)) <= 0.001) {
		tmp = 1.0 / (N - (N * ((-0.5 + ((0.08333333333333333 + (-0.041666666666666664 / N)) / N)) / N)));
	} else {
		tmp = (log(t_0) * log((N / (N + 1.0)))) / log((1.0 / t_0));
	}
	return tmp;
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = n * (n + 1.0d0)
    if ((log((n + 1.0d0)) - log(n)) <= 0.001d0) then
        tmp = 1.0d0 / (n - (n * (((-0.5d0) + ((0.08333333333333333d0 + ((-0.041666666666666664d0) / n)) / n)) / n)))
    else
        tmp = (log(t_0) * log((n / (n + 1.0d0)))) / log((1.0d0 / t_0))
    end if
    code = tmp
end function

Java

public static double code(double N) {
	double t_0 = N * (N + 1.0);
	double tmp;
	if ((Math.log((N + 1.0)) - Math.log(N)) <= 0.001) {
		tmp = 1.0 / (N - (N * ((-0.5 + ((0.08333333333333333 + (-0.041666666666666664 / N)) / N)) / N)));
	} else {
		tmp = (Math.log(t_0) * Math.log((N / (N + 1.0)))) / Math.log((1.0 / t_0));
	}
	return tmp;
}

Python

def code(N):
	t_0 = N * (N + 1.0)
	tmp = 0
	if (math.log((N + 1.0)) - math.log(N)) <= 0.001:
		tmp = 1.0 / (N - (N * ((-0.5 + ((0.08333333333333333 + (-0.041666666666666664 / N)) / N)) / N)))
	else:
		tmp = (math.log(t_0) * math.log((N / (N + 1.0)))) / math.log((1.0 / t_0))
	return tmp

Julia

function code(N)
	t_0 = Float64(N * Float64(N + 1.0))
	tmp = 0.0
	if (Float64(log(Float64(N + 1.0)) - log(N)) <= 0.001)
		tmp = Float64(1.0 / Float64(N - Float64(N * Float64(Float64(-0.5 + Float64(Float64(0.08333333333333333 + Float64(-0.041666666666666664 / N)) / N)) / N))));
	else
		tmp = Float64(Float64(log(t_0) * log(Float64(N / Float64(N + 1.0)))) / log(Float64(1.0 / t_0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(N)
	t_0 = N * (N + 1.0);
	tmp = 0.0;
	if ((log((N + 1.0)) - log(N)) <= 0.001)
		tmp = 1.0 / (N - (N * ((-0.5 + ((0.08333333333333333 + (-0.041666666666666664 / N)) / N)) / N)));
	else
		tmp = (log(t_0) * log((N / (N + 1.0)))) / log((1.0 / t_0));
	end
	tmp_2 = tmp;
end

Wolfram

code[N_] := Block[{t$95$0 = N[(N * N[(N + 1.0), $MachinePrecision]), $MachinePrecision]}, 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[(N[(-0.5 + N[(N[(0.08333333333333333 + N[(-0.041666666666666664 / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[t$95$0], $MachinePrecision] * N[Log[N[(N / N[(N + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Log[N[(1.0 / t$95$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)) < 1e-3

   1. Initial program 20.5%

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

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

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

      2. +-commutative N/A

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

      3. log1p-define N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\mathsf{log1p}\left(N\right)\right), \log \color{blue}{N}\right) \]

      4. log1p-lowering-log1p.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{log1p.f64}\left(N\right), \log \color{blue}{N}\right) \]

      5. log-lowering-log.f64 20.5%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{log1p.f64}\left(N\right), \mathsf{log.f64}\left(N\right)\right) \]

   3. Simplified 20.5%

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

   5. Taylor expanded in N around inf

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

   7. Simplified 99.6%

      \[\leadsto \color{blue}{\frac{1 + \frac{-0.5 + \frac{0.3333333333333333 - \frac{0.25}{N}}{N}}{N}}{N}} \]
   8. 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. sub-neg N/A

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

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

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

      10. distribute-neg-frac N/A

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

      11. /-lowering-/.f64 N/A

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

      12. metadata-eval 99.7%

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

   9. Applied egg-rr 99.7%

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

   10. Taylor expanded in N around -inf

       \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(-1 \cdot \left(N \cdot \left(-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N} - 1\right)\right)\right)}\right) \]
   11. 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

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

   12. Simplified 99.6%

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

       1. sub0-neg N/A

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

       2. *-commutative N/A

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

       3. distribute-rgt-in N/A

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

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

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

   14. Applied egg-rr 99.8%

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

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

   1. Initial program 92.2%

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

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

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

      2. +-commutative N/A

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

      3. log1p-define N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\mathsf{log1p}\left(N\right)\right), \log \color{blue}{N}\right) \]

      4. log1p-lowering-log1p.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{log1p.f64}\left(N\right), \log \color{blue}{N}\right) \]

      5. log-lowering-log.f64 92.3%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{log1p.f64}\left(N\right), \mathsf{log.f64}\left(N\right)\right) \]

   3. Simplified 92.3%

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

      1. flip-- N/A

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

      2. frac-2neg N/A

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

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

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

   6. Applied egg-rr 95.1%

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

4. Final simplification 99.5%

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

Alternative 2: 99.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;N \leq 940:\\ \;\;\;\;0 - \log \left(\frac{N}{N + 1}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N - N \cdot \frac{-0.5 + \frac{0.08333333333333333 + \frac{-0.041666666666666664}{N}}{N}}{N}}\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= N 940.0)
   (- 0.0 (log (/ N (+ N 1.0))))
   (/
    1.0
    (-
     N
     (*
      N
      (/
       (+ -0.5 (/ (+ 0.08333333333333333 (/ -0.041666666666666664 N)) N))
       N))))))

C

double code(double N) {
	double tmp;
	if (N <= 940.0) {
		tmp = 0.0 - log((N / (N + 1.0)));
	} else {
		tmp = 1.0 / (N - (N * ((-0.5 + ((0.08333333333333333 + (-0.041666666666666664 / N)) / N)) / N)));
	}
	return tmp;
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= 940.0d0) then
        tmp = 0.0d0 - log((n / (n + 1.0d0)))
    else
        tmp = 1.0d0 / (n - (n * (((-0.5d0) + ((0.08333333333333333d0 + ((-0.041666666666666664d0) / n)) / n)) / n)))
    end if
    code = tmp
end function

Java

public static double code(double N) {
	double tmp;
	if (N <= 940.0) {
		tmp = 0.0 - Math.log((N / (N + 1.0)));
	} else {
		tmp = 1.0 / (N - (N * ((-0.5 + ((0.08333333333333333 + (-0.041666666666666664 / N)) / N)) / N)));
	}
	return tmp;
}

Python

def code(N):
	tmp = 0
	if N <= 940.0:
		tmp = 0.0 - math.log((N / (N + 1.0)))
	else:
		tmp = 1.0 / (N - (N * ((-0.5 + ((0.08333333333333333 + (-0.041666666666666664 / N)) / N)) / N)))
	return tmp

Julia

function code(N)
	tmp = 0.0
	if (N <= 940.0)
		tmp = Float64(0.0 - log(Float64(N / Float64(N + 1.0))));
	else
		tmp = Float64(1.0 / Float64(N - Float64(N * Float64(Float64(-0.5 + Float64(Float64(0.08333333333333333 + Float64(-0.041666666666666664 / N)) / N)) / N))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(N)
	tmp = 0.0;
	if (N <= 940.0)
		tmp = 0.0 - log((N / (N + 1.0)));
	else
		tmp = 1.0 / (N - (N * ((-0.5 + ((0.08333333333333333 + (-0.041666666666666664 / N)) / N)) / N)));
	end
	tmp_2 = tmp;
end

Wolfram

code[N_] := If[LessEqual[N, 940.0], N[(0.0 - N[Log[N[(N / N[(N + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N - N[(N * N[(N[(-0.5 + N[(N[(0.08333333333333333 + N[(-0.041666666666666664 / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if N < 940

   1. Initial program 92.2%

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

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

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

      2. +-commutative N/A

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

      3. log1p-define N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\mathsf{log1p}\left(N\right)\right), \log \color{blue}{N}\right) \]

      4. log1p-lowering-log1p.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{log1p.f64}\left(N\right), \log \color{blue}{N}\right) \]

      5. log-lowering-log.f64 92.3%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{log1p.f64}\left(N\right), \mathsf{log.f64}\left(N\right)\right) \]

   3. Simplified 92.3%

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

      1. diff-log N/A

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

      2. clear-num N/A

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

      3. log-rec N/A

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

      4. diff-log N/A

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

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

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

      6. diff-log N/A

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

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

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

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

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

      9. +-lowering-+.f64 94.9%

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

   6. Applied egg-rr 94.9%

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

   if 940 < N

   1. Initial program 20.5%

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

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

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

      2. +-commutative N/A

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

      3. log1p-define N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\mathsf{log1p}\left(N\right)\right), \log \color{blue}{N}\right) \]

      4. log1p-lowering-log1p.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{log1p.f64}\left(N\right), \log \color{blue}{N}\right) \]

      5. log-lowering-log.f64 20.5%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{log1p.f64}\left(N\right), \mathsf{log.f64}\left(N\right)\right) \]

   3. Simplified 20.5%

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

   5. Taylor expanded in N around inf

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

   7. Simplified 99.6%

      \[\leadsto \color{blue}{\frac{1 + \frac{-0.5 + \frac{0.3333333333333333 - \frac{0.25}{N}}{N}}{N}}{N}} \]
   8. 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. sub-neg N/A

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

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

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

      10. distribute-neg-frac N/A

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

      11. /-lowering-/.f64 N/A

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

      12. metadata-eval 99.7%

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

   9. Applied egg-rr 99.7%

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

   10. Taylor expanded in N around -inf

       \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(-1 \cdot \left(N \cdot \left(-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N} - 1\right)\right)\right)}\right) \]
   11. 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

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

   12. Simplified 99.6%

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

       1. sub0-neg N/A

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

       2. *-commutative N/A

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

       3. distribute-rgt-in N/A

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

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

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

   14. Applied egg-rr 99.8%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;N \leq 940:\\ \;\;\;\;0 - \log \left(\frac{N}{N + 1}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N - N \cdot \frac{-0.5 + \frac{0.08333333333333333 + \frac{-0.041666666666666664}{N}}{N}}{N}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;N \leq 870:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N - N \cdot \frac{-0.5 + \frac{0.08333333333333333 + \frac{-0.041666666666666664}{N}}{N}}{N}}\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= N 870.0)
   (log (/ (+ N 1.0) N))
   (/
    1.0
    (-
     N
     (*
      N
      (/
       (+ -0.5 (/ (+ 0.08333333333333333 (/ -0.041666666666666664 N)) N))
       N))))))

C

double code(double N) {
	double tmp;
	if (N <= 870.0) {
		tmp = log(((N + 1.0) / N));
	} else {
		tmp = 1.0 / (N - (N * ((-0.5 + ((0.08333333333333333 + (-0.041666666666666664 / N)) / N)) / N)));
	}
	return tmp;
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= 870.0d0) then
        tmp = log(((n + 1.0d0) / n))
    else
        tmp = 1.0d0 / (n - (n * (((-0.5d0) + ((0.08333333333333333d0 + ((-0.041666666666666664d0) / n)) / n)) / n)))
    end if
    code = tmp
end function

Java

public static double code(double N) {
	double tmp;
	if (N <= 870.0) {
		tmp = Math.log(((N + 1.0) / N));
	} else {
		tmp = 1.0 / (N - (N * ((-0.5 + ((0.08333333333333333 + (-0.041666666666666664 / N)) / N)) / N)));
	}
	return tmp;
}

Python

def code(N):
	tmp = 0
	if N <= 870.0:
		tmp = math.log(((N + 1.0) / N))
	else:
		tmp = 1.0 / (N - (N * ((-0.5 + ((0.08333333333333333 + (-0.041666666666666664 / N)) / N)) / N)))
	return tmp

Julia

function code(N)
	tmp = 0.0
	if (N <= 870.0)
		tmp = log(Float64(Float64(N + 1.0) / N));
	else
		tmp = Float64(1.0 / Float64(N - Float64(N * Float64(Float64(-0.5 + Float64(Float64(0.08333333333333333 + Float64(-0.041666666666666664 / N)) / N)) / N))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(N)
	tmp = 0.0;
	if (N <= 870.0)
		tmp = log(((N + 1.0) / N));
	else
		tmp = 1.0 / (N - (N * ((-0.5 + ((0.08333333333333333 + (-0.041666666666666664 / N)) / N)) / N)));
	end
	tmp_2 = tmp;
end

Wolfram

code[N_] := If[LessEqual[N, 870.0], N[Log[N[(N[(N + 1.0), $MachinePrecision] / N), $MachinePrecision]], $MachinePrecision], N[(1.0 / N[(N - N[(N * N[(N[(-0.5 + N[(N[(0.08333333333333333 + N[(-0.041666666666666664 / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if N < 870

   1. Initial program 92.2%

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

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

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

      2. +-commutative N/A

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

      3. log1p-define N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\mathsf{log1p}\left(N\right)\right), \log \color{blue}{N}\right) \]

      4. log1p-lowering-log1p.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{log1p.f64}\left(N\right), \log \color{blue}{N}\right) \]

      5. log-lowering-log.f64 92.3%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{log1p.f64}\left(N\right), \mathsf{log.f64}\left(N\right)\right) \]

   3. Simplified 92.3%

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

      1. diff-log N/A

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

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

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

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

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

      4. +-lowering-+.f64 93.8%

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

   6. Applied egg-rr 93.8%

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

   if 870 < N

   1. Initial program 20.5%

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

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

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

      2. +-commutative N/A

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

      3. log1p-define N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\mathsf{log1p}\left(N\right)\right), \log \color{blue}{N}\right) \]

      4. log1p-lowering-log1p.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{log1p.f64}\left(N\right), \log \color{blue}{N}\right) \]

      5. log-lowering-log.f64 20.5%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{log1p.f64}\left(N\right), \mathsf{log.f64}\left(N\right)\right) \]

   3. Simplified 20.5%

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

   5. Taylor expanded in N around inf

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

   7. Simplified 99.6%

      \[\leadsto \color{blue}{\frac{1 + \frac{-0.5 + \frac{0.3333333333333333 - \frac{0.25}{N}}{N}}{N}}{N}} \]
   8. 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. sub-neg N/A

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

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

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

      10. distribute-neg-frac N/A

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

      11. /-lowering-/.f64 N/A

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

      12. metadata-eval 99.7%

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

   9. Applied egg-rr 99.7%

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

   10. Taylor expanded in N around -inf

       \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(-1 \cdot \left(N \cdot \left(-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N} - 1\right)\right)\right)}\right) \]
   11. 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

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

   12. Simplified 99.6%

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

       1. sub0-neg N/A

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

       2. *-commutative N/A

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

       3. distribute-rgt-in N/A

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

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

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

   14. Applied egg-rr 99.8%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;N \leq 870:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N - N \cdot \frac{-0.5 + \frac{0.08333333333333333 + \frac{-0.041666666666666664}{N}}{N}}{N}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.7% accurate, 12.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 24.9%

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

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

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

   2. +-commutative N/A

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

   3. log1p-define N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(\mathsf{log1p}\left(N\right)\right), \log \color{blue}{N}\right) \]

   4. log1p-lowering-log1p.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{log1p.f64}\left(N\right), \log \color{blue}{N}\right) \]

   5. log-lowering-log.f64 25.0%

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{log1p.f64}\left(N\right), \mathsf{log.f64}\left(N\right)\right) \]

3. Simplified 25.0%

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

5. Taylor expanded in N around inf

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

7. Simplified 96.6%

   \[\leadsto \color{blue}{\frac{1 + \frac{-0.5 + \frac{0.3333333333333333 - \frac{0.25}{N}}{N}}{N}}{N}} \]
8. 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. sub-neg N/A

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

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

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

   10. distribute-neg-frac N/A

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

   11. /-lowering-/.f64 N/A

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

   12. metadata-eval 96.7%

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

9. Applied egg-rr 96.7%

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

10. Taylor expanded in N around -inf

    \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(-1 \cdot \left(N \cdot \left(-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N} - 1\right)\right)\right)}\right) \]
11. 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

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

12. Simplified 97.0%

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

    1. sub0-neg N/A

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

    2. *-commutative N/A

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

    3. distribute-rgt-in N/A

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

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

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

14. Applied egg-rr 97.1%

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

15. Final simplification 97.1%

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

Alternative 5: 96.6% accurate, 12.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 24.9%

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

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

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

   2. +-commutative N/A

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

   3. log1p-define N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(\mathsf{log1p}\left(N\right)\right), \log \color{blue}{N}\right) \]

   4. log1p-lowering-log1p.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{log1p.f64}\left(N\right), \log \color{blue}{N}\right) \]

   5. log-lowering-log.f64 25.0%

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{log1p.f64}\left(N\right), \mathsf{log.f64}\left(N\right)\right) \]

3. Simplified 25.0%

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

5. Taylor expanded in N around inf

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

7. Simplified 96.6%

   \[\leadsto \color{blue}{\frac{1 + \frac{-0.5 + \frac{0.3333333333333333 - \frac{0.25}{N}}{N}}{N}}{N}} \]
8. 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. sub-neg N/A

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

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

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

   10. distribute-neg-frac N/A

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

   11. /-lowering-/.f64 N/A

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

   12. metadata-eval 96.7%

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

9. Applied egg-rr 96.7%

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

10. Taylor expanded in N around -inf

    \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(-1 \cdot \left(N \cdot \left(-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N} - 1\right)\right)\right)}\right) \]
11. 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

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

12. Simplified 97.0%

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

13. Final simplification 97.0%

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

Alternative 6: 96.6% accurate, 12.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 24.9%

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

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

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

   2. +-commutative N/A

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

   3. log1p-define N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(\mathsf{log1p}\left(N\right)\right), \log \color{blue}{N}\right) \]

   4. log1p-lowering-log1p.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{log1p.f64}\left(N\right), \log \color{blue}{N}\right) \]

   5. log-lowering-log.f64 25.0%

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{log1p.f64}\left(N\right), \mathsf{log.f64}\left(N\right)\right) \]

3. Simplified 25.0%

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

5. Taylor expanded in N around inf

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

7. Simplified 96.6%

   \[\leadsto \color{blue}{\frac{1 + \frac{-0.5 + \frac{0.3333333333333333 - \frac{0.25}{N}}{N}}{N}}{N}} \]
8. 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. sub-neg N/A

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

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

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

   10. distribute-neg-frac N/A

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

   11. /-lowering-/.f64 N/A

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

   12. metadata-eval 96.7%

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

9. Applied egg-rr 96.7%

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

10. Taylor expanded in N around -inf

    \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(-1 \cdot \left(N \cdot \left(-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N} - 1\right)\right)\right)}\right) \]
11. 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

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

12. Simplified 97.0%

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

    1. sub0-neg N/A

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

    2. *-commutative N/A

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

    3. associate-/r* N/A

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

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

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

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

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

    6. sub0-neg N/A

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

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

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

    8. +-commutative N/A

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

    9. sub0-neg N/A

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

    10. unsub-neg N/A

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

    11. --lowering--.f64 N/A

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

    12. /-lowering-/.f64 N/A

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

14. Applied egg-rr 96.9%

    \[\leadsto \color{blue}{\frac{\frac{1}{0 - N}}{-1 - \frac{0.5 + \frac{-0.08333333333333333 + \frac{0.041666666666666664}{N}}{N}}{N}}} \]

15. Final simplification 96.9%

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

Alternative 7: 96.6% accurate, 12.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 24.9%

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

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

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

   2. +-commutative N/A

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

   3. log1p-define N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(\mathsf{log1p}\left(N\right)\right), \log \color{blue}{N}\right) \]

   4. log1p-lowering-log1p.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{log1p.f64}\left(N\right), \log \color{blue}{N}\right) \]

   5. log-lowering-log.f64 25.0%

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{log1p.f64}\left(N\right), \mathsf{log.f64}\left(N\right)\right) \]

3. Simplified 25.0%

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

5. Taylor expanded in N around inf

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

7. Simplified 96.6%

   \[\leadsto \color{blue}{\frac{1 + \frac{-0.5 + \frac{0.3333333333333333 - \frac{0.25}{N}}{N}}{N}}{N}} \]
8. 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. sub-neg N/A

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

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

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

   10. distribute-neg-frac N/A

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

   11. /-lowering-/.f64 N/A

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

   12. metadata-eval 96.7%

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

9. Applied egg-rr 96.7%

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

10. Taylor expanded in N around -inf

    \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(-1 \cdot \left(N \cdot \left(-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N} - 1\right)\right)\right)}\right) \]
11. 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

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

12. Simplified 97.0%

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

    1. sub0-neg N/A

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

    2. associate-/r* N/A

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

    3. distribute-frac-neg2 N/A

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

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

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

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

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

14. Applied egg-rr 96.9%

    \[\leadsto \color{blue}{-\frac{\frac{1}{-1 - \frac{0.5 + \frac{-0.08333333333333333 + \frac{0.041666666666666664}{N}}{N}}{N}}}{N}} \]

15. Final simplification 96.9%

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

Alternative 8: 96.5% accurate, 12.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 24.9%

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

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

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

   2. +-commutative N/A

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

   3. log1p-define N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(\mathsf{log1p}\left(N\right)\right), \log \color{blue}{N}\right) \]

   4. log1p-lowering-log1p.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{log1p.f64}\left(N\right), \log \color{blue}{N}\right) \]

   5. log-lowering-log.f64 25.0%

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{log1p.f64}\left(N\right), \mathsf{log.f64}\left(N\right)\right) \]

3. Simplified 25.0%

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

5. Taylor expanded in N around inf

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

7. Simplified 96.6%

   \[\leadsto \color{blue}{\frac{1 + \frac{-0.5 + \frac{0.3333333333333333 - \frac{0.25}{N}}{N}}{N}}{N}} \]
8. 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. sub-neg N/A

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

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

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

   10. distribute-neg-frac N/A

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

   11. /-lowering-/.f64 N/A

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

   12. metadata-eval 96.7%

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

9. Applied egg-rr 96.7%

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

10. Taylor expanded in N around -inf

    \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(-1 \cdot \left(N \cdot \left(-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N} - 1\right)\right)\right)}\right) \]
11. 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

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

12. Simplified 97.0%

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

13. Taylor expanded in N around 0

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

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

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

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

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

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

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

    4. sub-neg N/A

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

    5. metadata-eval N/A

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

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

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

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

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

    8. +-commutative N/A

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

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

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

    10. unpow2 N/A

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

    11. *-lowering-*.f64 96.9%

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

15. Simplified 96.9%

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

16. Final simplification 96.9%

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

Alternative 9: 96.3% accurate, 12.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 24.9%

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

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

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

   2. +-commutative N/A

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

   3. log1p-define N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(\mathsf{log1p}\left(N\right)\right), \log \color{blue}{N}\right) \]

   4. log1p-lowering-log1p.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{log1p.f64}\left(N\right), \log \color{blue}{N}\right) \]

   5. log-lowering-log.f64 25.0%

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{log1p.f64}\left(N\right), \mathsf{log.f64}\left(N\right)\right) \]

3. Simplified 25.0%

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

5. Taylor expanded in N around inf

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

7. Simplified 96.6%

   \[\leadsto \color{blue}{\frac{1 + \frac{-0.5 + \frac{0.3333333333333333 - \frac{0.25}{N}}{N}}{N}}{N}} \]
8. 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. sub-neg N/A

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

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

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

   10. distribute-neg-frac N/A

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

   11. /-lowering-/.f64 N/A

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

   12. metadata-eval 96.7%

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

9. Applied egg-rr 96.7%

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

Alternative 10: 96.2% accurate, 13.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 24.9%

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

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

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

   2. +-commutative N/A

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

   3. log1p-define N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(\mathsf{log1p}\left(N\right)\right), \log \color{blue}{N}\right) \]

   4. log1p-lowering-log1p.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{log1p.f64}\left(N\right), \log \color{blue}{N}\right) \]

   5. log-lowering-log.f64 25.0%

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{log1p.f64}\left(N\right), \mathsf{log.f64}\left(N\right)\right) \]

3. Simplified 25.0%

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

5. Taylor expanded in N around inf

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

7. Simplified 96.6%

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

Alternative 11: 95.4% 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 24.9%

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

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

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

   2. +-commutative N/A

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

   3. log1p-define N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(\mathsf{log1p}\left(N\right)\right), \log \color{blue}{N}\right) \]

   4. log1p-lowering-log1p.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{log1p.f64}\left(N\right), \log \color{blue}{N}\right) \]

   5. log-lowering-log.f64 25.0%

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{log1p.f64}\left(N\right), \mathsf{log.f64}\left(N\right)\right) \]

3. Simplified 25.0%

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

5. Taylor expanded in N around inf

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

7. Simplified 96.6%

   \[\leadsto \color{blue}{\frac{1 + \frac{-0.5 + \frac{0.3333333333333333 - \frac{0.25}{N}}{N}}{N}}{N}} \]
8. 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. sub-neg N/A

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

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

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

   10. distribute-neg-frac N/A

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

   11. /-lowering-/.f64 N/A

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

   12. metadata-eval 96.7%

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

9. Applied egg-rr 96.7%

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

10. Taylor expanded in N around inf

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

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

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

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

    4. associate-*r/ N/A

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

    5. metadata-eval N/A

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

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

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

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

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

    8. unpow2 N/A

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

    9. *-lowering-*.f64 95.6%

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

12. Simplified 95.6%

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

Alternative 12: 95.0% accurate, 15.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 24.9%

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

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

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

   2. +-commutative N/A

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

   3. log1p-define N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(\mathsf{log1p}\left(N\right)\right), \log \color{blue}{N}\right) \]

   4. log1p-lowering-log1p.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{log1p.f64}\left(N\right), \log \color{blue}{N}\right) \]

   5. log-lowering-log.f64 25.0%

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{log1p.f64}\left(N\right), \mathsf{log.f64}\left(N\right)\right) \]

3. Simplified 25.0%

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

5. Taylor expanded in N around inf

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

7. Simplified 96.6%

   \[\leadsto \color{blue}{\frac{1 + \frac{-0.5 + \frac{0.3333333333333333 - \frac{0.25}{N}}{N}}{N}}{N}} \]
8. 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. sub-neg N/A

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

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

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

   10. distribute-neg-frac N/A

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

   11. /-lowering-/.f64 N/A

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

   12. metadata-eval 96.7%

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

9. Applied egg-rr 96.7%

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

10. Taylor expanded in N around -inf

    \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \color{blue}{\left(1 + -1 \cdot \frac{\frac{1}{2} - \frac{1}{3} \cdot \frac{1}{N}}{N}\right)}\right)\right) \]
11. 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 95.2%

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

12. Simplified 95.2%

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

13. Final simplification 95.2%

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

Alternative 13: 95.0% accurate, 18.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 24.9%

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

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

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

   2. +-commutative N/A

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

   3. log1p-define N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(\mathsf{log1p}\left(N\right)\right), \log \color{blue}{N}\right) \]

   4. log1p-lowering-log1p.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{log1p.f64}\left(N\right), \log \color{blue}{N}\right) \]

   5. log-lowering-log.f64 25.0%

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{log1p.f64}\left(N\right), \mathsf{log.f64}\left(N\right)\right) \]

3. Simplified 25.0%

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

5. Taylor expanded in N around inf

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

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

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

7. Simplified 95.2%

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

Alternative 14: 93.0% 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 24.9%

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

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

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

   2. +-commutative N/A

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

   3. log1p-define N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(\mathsf{log1p}\left(N\right)\right), \log \color{blue}{N}\right) \]

   4. log1p-lowering-log1p.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{log1p.f64}\left(N\right), \log \color{blue}{N}\right) \]

   5. log-lowering-log.f64 25.0%

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{log1p.f64}\left(N\right), \mathsf{log.f64}\left(N\right)\right) \]

3. Simplified 25.0%

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

5. Taylor expanded in N around inf

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

7. Simplified 96.6%

   \[\leadsto \color{blue}{\frac{1 + \frac{-0.5 + \frac{0.3333333333333333 - \frac{0.25}{N}}{N}}{N}}{N}} \]
8. 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. sub-neg N/A

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

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

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

   10. distribute-neg-frac N/A

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

   11. /-lowering-/.f64 N/A

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

   12. metadata-eval 96.7%

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

9. Applied egg-rr 96.7%

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

10. Taylor expanded in N around inf

    \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(N \cdot \left(1 + \frac{1}{2} \cdot \frac{1}{N}\right)\right)}\right) \]
11. 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 92.9%

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

12. Simplified 92.9%

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

Alternative 15: 92.4% 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 24.9%

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

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

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

   2. +-commutative N/A

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

   3. log1p-define N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(\mathsf{log1p}\left(N\right)\right), \log \color{blue}{N}\right) \]

   4. log1p-lowering-log1p.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{log1p.f64}\left(N\right), \log \color{blue}{N}\right) \]

   5. log-lowering-log.f64 25.0%

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{log1p.f64}\left(N\right), \mathsf{log.f64}\left(N\right)\right) \]

3. Simplified 25.0%

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

5. Taylor expanded in N around inf

   \[\leadsto \color{blue}{\frac{1 - \frac{1}{2} \cdot \frac{1}{N}}{N}} \]
6. 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. --lowering--.f64 N/A

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

   3. associate-*r/ N/A

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

   4. metadata-eval N/A

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

   5. /-lowering-/.f64 92.3%

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

7. Simplified 92.3%

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

Alternative 16: 92.2% accurate, 29.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 24.9%

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

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

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

   2. +-commutative N/A

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

   3. log1p-define N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(\mathsf{log1p}\left(N\right)\right), \log \color{blue}{N}\right) \]

   4. log1p-lowering-log1p.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{log1p.f64}\left(N\right), \log \color{blue}{N}\right) \]

   5. log-lowering-log.f64 25.0%

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{log1p.f64}\left(N\right), \mathsf{log.f64}\left(N\right)\right) \]

3. Simplified 25.0%

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

5. Taylor expanded in N around inf

   \[\leadsto \color{blue}{\frac{1 - \frac{1}{2} \cdot \frac{1}{N}}{N}} \]
6. 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. --lowering--.f64 N/A

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

   3. associate-*r/ N/A

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

   4. metadata-eval N/A

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

   5. /-lowering-/.f64 92.3%

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

7. Simplified 92.3%

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

8. Taylor expanded in N around 0

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

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

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

   2. sub-neg N/A

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

   3. metadata-eval N/A

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

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

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

   5. unpow2 N/A

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

   6. *-lowering-*.f64 92.1%

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

10. Simplified 92.1%

    \[\leadsto \color{blue}{\frac{N + -0.5}{N \cdot N}} \]
11. Add Preprocessing

Alternative 17: 84.6% 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 24.9%

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

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

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

   2. +-commutative N/A

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

   3. log1p-define N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(\mathsf{log1p}\left(N\right)\right), \log \color{blue}{N}\right) \]

   4. log1p-lowering-log1p.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{log1p.f64}\left(N\right), \log \color{blue}{N}\right) \]

   5. log-lowering-log.f64 25.0%

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{log1p.f64}\left(N\right), \mathsf{log.f64}\left(N\right)\right) \]

3. Simplified 25.0%

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

5. Taylor expanded in N around inf

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

   1. /-lowering-/.f64 83.7%

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

7. Simplified 83.7%

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

Alternative 18: 3.3% accurate, 205.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (N) :precision binary64 0.0)

C

double code(double N) {
	return 0.0;
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    code = 0.0d0
end function

Java

public static double code(double N) {
	return 0.0;
}

Python

def code(N):
	return 0.0

Julia

function code(N)
	return 0.0
end

MATLAB

function tmp = code(N)
	tmp = 0.0;
end

Wolfram

code[N_] := 0.0

Derivation

1. Initial program 24.9%

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

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

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

   2. +-commutative N/A

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

   3. log1p-define N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(\mathsf{log1p}\left(N\right)\right), \log \color{blue}{N}\right) \]

   4. log1p-lowering-log1p.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{log1p.f64}\left(N\right), \log \color{blue}{N}\right) \]

   5. log-lowering-log.f64 25.0%

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{log1p.f64}\left(N\right), \mathsf{log.f64}\left(N\right)\right) \]

3. Simplified 25.0%

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

   1. flip-- N/A

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

   2. div-sub N/A

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

   3. associate-/l* N/A

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

   4. div-inv N/A

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

   5. prod-diff N/A

      \[\leadsto \mathsf{fma}\left(\log \left(1 + N\right), \frac{\log \left(1 + N\right)}{\log \left(1 + N\right) + \log N}, \mathsf{neg}\left(\frac{1}{\log \left(1 + N\right) + \log N} \cdot \left(\log N \cdot \log N\right)\right)\right) + \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(\frac{1}{\log \left(1 + N\right) + \log N}\right), \log N \cdot \log N, \frac{1}{\log \left(1 + N\right) + \log N} \cdot \left(\log N \cdot \log N\right)\right)} \]

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

      \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{fma}\left(\log \left(1 + N\right), \frac{\log \left(1 + N\right)}{\log \left(1 + N\right) + \log N}, \mathsf{neg}\left(\frac{1}{\log \left(1 + N\right) + \log N} \cdot \left(\log N \cdot \log N\right)\right)\right)\right), \color{blue}{\left(\mathsf{fma}\left(\mathsf{neg}\left(\frac{1}{\log \left(1 + N\right) + \log N}\right), \log N \cdot \log N, \frac{1}{\log \left(1 + N\right) + \log N} \cdot \left(\log N \cdot \log N\right)\right)\right)}\right) \]

6. Applied egg-rr 26.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{log1p}\left(N\right), \frac{\mathsf{log1p}\left(N\right)}{\log \left(N \cdot \left(1 + N\right)\right)}, -\frac{1}{\log \left(N \cdot \left(1 + N\right)\right)} \cdot {\log N}^{2}\right) + \mathsf{fma}\left(-\frac{1}{\log \left(N \cdot \left(1 + N\right)\right)}, {\log N}^{2}, \frac{1}{\log \left(N \cdot \left(1 + N\right)\right)} \cdot {\log N}^{2}\right)} \]

7. Taylor expanded in N around inf

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

   1. distribute-rgt-out N/A

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

   2. metadata-eval N/A

      \[\leadsto \log \left(\frac{1}{N}\right) \cdot 0 \]

   3. mul0-rgt 3.3%

      \[\leadsto 0 \]

9. Simplified 3.3%

   \[\leadsto \color{blue}{0} \]
10. 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 2024160 
(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)))