2log (problem 3.3.6)

Percentage Accurate: 24.0% → 99.8%

Time: 10.1s

Alternatives: 9

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: 24.0% 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, 1.8× 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]

Derivation

1. Initial program 24.4%

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

   1. diff-log N/A

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

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

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

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

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

   4. +-lowering-+.f64 27.4

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

4. Applied egg-rr 27.4%

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

5. Taylor expanded in N around inf

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

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

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

   2. /-lowering-/.f64 27.4

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

7. Simplified 27.4%

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

   1. accelerator-lowering-log1p.f64 N/A

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\frac{1}{N}\right)} \]

   2. /-lowering-/.f64 99.8

      \[\leadsto \mathsf{log1p}\left(\color{blue}{\frac{1}{N}}\right) \]

9. Applied egg-rr 99.8%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(\frac{1}{N}\right)} \]
10. Add Preprocessing

Alternative 2: 96.6% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 24.4%

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

3. Taylor expanded in N around inf

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

5. Simplified 97.0%

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

   1. clear-num N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

   9. /-lowering-/.f64 97.0

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

7. Applied egg-rr 97.0%

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

8. Taylor expanded in N around -inf

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

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

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

10. Simplified 97.4%

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

11. Taylor expanded in N around inf

    \[\leadsto \frac{1}{\color{blue}{N \cdot \left(\left(1 + \left(\frac{1}{2} \cdot \frac{1}{N} + \frac{1}{24} \cdot \frac{1}{{N}^{3}}\right)\right) - \frac{\frac{1}{12}}{{N}^{2}}\right)}} \]
12. Step-by-step derivation

    1. sub-neg N/A

       \[\leadsto \frac{1}{N \cdot \color{blue}{\left(\left(1 + \left(\frac{1}{2} \cdot \frac{1}{N} + \frac{1}{24} \cdot \frac{1}{{N}^{3}}\right)\right) + \left(\mathsf{neg}\left(\frac{\frac{1}{12}}{{N}^{2}}\right)\right)\right)}} \]

    2. distribute-lft-in N/A

       \[\leadsto \frac{1}{\color{blue}{N \cdot \left(1 + \left(\frac{1}{2} \cdot \frac{1}{N} + \frac{1}{24} \cdot \frac{1}{{N}^{3}}\right)\right) + N \cdot \left(\mathsf{neg}\left(\frac{\frac{1}{12}}{{N}^{2}}\right)\right)}} \]

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

       \[\leadsto \frac{1}{N \cdot \left(1 + \left(\frac{1}{2} \cdot \frac{1}{N} + \frac{1}{24} \cdot \frac{1}{{N}^{3}}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(N \cdot \frac{\frac{1}{12}}{{N}^{2}}\right)\right)}} \]

    4. associate-/l* N/A

       \[\leadsto \frac{1}{N \cdot \left(1 + \left(\frac{1}{2} \cdot \frac{1}{N} + \frac{1}{24} \cdot \frac{1}{{N}^{3}}\right)\right) + \left(\mathsf{neg}\left(\color{blue}{\frac{N \cdot \frac{1}{12}}{{N}^{2}}}\right)\right)} \]

    5. *-commutative N/A

       \[\leadsto \frac{1}{N \cdot \left(1 + \left(\frac{1}{2} \cdot \frac{1}{N} + \frac{1}{24} \cdot \frac{1}{{N}^{3}}\right)\right) + \left(\mathsf{neg}\left(\frac{\color{blue}{\frac{1}{12} \cdot N}}{{N}^{2}}\right)\right)} \]

    6. unpow2 N/A

       \[\leadsto \frac{1}{N \cdot \left(1 + \left(\frac{1}{2} \cdot \frac{1}{N} + \frac{1}{24} \cdot \frac{1}{{N}^{3}}\right)\right) + \left(\mathsf{neg}\left(\frac{\frac{1}{12} \cdot N}{\color{blue}{N \cdot N}}\right)\right)} \]

    7. associate-/r* N/A

       \[\leadsto \frac{1}{N \cdot \left(1 + \left(\frac{1}{2} \cdot \frac{1}{N} + \frac{1}{24} \cdot \frac{1}{{N}^{3}}\right)\right) + \left(\mathsf{neg}\left(\color{blue}{\frac{\frac{\frac{1}{12} \cdot N}{N}}{N}}\right)\right)} \]

    8. associate-/l* N/A

       \[\leadsto \frac{1}{N \cdot \left(1 + \left(\frac{1}{2} \cdot \frac{1}{N} + \frac{1}{24} \cdot \frac{1}{{N}^{3}}\right)\right) + \left(\mathsf{neg}\left(\frac{\color{blue}{\frac{1}{12} \cdot \frac{N}{N}}}{N}\right)\right)} \]

    9. *-lft-identity N/A

       \[\leadsto \frac{1}{N \cdot \left(1 + \left(\frac{1}{2} \cdot \frac{1}{N} + \frac{1}{24} \cdot \frac{1}{{N}^{3}}\right)\right) + \left(\mathsf{neg}\left(\frac{\frac{1}{12} \cdot \frac{N}{N}}{\color{blue}{1 \cdot N}}\right)\right)} \]

    10. times-frac N/A

        \[\leadsto \frac{1}{N \cdot \left(1 + \left(\frac{1}{2} \cdot \frac{1}{N} + \frac{1}{24} \cdot \frac{1}{{N}^{3}}\right)\right) + \left(\mathsf{neg}\left(\color{blue}{\frac{\frac{1}{12}}{1} \cdot \frac{\frac{N}{N}}{N}}\right)\right)} \]

    11. metadata-eval N/A

        \[\leadsto \frac{1}{N \cdot \left(1 + \left(\frac{1}{2} \cdot \frac{1}{N} + \frac{1}{24} \cdot \frac{1}{{N}^{3}}\right)\right) + \left(\mathsf{neg}\left(\color{blue}{\frac{1}{12}} \cdot \frac{\frac{N}{N}}{N}\right)\right)} \]

    12. *-inverses N/A

        \[\leadsto \frac{1}{N \cdot \left(1 + \left(\frac{1}{2} \cdot \frac{1}{N} + \frac{1}{24} \cdot \frac{1}{{N}^{3}}\right)\right) + \left(\mathsf{neg}\left(\frac{1}{12} \cdot \frac{\color{blue}{1}}{N}\right)\right)} \]

    13. accelerator-lowering-fma.f64 N/A

        \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(N, 1 + \left(\frac{1}{2} \cdot \frac{1}{N} + \frac{1}{24} \cdot \frac{1}{{N}^{3}}\right), \mathsf{neg}\left(\frac{1}{12} \cdot \frac{1}{N}\right)\right)}} \]

13. Simplified 97.4%

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

Alternative 3: 96.6% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 24.4%

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

3. Taylor expanded in N around inf

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

5. Simplified 97.0%

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

   1. clear-num N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

   9. /-lowering-/.f64 97.0

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

7. Applied egg-rr 97.0%

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

8. Taylor expanded in N around -inf

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

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

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

10. Simplified 97.4%

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

    1. sub-neg N/A

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

    2. neg-mul-1 N/A

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

    3. +-commutative N/A

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

    4. neg-mul-1 N/A

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

    5. div-inv N/A

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

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

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

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

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

12. Applied egg-rr 97.4%

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

13. Final simplification 97.4%

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

Alternative 4: 96.6% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 24.4%

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

3. Taylor expanded in N around inf

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

5. Simplified 97.0%

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

   1. clear-num N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

   9. /-lowering-/.f64 97.0

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

7. Applied egg-rr 97.0%

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

8. Taylor expanded in N around -inf

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

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

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

10. Simplified 97.4%

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

11. Taylor expanded in N around 0

    \[\leadsto \frac{1}{\left(-1 - \color{blue}{\frac{\frac{1}{24} + N \cdot \left(\frac{1}{2} \cdot N - \frac{1}{12}\right)}{{N}^{3}}}\right) \cdot \left(0 - N\right)} \]
12. Step-by-step derivation

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

       \[\leadsto \frac{1}{\left(-1 - \color{blue}{\frac{\frac{1}{24} + N \cdot \left(\frac{1}{2} \cdot N - \frac{1}{12}\right)}{{N}^{3}}}\right) \cdot \left(0 - N\right)} \]

    2. +-commutative N/A

       \[\leadsto \frac{1}{\left(-1 - \frac{\color{blue}{N \cdot \left(\frac{1}{2} \cdot N - \frac{1}{12}\right) + \frac{1}{24}}}{{N}^{3}}\right) \cdot \left(0 - N\right)} \]

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

       \[\leadsto \frac{1}{\left(-1 - \frac{\color{blue}{\mathsf{fma}\left(N, \frac{1}{2} \cdot N - \frac{1}{12}, \frac{1}{24}\right)}}{{N}^{3}}\right) \cdot \left(0 - N\right)} \]

    4. sub-neg N/A

       \[\leadsto \frac{1}{\left(-1 - \frac{\mathsf{fma}\left(N, \color{blue}{\frac{1}{2} \cdot N + \left(\mathsf{neg}\left(\frac{1}{12}\right)\right)}, \frac{1}{24}\right)}{{N}^{3}}\right) \cdot \left(0 - N\right)} \]

    5. *-commutative N/A

       \[\leadsto \frac{1}{\left(-1 - \frac{\mathsf{fma}\left(N, \color{blue}{N \cdot \frac{1}{2}} + \left(\mathsf{neg}\left(\frac{1}{12}\right)\right), \frac{1}{24}\right)}{{N}^{3}}\right) \cdot \left(0 - N\right)} \]

    6. metadata-eval N/A

       \[\leadsto \frac{1}{\left(-1 - \frac{\mathsf{fma}\left(N, N \cdot \frac{1}{2} + \color{blue}{\frac{-1}{12}}, \frac{1}{24}\right)}{{N}^{3}}\right) \cdot \left(0 - N\right)} \]

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

       \[\leadsto \frac{1}{\left(-1 - \frac{\mathsf{fma}\left(N, \color{blue}{\mathsf{fma}\left(N, \frac{1}{2}, \frac{-1}{12}\right)}, \frac{1}{24}\right)}{{N}^{3}}\right) \cdot \left(0 - N\right)} \]

    8. cube-mult N/A

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

    9. unpow2 N/A

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

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

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

    11. unpow2 N/A

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

    12. *-lowering-*.f64 97.4

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

13. Simplified 97.4%

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

14. Final simplification 97.4%

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

Alternative 5: 96.5% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 24.4%

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

3. Taylor expanded in N around inf

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

5. Simplified 97.0%

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

   1. clear-num N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

   9. /-lowering-/.f64 97.0

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

7. Applied egg-rr 97.0%

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

8. Taylor expanded in N around -inf

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

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

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

10. Simplified 97.4%

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

11. Taylor expanded in N around 0

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

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

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

    2. +-commutative N/A

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

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

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

    4. sub-neg N/A

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

    5. metadata-eval N/A

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

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

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

    7. +-commutative N/A

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

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

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

    9. unpow2 N/A

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

    10. *-lowering-*.f64 97.3

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

13. Simplified 97.3%

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

Alternative 6: 95.5% accurate, 7.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 24.4%

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

3. Taylor expanded in N around inf

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

5. Simplified 97.0%

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

   1. clear-num N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

   9. /-lowering-/.f64 97.0

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

7. Applied egg-rr 97.0%

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

8. Taylor expanded in N around -inf

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

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

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

10. Simplified 97.4%

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

11. Taylor expanded in N around inf

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

    1. sub-neg N/A

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

    2. distribute-lft-in N/A

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

    3. distribute-rgt-in N/A

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

    4. *-lft-identity N/A

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

    5. associate-*l* N/A

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

    6. lft-mult-inverse N/A

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

    7. metadata-eval N/A

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

    8. +-commutative N/A

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

    9. distribute-rgt-neg-out N/A

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

    10. associate-/l* N/A

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

    11. *-commutative N/A

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

    12. unpow2 N/A

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

    13. associate-/r* N/A

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

    14. associate-/l* N/A

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

    15. *-lft-identity N/A

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

    16. times-frac N/A

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

    17. metadata-eval N/A

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

    18. *-inverses N/A

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

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

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

13. Simplified 95.8%

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

14. Final simplification 95.8%

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

Alternative 7: 92.9% accurate, 13.8× 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 24.4%

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

3. Taylor expanded in N around inf

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

5. Simplified 97.0%

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

   1. clear-num N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

   9. /-lowering-/.f64 97.0

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

7. Applied egg-rr 97.0%

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

8. Taylor expanded in N around inf

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

   1. distribute-rgt-in N/A

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

   2. *-lft-identity N/A

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

   3. associate-*l* N/A

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

   4. lft-mult-inverse N/A

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

   5. metadata-eval N/A

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

   6. +-lowering-+.f64 92.8

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

10. Simplified 92.8%

    \[\leadsto \frac{1}{\color{blue}{N + 0.5}} \]
11. 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 24.4%

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

3. Taylor expanded in N around inf

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

   1. /-lowering-/.f64 83.9

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

5. Simplified 83.9%

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

Alternative 9: 9.9% accurate, 207.0× 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 24.4%

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

3. Taylor expanded in N around inf

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

5. Simplified 97.0%

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

   1. clear-num N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

   9. /-lowering-/.f64 97.0

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

7. Applied egg-rr 97.0%

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

8. Taylor expanded in N around inf

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

   1. distribute-rgt-in N/A

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

   2. *-lft-identity N/A

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

   3. associate-*l* N/A

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

   4. lft-mult-inverse N/A

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

   5. metadata-eval N/A

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

   6. +-lowering-+.f64 92.8

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

10. Simplified 92.8%

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

11. Taylor expanded in N around 0

    \[\leadsto \color{blue}{2} \]
12. Step-by-step derivation

13. Simplified 9.9%

    \[\leadsto \color{blue}{2} \]
14. Add Preprocessing

Developer Target 1: 99.8% accurate, 1.8× 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]

Developer Target 2: 26.7% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Developer Target 3: 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 2024194 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  :pre (and (> N 1.0) (< N 1e+40))

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

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

  :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)))