2log (problem 3.3.6)

Percentage Accurate: 23.7% → 99.3%

Time: 9.4s

Alternatives: 10

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: 99.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 0.001:\\ \;\;\;\;\frac{1}{N \cdot \left(\frac{0.041666666666666664}{{N}^{3}} - \left(-1 + \frac{\frac{0.08333333333333333}{N} + -0.5}{N}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= (- (log (+ N 1.0)) (log N)) 0.001)
   (/
    1.0
    (*
     N
     (-
      (/ 0.041666666666666664 (pow N 3.0))
      (+ -1.0 (/ (+ (/ 0.08333333333333333 N) -0.5) N)))))
   (log (/ (+ N 1.0) N))))

C

double code(double N) {
	double tmp;
	if ((log((N + 1.0)) - log(N)) <= 0.001) {
		tmp = 1.0 / (N * ((0.041666666666666664 / pow(N, 3.0)) - (-1.0 + (((0.08333333333333333 / N) + -0.5) / N))));
	} else {
		tmp = log(((N + 1.0) / N));
	}
	return tmp;
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((log((n + 1.0d0)) - log(n)) <= 0.001d0) then
        tmp = 1.0d0 / (n * ((0.041666666666666664d0 / (n ** 3.0d0)) - ((-1.0d0) + (((0.08333333333333333d0 / n) + (-0.5d0)) / n))))
    else
        tmp = log(((n + 1.0d0) / n))
    end if
    code = tmp
end function

Java

public static double code(double N) {
	double tmp;
	if ((Math.log((N + 1.0)) - Math.log(N)) <= 0.001) {
		tmp = 1.0 / (N * ((0.041666666666666664 / Math.pow(N, 3.0)) - (-1.0 + (((0.08333333333333333 / N) + -0.5) / N))));
	} else {
		tmp = Math.log(((N + 1.0) / N));
	}
	return tmp;
}

Python

def code(N):
	tmp = 0
	if (math.log((N + 1.0)) - math.log(N)) <= 0.001:
		tmp = 1.0 / (N * ((0.041666666666666664 / math.pow(N, 3.0)) - (-1.0 + (((0.08333333333333333 / N) + -0.5) / N))))
	else:
		tmp = math.log(((N + 1.0) / N))
	return tmp

Julia

function code(N)
	tmp = 0.0
	if (Float64(log(Float64(N + 1.0)) - log(N)) <= 0.001)
		tmp = Float64(1.0 / Float64(N * Float64(Float64(0.041666666666666664 / (N ^ 3.0)) - Float64(-1.0 + Float64(Float64(Float64(0.08333333333333333 / N) + -0.5) / N)))));
	else
		tmp = log(Float64(Float64(N + 1.0) / N));
	end
	return tmp
end

MATLAB

function tmp_2 = code(N)
	tmp = 0.0;
	if ((log((N + 1.0)) - log(N)) <= 0.001)
		tmp = 1.0 / (N * ((0.041666666666666664 / (N ^ 3.0)) - (-1.0 + (((0.08333333333333333 / N) + -0.5) / N))));
	else
		tmp = log(((N + 1.0) / N));
	end
	tmp_2 = tmp;
end

Wolfram

code[N_] := If[LessEqual[N[(N[Log[N[(N + 1.0), $MachinePrecision]], $MachinePrecision] - N[Log[N], $MachinePrecision]), $MachinePrecision], 0.001], N[(1.0 / N[(N * N[(N[(0.041666666666666664 / N[Power[N, 3.0], $MachinePrecision]), $MachinePrecision] - N[(-1.0 + N[(N[(N[(0.08333333333333333 / N), $MachinePrecision] + -0.5), $MachinePrecision] / N), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[N[(N[(N + 1.0), $MachinePrecision] / N), $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 17.1%

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

      1. +-commutative 17.1%

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

      2. log1p-define 17.1%

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

   3. Simplified 17.1%

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

   5. Taylor expanded in N around inf 99.6%

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

   7. Simplified 99.6%

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

      1. clear-num 99.7%

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

      2. inv-pow 99.7%

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

      3. associate--l+ 99.7%

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

      4. div-inv 99.7%

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

      5. pow-flip 99.7%

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

      6. metadata-eval 99.7%

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

   9. Applied egg-rr 99.7%

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

       1. unpow-1 99.7%

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

       2. +-commutative 99.7%

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

       3. sub-neg 99.7%

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

       4. associate-+l+ 99.7%

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

       5. +-commutative 99.7%

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

       6. distribute-lft-neg-in 99.7%

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

       7. metadata-eval 99.7%

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

   11. Simplified 99.7%

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

   12. Taylor expanded in N around -inf 99.7%

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

   13. Taylor expanded in N around inf 99.7%

       \[\leadsto \frac{1}{-1 \cdot \color{blue}{\left(N \cdot \left(\frac{0.08333333333333333}{{N}^{2}} - \left(1 + \left(0.5 \cdot \frac{1}{N} + 0.041666666666666664 \cdot \frac{1}{{N}^{3}}\right)\right)\right)\right)}} \]
   14. Step-by-step derivation

       1. associate-+r+ 99.7%

          \[\leadsto \frac{1}{-1 \cdot \left(N \cdot \left(\frac{0.08333333333333333}{{N}^{2}} - \color{blue}{\left(\left(1 + 0.5 \cdot \frac{1}{N}\right) + 0.041666666666666664 \cdot \frac{1}{{N}^{3}}\right)}\right)\right)} \]

       2. associate--r+ 99.7%

          \[\leadsto \frac{1}{-1 \cdot \left(N \cdot \color{blue}{\left(\left(\frac{0.08333333333333333}{{N}^{2}} - \left(1 + 0.5 \cdot \frac{1}{N}\right)\right) - 0.041666666666666664 \cdot \frac{1}{{N}^{3}}\right)}\right)} \]

   15. Simplified 99.7%

       \[\leadsto \frac{1}{-1 \cdot \color{blue}{\left(N \cdot \left(\left(\frac{\frac{0.08333333333333333}{N} + -0.5}{N} - 1\right) - \frac{0.041666666666666664}{{N}^{3}}\right)\right)}} \]

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

   1. Initial program 90.7%

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

      1. +-commutative 90.7%

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

      2. log1p-define 91.0%

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

   3. Simplified 91.0%

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

      1. add-log-exp 91.0%

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

      2. log1p-expm1-u 90.8%

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

      3. log1p-undefine 91.0%

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

      4. diff-log 90.6%

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

      5. log1p-undefine 90.3%

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

      6. rem-exp-log 91.3%

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

      7. +-commutative 91.3%

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

      8. add-exp-log 91.6%

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

      9. log1p-undefine 91.2%

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

      10. log1p-expm1-u 91.6%

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

      11. add-exp-log 94.2%

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

   6. Applied egg-rr 94.2%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 0.001:\\ \;\;\;\;\frac{1}{N \cdot \left(\frac{0.041666666666666664}{{N}^{3}} - \left(-1 + \frac{\frac{0.08333333333333333}{N} + -0.5}{N}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;N \leq 920:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{N \cdot \left(-1 - \frac{\frac{\frac{0.041666666666666664}{N} + -0.08333333333333333}{N} + 0.5}{N}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= N 920.0)
   (log (/ (+ N 1.0) N))
   (/
    -1.0
    (*
     N
     (-
      -1.0
      (/
       (+ (/ (+ (/ 0.041666666666666664 N) -0.08333333333333333) N) 0.5)
       N))))))

C

double code(double N) {
	double tmp;
	if (N <= 920.0) {
		tmp = log(((N + 1.0) / N));
	} else {
		tmp = -1.0 / (N * (-1.0 - (((((0.041666666666666664 / N) + -0.08333333333333333) / N) + 0.5) / N)));
	}
	return tmp;
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= 920.0d0) then
        tmp = log(((n + 1.0d0) / n))
    else
        tmp = (-1.0d0) / (n * ((-1.0d0) - (((((0.041666666666666664d0 / n) + (-0.08333333333333333d0)) / n) + 0.5d0) / n)))
    end if
    code = tmp
end function

Java

public static double code(double N) {
	double tmp;
	if (N <= 920.0) {
		tmp = Math.log(((N + 1.0) / N));
	} else {
		tmp = -1.0 / (N * (-1.0 - (((((0.041666666666666664 / N) + -0.08333333333333333) / N) + 0.5) / N)));
	}
	return tmp;
}

Python

def code(N):
	tmp = 0
	if N <= 920.0:
		tmp = math.log(((N + 1.0) / N))
	else:
		tmp = -1.0 / (N * (-1.0 - (((((0.041666666666666664 / N) + -0.08333333333333333) / N) + 0.5) / N)))
	return tmp

Julia

function code(N)
	tmp = 0.0
	if (N <= 920.0)
		tmp = log(Float64(Float64(N + 1.0) / N));
	else
		tmp = Float64(-1.0 / Float64(N * Float64(-1.0 - Float64(Float64(Float64(Float64(Float64(0.041666666666666664 / N) + -0.08333333333333333) / N) + 0.5) / N))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(N)
	tmp = 0.0;
	if (N <= 920.0)
		tmp = log(((N + 1.0) / N));
	else
		tmp = -1.0 / (N * (-1.0 - (((((0.041666666666666664 / N) + -0.08333333333333333) / N) + 0.5) / N)));
	end
	tmp_2 = tmp;
end

Wolfram

code[N_] := If[LessEqual[N, 920.0], N[Log[N[(N[(N + 1.0), $MachinePrecision] / N), $MachinePrecision]], $MachinePrecision], N[(-1.0 / N[(N * N[(-1.0 - N[(N[(N[(N[(N[(0.041666666666666664 / N), $MachinePrecision] + -0.08333333333333333), $MachinePrecision] / N), $MachinePrecision] + 0.5), $MachinePrecision] / N), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if N < 920

   1. Initial program 90.7%

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

      1. +-commutative 90.7%

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

      2. log1p-define 91.0%

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

   3. Simplified 91.0%

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

      1. add-log-exp 91.0%

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

      2. log1p-expm1-u 90.8%

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

      3. log1p-undefine 91.0%

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

      4. diff-log 90.6%

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

      5. log1p-undefine 90.3%

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

      6. rem-exp-log 91.3%

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

      7. +-commutative 91.3%

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

      8. add-exp-log 91.6%

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

      9. log1p-undefine 91.2%

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

      10. log1p-expm1-u 91.6%

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

      11. add-exp-log 94.2%

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

   6. Applied egg-rr 94.2%

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

   if 920 < N

   1. Initial program 17.1%

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

      1. +-commutative 17.1%

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

      2. log1p-define 17.1%

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

   3. Simplified 17.1%

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

   5. Taylor expanded in N around inf 99.6%

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

   7. Simplified 99.6%

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

      1. clear-num 99.7%

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

      2. inv-pow 99.7%

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

      3. associate--l+ 99.7%

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

      4. div-inv 99.7%

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

      5. pow-flip 99.7%

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

      6. metadata-eval 99.7%

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

   9. Applied egg-rr 99.7%

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

       1. unpow-1 99.7%

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

       2. +-commutative 99.7%

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

       3. sub-neg 99.7%

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

       4. associate-+l+ 99.7%

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

       5. +-commutative 99.7%

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

       6. distribute-lft-neg-in 99.7%

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

       7. metadata-eval 99.7%

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

   11. Simplified 99.7%

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

   12. Taylor expanded in N around -inf 99.7%

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

   13. Taylor expanded in N around -inf 99.7%

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

       1. associate-*r* 99.7%

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

       2. neg-mul-1 99.7%

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

       3. associate-*r/ 99.7%

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

   15. Simplified 99.7%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;N \leq 920:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{N \cdot \left(-1 - \frac{\frac{\frac{0.041666666666666664}{N} + -0.08333333333333333}{N} + 0.5}{N}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.6% accurate, 10.8× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 23.4%

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

   1. +-commutative 23.4%

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

   2. log1p-define 23.4%

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

3. Simplified 23.4%

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

5. Taylor expanded in N around inf 96.0%

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

7. Simplified 96.1%

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

   1. clear-num 96.1%

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

   2. inv-pow 96.1%

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

   3. associate--l+ 96.1%

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

   4. div-inv 96.1%

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

   5. pow-flip 96.1%

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

   6. metadata-eval 96.1%

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

9. Applied egg-rr 96.1%

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

    1. unpow-1 96.1%

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

    2. +-commutative 96.1%

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

    3. sub-neg 96.1%

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

    4. associate-+l+ 96.1%

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

    5. +-commutative 96.1%

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

    6. distribute-lft-neg-in 96.1%

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

    7. metadata-eval 96.1%

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

11. Simplified 96.1%

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

12. Taylor expanded in N around -inf 96.5%

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

    1. div-sub 96.5%

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

    2. un-div-inv 96.5%

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

14. Applied egg-rr 96.5%

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

15. Final simplification 96.5%

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

Alternative 4: 96.6% accurate, 12.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 23.4%

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

   1. +-commutative 23.4%

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

   2. log1p-define 23.4%

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

3. Simplified 23.4%

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

5. Taylor expanded in N around inf 96.0%

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

7. Simplified 96.1%

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

   1. clear-num 96.1%

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

   2. inv-pow 96.1%

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

   3. associate--l+ 96.1%

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

   4. div-inv 96.1%

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

   5. pow-flip 96.1%

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

   6. metadata-eval 96.1%

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

9. Applied egg-rr 96.1%

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

    1. unpow-1 96.1%

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

    2. +-commutative 96.1%

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

    3. sub-neg 96.1%

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

    4. associate-+l+ 96.1%

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

    5. +-commutative 96.1%

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

    6. distribute-lft-neg-in 96.1%

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

    7. metadata-eval 96.1%

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

11. Simplified 96.1%

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

12. Taylor expanded in N around -inf 96.5%

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

13. Taylor expanded in N around -inf 96.5%

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

    1. associate-*r* 96.5%

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

    2. neg-mul-1 96.5%

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

    3. associate-*r/ 96.5%

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

15. Simplified 96.5%

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

16. Final simplification 96.5%

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

Alternative 5: 96.2% accurate, 13.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 23.4%

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

   1. +-commutative 23.4%

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

   2. log1p-define 23.4%

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

3. Simplified 23.4%

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

5. Taylor expanded in N around -inf 96.1%

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

   1. mul-1-neg 96.1%

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

   2. distribute-neg-frac2 96.1%

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

7. Simplified 96.1%

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

8. Final simplification 96.1%

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

Alternative 6: 95.4% accurate, 15.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 23.4%

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

   1. +-commutative 23.4%

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

   2. log1p-define 23.4%

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

3. Simplified 23.4%

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

5. Taylor expanded in N around inf 96.0%

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

7. Simplified 96.1%

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

   1. clear-num 96.1%

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

   2. inv-pow 96.1%

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

   3. associate--l+ 96.1%

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

   4. div-inv 96.1%

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

   5. pow-flip 96.1%

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

   6. metadata-eval 96.1%

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

9. Applied egg-rr 96.1%

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

    1. unpow-1 96.1%

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

    2. +-commutative 96.1%

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

    3. sub-neg 96.1%

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

    4. associate-+l+ 96.1%

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

    5. +-commutative 96.1%

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

    6. distribute-lft-neg-in 96.1%

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

    7. metadata-eval 96.1%

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

11. Simplified 96.1%

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

12. Taylor expanded in N around -inf 96.5%

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

13. Taylor expanded in N around inf 94.9%

    \[\leadsto \frac{1}{-1 \cdot \color{blue}{\left(N \cdot \left(\frac{0.08333333333333333}{{N}^{2}} - \left(1 + 0.5 \cdot \frac{1}{N}\right)\right)\right)}} \]
14. Step-by-step derivation

    1. +-commutative 94.9%

       \[\leadsto \frac{1}{-1 \cdot \left(N \cdot \left(\frac{0.08333333333333333}{{N}^{2}} - \color{blue}{\left(0.5 \cdot \frac{1}{N} + 1\right)}\right)\right)} \]

    2. associate--r+ 94.9%

       \[\leadsto \frac{1}{-1 \cdot \left(N \cdot \color{blue}{\left(\left(\frac{0.08333333333333333}{{N}^{2}} - 0.5 \cdot \frac{1}{N}\right) - 1\right)}\right)} \]

    3. unpow2 94.9%

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

    4. associate-/r* 94.9%

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

    5. metadata-eval 94.9%

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

    6. associate-*r/ 94.9%

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

    7. associate-*r/ 94.9%

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

    8. metadata-eval 94.9%

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

    9. div-sub 94.9%

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

    10. sub-neg 94.9%

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

    11. associate-*r/ 94.9%

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

    12. metadata-eval 94.9%

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

    13. metadata-eval 94.9%

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

15. Simplified 94.9%

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

16. Final simplification 94.9%

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

Alternative 7: 94.9% 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 23.4%

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

   1. +-commutative 23.4%

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

   2. log1p-define 23.4%

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

3. Simplified 23.4%

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

5. Taylor expanded in N around inf 94.4%

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

   1. associate--l+ 94.4%

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

   2. unpow2 94.4%

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

   3. associate-/r* 94.4%

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

   4. metadata-eval 94.4%

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

   5. associate-*r/ 94.4%

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

   6. associate-*r/ 94.4%

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

   7. metadata-eval 94.4%

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

   8. div-sub 94.4%

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

   9. sub-neg 94.4%

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

   10. metadata-eval 94.4%

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

   11. +-commutative 94.4%

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

   12. associate-*r/ 94.4%

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

   13. metadata-eval 94.4%

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

7. Simplified 94.4%

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

8. Final simplification 94.4%

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

Alternative 8: 92.9% accurate, 41.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 23.4%

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

   1. +-commutative 23.4%

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

   2. log1p-define 23.4%

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

3. Simplified 23.4%

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

5. Taylor expanded in N around inf 96.0%

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

7. Simplified 96.1%

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

   1. clear-num 96.1%

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

   2. inv-pow 96.1%

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

   3. associate--l+ 96.1%

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

   4. div-inv 96.1%

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

   5. pow-flip 96.1%

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

   6. metadata-eval 96.1%

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

9. Applied egg-rr 96.1%

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

    1. unpow-1 96.1%

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

    2. +-commutative 96.1%

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

    3. sub-neg 96.1%

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

    4. associate-+l+ 96.1%

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

    5. +-commutative 96.1%

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

    6. distribute-lft-neg-in 96.1%

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

    7. metadata-eval 96.1%

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

11. Simplified 96.1%

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

12. Taylor expanded in N around inf 92.3%

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

    1. distribute-lft-in 92.4%

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

    2. *-rgt-identity 92.4%

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

    3. associate-*r/ 92.4%

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

    4. metadata-eval 92.4%

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

    5. associate-*r/ 92.4%

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

    6. *-commutative 92.4%

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

    7. associate-*r/ 92.4%

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

    8. *-inverses 92.4%

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

    9. metadata-eval 92.4%

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

14. Simplified 92.4%

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

15. Final simplification 92.4%

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

Alternative 9: 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 23.4%

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

   1. +-commutative 23.4%

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

   2. log1p-define 23.4%

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

3. Simplified 23.4%

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

5. Taylor expanded in N around inf 85.0%

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

6. Final simplification 85.0%

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

Alternative 10: 9.9% accurate, 205.0× speedup

Math

\[\begin{array}{l} \\ 2 \end{array} \]

FPCore

(FPCore (N) :precision binary64 2.0)

C

double code(double N) {
	return 2.0;
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    code = 2.0d0
end function

Java

public static double code(double N) {
	return 2.0;
}

Python

def code(N):
	return 2.0

Julia

function code(N)
	return 2.0
end

MATLAB

function tmp = code(N)
	tmp = 2.0;
end

Wolfram

code[N_] := 2.0

Derivation

1. Initial program 23.4%

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

   1. +-commutative 23.4%

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

   2. log1p-define 23.4%

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

3. Simplified 23.4%

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

5. Taylor expanded in N around inf 96.0%

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

7. Simplified 96.1%

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

   1. clear-num 96.1%

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

   2. inv-pow 96.1%

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

   3. associate--l+ 96.1%

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

   4. div-inv 96.1%

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

   5. pow-flip 96.1%

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

   6. metadata-eval 96.1%

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

9. Applied egg-rr 96.1%

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

    1. unpow-1 96.1%

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

    2. +-commutative 96.1%

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

    3. sub-neg 96.1%

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

    4. associate-+l+ 96.1%

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

    5. +-commutative 96.1%

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

    6. distribute-lft-neg-in 96.1%

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

    7. metadata-eval 96.1%

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

11. Simplified 96.1%

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

12. Taylor expanded in N around inf 92.3%

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

    1. distribute-lft-in 92.4%

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

    2. *-rgt-identity 92.4%

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

    3. associate-*r/ 92.4%

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

    4. metadata-eval 92.4%

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

    5. associate-*r/ 92.4%

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

    6. *-commutative 92.4%

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

    7. associate-*r/ 92.4%

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

    8. *-inverses 92.4%

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

    9. metadata-eval 92.4%

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

14. Simplified 92.4%

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

15. Taylor expanded in N around 0 9.9%

    \[\leadsto \color{blue}{2} \]

16. Final simplification 9.9%

    \[\leadsto 2 \]
17. Add Preprocessing

Developer target: 99.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Reproduce

herbie shell --seed 2024095 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  :pre (and (> N 1.0) (< N 1e+40))

  :alt
  (log1p (/ 1.0 N))

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