2log (problem 3.3.6)

Percentage Accurate: 24.1% → 99.8%

Time: 8.0s

Alternatives: 8

Speedup: 17.3×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 24.1% 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: 99.8% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (N) :precision binary64 (log1p (pow N -1.0)))

C

double code(double N) {
	return log1p(pow(N, -1.0));
}

Java

public static double code(double N) {
	return Math.log1p(Math.pow(N, -1.0));
}

Python

def code(N):
	return math.log1p(math.pow(N, -1.0))

Julia

function code(N)
	return log1p((N ^ -1.0))
end

Wolfram

code[N_] := N[Log[1 + N[Power[N, -1.0], $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 23.7%

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

   1. lift--.f64 N/A

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

   2. lift-log.f64 N/A

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

   3. lift-log.f64 N/A

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

   4. diff-log N/A

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

   5. lower-log.f64 N/A

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

   6. lower-/.f64 26.4

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

   7. lift-+.f64 N/A

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

   8. +-commutative N/A

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

   9. lower-+.f64 26.4

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

4. Applied rewrites 26.4%

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

   1. lift-/.f64 N/A

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

   2. frac-2neg N/A

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

   3. div-inv N/A

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

   4. metadata-eval N/A

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

   5. frac-2neg N/A

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

   6. lower-*.f64 N/A

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

   7. lift-+.f64 N/A

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

   8. +-commutative N/A

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

   9. distribute-neg-in N/A

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

   10. neg-mul-1 N/A

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

   11. metadata-eval N/A

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

   12. lower-fma.f64 N/A

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

   13. lower-/.f64 26.1

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

6. Applied rewrites 26.1%

   \[\leadsto \log \color{blue}{\left(\mathsf{fma}\left(-1, N, -1\right) \cdot \frac{-1}{N}\right)} \]
7. Step-by-step derivation

   1. lift-log.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lift-fma.f64 N/A

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

   5. neg-mul-1 N/A

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

   6. lift-neg.f64 N/A

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

   7. distribute-lft-in N/A

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

   8. lift-/.f64 N/A

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

   9. frac-2neg N/A

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

   10. metadata-eval N/A

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

   11. lift-neg.f64 N/A

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

   12. inv-pow N/A

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

   13. pow-plus N/A

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

   14. metadata-eval N/A

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

   15. metadata-eval N/A

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

   16. lift-/.f64 N/A

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

   17. frac-2neg N/A

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

   18. metadata-eval N/A

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

   19. lift-neg.f64 N/A

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

   20. inv-pow N/A

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

   21. metadata-eval N/A

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

   22. unpow-prod-down N/A

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

   23. lift-neg.f64 N/A

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

   24. neg-mul-1 N/A

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

   25. *-commutative N/A

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

   26. associate-*l* N/A

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

   27. metadata-eval N/A

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

   28. *-rgt-identity N/A

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

8. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\mathsf{log1p}\left({N}^{-1}\right)} \]
9. Add Preprocessing

Alternative 2: 99.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\log \left(N - -1\right) - \log N \leq 0.0002:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\frac{0.5 - \frac{0.08333333333333333 - \frac{0.041666666666666664}{N}}{N}}{N}, N, N\right)}\\ \mathbf{else}:\\ \;\;\;\;-\log \left(\frac{N}{N - -1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= (- (log (- N -1.0)) (log N)) 0.0002)
   (/
    1.0
    (fma
     (/ (- 0.5 (/ (- 0.08333333333333333 (/ 0.041666666666666664 N)) N)) N)
     N
     N))
   (- (log (/ N (- N -1.0))))))

C

double code(double N) {
	double tmp;
	if ((log((N - -1.0)) - log(N)) <= 0.0002) {
		tmp = 1.0 / fma(((0.5 - ((0.08333333333333333 - (0.041666666666666664 / N)) / N)) / N), N, N);
	} else {
		tmp = -log((N / (N - -1.0)));
	}
	return tmp;
}

Julia

function code(N)
	tmp = 0.0
	if (Float64(log(Float64(N - -1.0)) - log(N)) <= 0.0002)
		tmp = Float64(1.0 / fma(Float64(Float64(0.5 - Float64(Float64(0.08333333333333333 - Float64(0.041666666666666664 / N)) / N)) / N), N, N));
	else
		tmp = Float64(-log(Float64(N / Float64(N - -1.0))));
	end
	return tmp
end

Wolfram

code[N_] := If[LessEqual[N[(N[Log[N[(N - -1.0), $MachinePrecision]], $MachinePrecision] - N[Log[N], $MachinePrecision]), $MachinePrecision], 0.0002], N[(1.0 / N[(N[(N[(0.5 - N[(N[(0.08333333333333333 - N[(0.041666666666666664 / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision] * N + N), $MachinePrecision]), $MachinePrecision], (-N[Log[N[(N / N[(N - -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (log.f64 (+.f64 N #s(literal 1 binary64))) (log.f64 N)) < 2.0000000000000001e-4

   1. Initial program 17.8%

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

   3. Taylor expanded in N around inf

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

   5. Applied rewrites 99.9%

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

   7. Applied rewrites 99.9%

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

   8. Taylor expanded in N around -inf

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

   10. Applied rewrites 99.8%

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

   12. Applied rewrites 99.9%

       \[\leadsto \frac{1}{\mathsf{fma}\left(\frac{0.5 - \frac{0.08333333333333333 - \frac{0.041666666666666664}{N}}{N}}{N}, N, N\right)} \]

   if 2.0000000000000001e-4 < (-.f64 (log.f64 (+.f64 N #s(literal 1 binary64))) (log.f64 N))

   1. Initial program 93.6%

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

      1. lift--.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. diff-log N/A

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

      5. lower-log.f64 N/A

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

      6. lower-/.f64 95.0

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 95.0

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

   4. Applied rewrites 95.0%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. div-inv N/A

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

      4. metadata-eval N/A

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

      5. frac-2neg N/A

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

      6. lower-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. distribute-neg-in N/A

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

      10. neg-mul-1 N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 94.5

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

   6. Applied rewrites 94.5%

      \[\leadsto \log \color{blue}{\left(\mathsf{fma}\left(-1, N, -1\right) \cdot \frac{-1}{N}\right)} \]
   7. Step-by-step derivation

      1. lift-log.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. clear-num N/A

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

      6. log-rec N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 96.5

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

      11. lift-fma.f64 N/A

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

      12. neg-mul-1 N/A

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

      13. lift-neg.f64 N/A

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

      14. +-commutative N/A

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

      15. lift-neg.f64 N/A

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

      16. unsub-neg N/A

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

      17. lower--.f64 96.5

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

   8. Applied rewrites 96.5%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(N - -1\right) - \log N \leq 0.0002:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\frac{0.5 - \frac{0.08333333333333333 - \frac{0.041666666666666664}{N}}{N}}{N}, N, N\right)}\\ \mathbf{else}:\\ \;\;\;\;-\log \left(\frac{N}{N - -1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\log \left(N - -1\right) - \log N \leq 0.0002:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\frac{0.5 - \frac{0.08333333333333333 - \frac{0.041666666666666664}{N}}{N}}{N}, N, N\right)}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{N - -1}{N}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= (- (log (- N -1.0)) (log N)) 0.0002)
   (/
    1.0
    (fma
     (/ (- 0.5 (/ (- 0.08333333333333333 (/ 0.041666666666666664 N)) N)) N)
     N
     N))
   (log (/ (- N -1.0) N))))

C

double code(double N) {
	double tmp;
	if ((log((N - -1.0)) - log(N)) <= 0.0002) {
		tmp = 1.0 / fma(((0.5 - ((0.08333333333333333 - (0.041666666666666664 / N)) / N)) / N), N, N);
	} else {
		tmp = log(((N - -1.0) / N));
	}
	return tmp;
}

Julia

function code(N)
	tmp = 0.0
	if (Float64(log(Float64(N - -1.0)) - log(N)) <= 0.0002)
		tmp = Float64(1.0 / fma(Float64(Float64(0.5 - Float64(Float64(0.08333333333333333 - Float64(0.041666666666666664 / N)) / N)) / N), N, N));
	else
		tmp = log(Float64(Float64(N - -1.0) / N));
	end
	return tmp
end

Wolfram

code[N_] := If[LessEqual[N[(N[Log[N[(N - -1.0), $MachinePrecision]], $MachinePrecision] - N[Log[N], $MachinePrecision]), $MachinePrecision], 0.0002], N[(1.0 / N[(N[(N[(0.5 - N[(N[(0.08333333333333333 - N[(0.041666666666666664 / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision] * N + N), $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)) < 2.0000000000000001e-4

   1. Initial program 17.8%

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

   3. Taylor expanded in N around inf

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

   5. Applied rewrites 99.9%

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

   7. Applied rewrites 99.9%

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

   8. Taylor expanded in N around -inf

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

   10. Applied rewrites 99.8%

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

   12. Applied rewrites 99.9%

       \[\leadsto \frac{1}{\mathsf{fma}\left(\frac{0.5 - \frac{0.08333333333333333 - \frac{0.041666666666666664}{N}}{N}}{N}, N, N\right)} \]

   if 2.0000000000000001e-4 < (-.f64 (log.f64 (+.f64 N #s(literal 1 binary64))) (log.f64 N))

   1. Initial program 93.6%

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

      1. lift--.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. diff-log N/A

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

      5. lower-log.f64 N/A

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

      6. lower-/.f64 95.0

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 95.0

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

   4. Applied rewrites 95.0%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(N - -1\right) - \log N \leq 0.0002:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\frac{0.5 - \frac{0.08333333333333333 - \frac{0.041666666666666664}{N}}{N}}{N}, N, N\right)}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{N - -1}{N}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.6% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 23.7%

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

3. Taylor expanded in N around inf

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

5. Applied rewrites 95.9%

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

7. Applied rewrites 95.9%

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

8. Taylor expanded in N around -inf

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

10. Applied rewrites 96.2%

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

12. Applied rewrites 96.3%

    \[\leadsto \frac{1}{\mathsf{fma}\left(\frac{0.5 - \frac{0.08333333333333333 - \frac{0.041666666666666664}{N}}{N}}{N}, N, N\right)} \]
13. Add Preprocessing

Alternative 5: 96.4% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 23.7%

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

3. Taylor expanded in N around inf

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

5. Applied rewrites 95.9%

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

7. Applied rewrites 95.9%

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

8. Taylor expanded in N around -inf

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

10. Applied rewrites 96.2%

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

11. Taylor expanded in N around 0

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

13. Applied rewrites 96.1%

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

Alternative 6: 94.8% accurate, 5.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 23.7%

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

3. Taylor expanded in N around inf

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

   1. lower-/.f64 N/A

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

   2. associate--l+ N/A

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

   3. +-commutative N/A

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

   4. unpow2 N/A

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

   5. associate-/r* N/A

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

   6. metadata-eval N/A

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

   7. associate-*r/ N/A

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

   8. associate-*r/ N/A

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

   9. metadata-eval N/A

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

   10. div-sub N/A

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

   11. metadata-eval N/A

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

   12. sub-neg N/A

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

   13. lower--.f64 N/A

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

   14. lower-/.f64 N/A

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

   15. lower--.f64 N/A

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

   16. associate-*r/ N/A

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

   17. metadata-eval N/A

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

   18. lower-/.f64 94.8

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

5. Applied rewrites 94.8%

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

Alternative 7: 92.9% accurate, 13.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 23.7%

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

3. Taylor expanded in N around inf

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

5. Applied rewrites 95.9%

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

7. Applied rewrites 95.9%

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

8. Taylor expanded in N around inf

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

10. Applied rewrites 93.2%

    \[\leadsto \frac{1}{\mathsf{fma}\left(N, \color{blue}{\frac{0.5}{N}}, N\right)} \]
11. Step-by-step derivation

12. Applied rewrites 93.2%

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

13. Final simplification 93.2%

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

Alternative 8: 84.3% accurate, 17.3× 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 23.7%

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

3. Taylor expanded in N around inf

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

   1. lower-/.f64 84.6

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

5. Applied rewrites 84.6%

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

Developer Target 1: 96.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \left(\left(\frac{1}{N} + \frac{-1}{2 \cdot {N}^{2}}\right) + \frac{1}{3 \cdot {N}^{3}}\right) + \frac{-1}{4 \cdot {N}^{4}} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (+
  (+ (+ (/ 1.0 N) (/ -1.0 (* 2.0 (pow N 2.0)))) (/ 1.0 (* 3.0 (pow N 3.0))))
  (/ -1.0 (* 4.0 (pow N 4.0)))))

C

double code(double N) {
	return (((1.0 / N) + (-1.0 / (2.0 * pow(N, 2.0)))) + (1.0 / (3.0 * pow(N, 3.0)))) + (-1.0 / (4.0 * pow(N, 4.0)));
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    code = (((1.0d0 / n) + ((-1.0d0) / (2.0d0 * (n ** 2.0d0)))) + (1.0d0 / (3.0d0 * (n ** 3.0d0)))) + ((-1.0d0) / (4.0d0 * (n ** 4.0d0)))
end function

Java

public static double code(double N) {
	return (((1.0 / N) + (-1.0 / (2.0 * Math.pow(N, 2.0)))) + (1.0 / (3.0 * Math.pow(N, 3.0)))) + (-1.0 / (4.0 * Math.pow(N, 4.0)));
}

Python

def code(N):
	return (((1.0 / N) + (-1.0 / (2.0 * math.pow(N, 2.0)))) + (1.0 / (3.0 * math.pow(N, 3.0)))) + (-1.0 / (4.0 * math.pow(N, 4.0)))

Julia

function code(N)
	return Float64(Float64(Float64(Float64(1.0 / N) + Float64(-1.0 / Float64(2.0 * (N ^ 2.0)))) + Float64(1.0 / Float64(3.0 * (N ^ 3.0)))) + Float64(-1.0 / Float64(4.0 * (N ^ 4.0))))
end

MATLAB

function tmp = code(N)
	tmp = (((1.0 / N) + (-1.0 / (2.0 * (N ^ 2.0)))) + (1.0 / (3.0 * (N ^ 3.0)))) + (-1.0 / (4.0 * (N ^ 4.0)));
end

Wolfram

code[N_] := N[(N[(N[(N[(1.0 / N), $MachinePrecision] + N[(-1.0 / N[(2.0 * N[Power[N, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(3.0 * N[Power[N, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[(4.0 * N[Power[N, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024296 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  :pre (and (> N 1.0) (< N 1e+40))

  :alt
  (! :herbie-platform default (+ (/ 1 N) (/ -1 (* 2 (pow N 2))) (/ 1 (* 3 (pow N 3))) (/ -1 (* 4 (pow N 4)))))

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