2log (problem 3.3.6)

Percentage Accurate: 24.2% → 99.8%

Time: 7.8s

Alternatives: 9

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: 24.2% 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, 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]

Derivation

1. Initial program 26.4%

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

   1. diff-log 28.8%

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

4. Applied egg-rr 28.8%

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

   1. *-lft-identity 28.8%

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

   2. associate-*l/ 28.5%

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

   3. distribute-lft-in 28.6%

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

   4. lft-mult-inverse 28.8%

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

   5. *-rgt-identity 28.8%

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

   6. log1p-define 99.8%

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

6. Simplified 99.8%

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

Alternative 2: 96.2% accurate, 12.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 26.4%

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

   1. diff-log 28.8%

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

4. Applied egg-rr 28.8%

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

   1. *-lft-identity 28.8%

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

   2. associate-*l/ 28.5%

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

   3. distribute-lft-in 28.6%

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

   4. lft-mult-inverse 28.8%

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

   5. *-rgt-identity 28.8%

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

   6. log1p-define 99.8%

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

6. Simplified 99.8%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(\frac{1}{N}\right)} \]
7. Step-by-step derivation

   1. add-sqr-sqrt 99.2%

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

   2. sqrt-unprod 99.8%

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

   3. inv-pow 99.8%

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

   4. inv-pow 99.8%

      \[\leadsto \mathsf{log1p}\left(\sqrt{{N}^{-1} \cdot \color{blue}{{N}^{-1}}}\right) \]

   5. pow-prod-up 99.7%

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

   6. metadata-eval 99.7%

      \[\leadsto \mathsf{log1p}\left(\sqrt{{N}^{\color{blue}{-2}}}\right) \]

8. Applied egg-rr 99.7%

   \[\leadsto \mathsf{log1p}\left(\color{blue}{\sqrt{{N}^{-2}}}\right) \]

9. Taylor expanded in N around inf 95.6%

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

11. Simplified 95.6%

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

    1. clear-num 95.6%

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

    2. inv-pow 95.6%

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

13. Applied egg-rr 95.6%

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

15. Simplified 95.6%

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

16. Final simplification 95.6%

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

Alternative 3: 96.1% accurate, 13.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 26.4%

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

   1. diff-log 28.8%

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

4. Applied egg-rr 28.8%

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

   1. *-lft-identity 28.8%

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

   2. associate-*l/ 28.5%

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

   3. distribute-lft-in 28.6%

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

   4. lft-mult-inverse 28.8%

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

   5. *-rgt-identity 28.8%

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

   6. log1p-define 99.8%

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

6. Simplified 99.8%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(\frac{1}{N}\right)} \]
7. Step-by-step derivation

   1. add-sqr-sqrt 99.2%

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

   2. sqrt-unprod 99.8%

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

   3. inv-pow 99.8%

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

   4. inv-pow 99.8%

      \[\leadsto \mathsf{log1p}\left(\sqrt{{N}^{-1} \cdot \color{blue}{{N}^{-1}}}\right) \]

   5. pow-prod-up 99.7%

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

   6. metadata-eval 99.7%

      \[\leadsto \mathsf{log1p}\left(\sqrt{{N}^{\color{blue}{-2}}}\right) \]

8. Applied egg-rr 99.7%

   \[\leadsto \mathsf{log1p}\left(\color{blue}{\sqrt{{N}^{-2}}}\right) \]

9. Taylor expanded in N around inf 95.6%

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

11. Simplified 95.6%

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

12. Final simplification 95.6%

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

Alternative 4: 95.4% accurate, 15.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 26.4%

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

3. Taylor expanded in N around inf 94.2%

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

   1. associate--l+ 94.2%

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

   2. unpow2 94.2%

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

   3. associate-/r* 94.2%

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

   4. metadata-eval 94.2%

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

   5. associate-*r/ 94.2%

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

   6. associate-*r/ 94.2%

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

   7. metadata-eval 94.2%

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

   8. div-sub 94.2%

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

   9. sub-neg 94.2%

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

   10. metadata-eval 94.2%

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

   11. +-commutative 94.2%

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

   12. associate-*r/ 94.2%

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

   13. metadata-eval 94.2%

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

5. Simplified 94.2%

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

   1. clear-num 94.2%

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

   2. inv-pow 94.2%

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

7. Applied egg-rr 94.2%

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

   1. unpow-1 94.2%

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

   2. remove-double-neg 94.2%

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

   3. neg-mul-1 94.2%

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

   4. associate-*r/ 94.2%

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

   5. neg-mul-1 94.2%

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

   6. distribute-neg-in 94.2%

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

   7. metadata-eval 94.2%

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

   8. metadata-eval 94.2%

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

   9. associate-*r/ 94.2%

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

   10. sub-neg 94.2%

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

   11. associate-*r/ 94.2%

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

   12. metadata-eval 94.2%

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

   13. unsub-neg 94.2%

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

   14. unsub-neg 94.2%

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

   15. unsub-neg 94.2%

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

9. Simplified 94.2%

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

10. Taylor expanded in N around inf 94.7%

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

    1. associate--l+ 94.7%

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

    2. associate-*r/ 94.7%

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

    3. metadata-eval 94.7%

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

    4. unpow2 94.7%

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

    5. associate-/r* 94.7%

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

    6. metadata-eval 94.7%

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

    7. associate-*r/ 94.7%

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

    8. div-sub 94.7%

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

    9. cancel-sign-sub-inv 94.7%

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

    10. associate-*r/ 94.7%

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

    11. metadata-eval 94.7%

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

    12. metadata-eval 94.7%

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

12. Simplified 94.7%

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

    1. +-commutative 94.7%

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

    2. distribute-rgt-in 94.8%

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

    3. *-un-lft-identity 94.8%

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

14. Applied egg-rr 94.8%

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

15. Final simplification 94.8%

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

Alternative 5: 95.3% accurate, 15.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 26.4%

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

3. Taylor expanded in N around inf 94.2%

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

   1. associate--l+ 94.2%

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

   2. unpow2 94.2%

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

   3. associate-/r* 94.2%

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

   4. metadata-eval 94.2%

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

   5. associate-*r/ 94.2%

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

   6. associate-*r/ 94.2%

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

   7. metadata-eval 94.2%

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

   8. div-sub 94.2%

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

   9. sub-neg 94.2%

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

   10. metadata-eval 94.2%

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

   11. +-commutative 94.2%

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

   12. associate-*r/ 94.2%

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

   13. metadata-eval 94.2%

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

5. Simplified 94.2%

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

   1. clear-num 94.2%

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

   2. inv-pow 94.2%

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

7. Applied egg-rr 94.2%

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

   1. unpow-1 94.2%

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

   2. remove-double-neg 94.2%

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

   3. neg-mul-1 94.2%

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

   4. associate-*r/ 94.2%

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

   5. neg-mul-1 94.2%

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

   6. distribute-neg-in 94.2%

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

   7. metadata-eval 94.2%

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

   8. metadata-eval 94.2%

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

   9. associate-*r/ 94.2%

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

   10. sub-neg 94.2%

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

   11. associate-*r/ 94.2%

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

   12. metadata-eval 94.2%

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

   13. unsub-neg 94.2%

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

   14. unsub-neg 94.2%

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

   15. unsub-neg 94.2%

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

9. Simplified 94.2%

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

10. Taylor expanded in N around inf 94.7%

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

    1. associate--l+ 94.7%

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

    2. associate-*r/ 94.7%

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

    3. metadata-eval 94.7%

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

    4. unpow2 94.7%

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

    5. associate-/r* 94.7%

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

    6. metadata-eval 94.7%

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

    7. associate-*r/ 94.7%

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

    8. div-sub 94.7%

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

    9. cancel-sign-sub-inv 94.7%

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

    10. associate-*r/ 94.7%

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

    11. metadata-eval 94.7%

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

    12. metadata-eval 94.7%

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

12. Simplified 94.7%

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

13. Final simplification 94.7%

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

Alternative 6: 95.3% accurate, 18.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 26.4%

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

3. Taylor expanded in N around inf 94.2%

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

   1. associate--l+ 94.2%

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

   2. unpow2 94.2%

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

   3. associate-/r* 94.2%

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

   4. metadata-eval 94.2%

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

   5. associate-*r/ 94.2%

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

   6. associate-*r/ 94.2%

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

   7. metadata-eval 94.2%

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

   8. div-sub 94.2%

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

   9. sub-neg 94.2%

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

   10. metadata-eval 94.2%

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

   11. +-commutative 94.2%

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

   12. associate-*r/ 94.2%

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

   13. metadata-eval 94.2%

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

5. Simplified 94.2%

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

   1. clear-num 94.2%

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

   2. inv-pow 94.2%

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

7. Applied egg-rr 94.2%

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

   1. unpow-1 94.2%

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

   2. remove-double-neg 94.2%

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

   3. neg-mul-1 94.2%

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

   4. associate-*r/ 94.2%

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

   5. neg-mul-1 94.2%

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

   6. distribute-neg-in 94.2%

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

   7. metadata-eval 94.2%

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

   8. metadata-eval 94.2%

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

   9. associate-*r/ 94.2%

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

   10. sub-neg 94.2%

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

   11. associate-*r/ 94.2%

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

   12. metadata-eval 94.2%

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

   13. unsub-neg 94.2%

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

   14. unsub-neg 94.2%

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

   15. unsub-neg 94.2%

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

9. Simplified 94.2%

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

10. Taylor expanded in N around inf 94.7%

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

    1. associate--l+ 94.7%

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

    2. associate-*r/ 94.7%

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

    3. metadata-eval 94.7%

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

    4. unpow2 94.7%

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

    5. associate-/r* 94.7%

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

    6. metadata-eval 94.7%

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

    7. associate-*r/ 94.7%

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

    8. div-sub 94.7%

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

    9. cancel-sign-sub-inv 94.7%

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

    10. associate-*r/ 94.7%

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

    11. metadata-eval 94.7%

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

    12. metadata-eval 94.7%

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

12. Simplified 94.7%

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

13. Taylor expanded in N around 0 94.7%

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

14. Final simplification 94.7%

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

Alternative 7: 92.8% 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 26.4%

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

3. Taylor expanded in N around inf 94.2%

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

   1. associate--l+ 94.2%

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

   2. unpow2 94.2%

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

   3. associate-/r* 94.2%

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

   4. metadata-eval 94.2%

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

   5. associate-*r/ 94.2%

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

   6. associate-*r/ 94.2%

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

   7. metadata-eval 94.2%

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

   8. div-sub 94.2%

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

   9. sub-neg 94.2%

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

   10. metadata-eval 94.2%

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

   11. +-commutative 94.2%

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

   12. associate-*r/ 94.2%

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

   13. metadata-eval 94.2%

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

5. Simplified 94.2%

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

   1. clear-num 94.2%

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

   2. inv-pow 94.2%

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

7. Applied egg-rr 94.2%

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

   1. unpow-1 94.2%

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

   2. remove-double-neg 94.2%

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

   3. neg-mul-1 94.2%

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

   4. associate-*r/ 94.2%

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

   5. neg-mul-1 94.2%

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

   6. distribute-neg-in 94.2%

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

   7. metadata-eval 94.2%

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

   8. metadata-eval 94.2%

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

   9. associate-*r/ 94.2%

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

   10. sub-neg 94.2%

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

   11. associate-*r/ 94.2%

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

   12. metadata-eval 94.2%

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

   13. unsub-neg 94.2%

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

   14. unsub-neg 94.2%

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

   15. unsub-neg 94.2%

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

9. Simplified 94.2%

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

10. Taylor expanded in N around inf 91.7%

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

    1. distribute-rgt-in 91.8%

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

    2. *-lft-identity 91.8%

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

    3. associate-*l* 91.8%

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

    4. unpow-1 91.8%

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

    5. pow-plus 91.8%

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

    6. metadata-eval 91.8%

       \[\leadsto \frac{1}{N + 0.5 \cdot {N}^{\color{blue}{0}}} \]

    7. metadata-eval 91.8%

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

    8. metadata-eval 91.8%

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

12. Simplified 91.8%

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

Alternative 8: 84.2% 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 26.4%

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

3. Taylor expanded in N around inf 82.4%

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

Alternative 9: 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 26.4%

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

3. Taylor expanded in N around inf 94.2%

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

   1. associate--l+ 94.2%

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

   2. unpow2 94.2%

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

   3. associate-/r* 94.2%

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

   4. metadata-eval 94.2%

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

   5. associate-*r/ 94.2%

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

   6. associate-*r/ 94.2%

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

   7. metadata-eval 94.2%

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

   8. div-sub 94.2%

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

   9. sub-neg 94.2%

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

   10. metadata-eval 94.2%

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

   11. +-commutative 94.2%

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

   12. associate-*r/ 94.2%

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

   13. metadata-eval 94.2%

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

5. Simplified 94.2%

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

   1. clear-num 94.2%

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

   2. inv-pow 94.2%

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

7. Applied egg-rr 94.2%

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

   1. unpow-1 94.2%

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

   2. remove-double-neg 94.2%

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

   3. neg-mul-1 94.2%

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

   4. associate-*r/ 94.2%

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

   5. neg-mul-1 94.2%

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

   6. distribute-neg-in 94.2%

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

   7. metadata-eval 94.2%

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

   8. metadata-eval 94.2%

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

   9. associate-*r/ 94.2%

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

   10. sub-neg 94.2%

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

   11. associate-*r/ 94.2%

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

   12. metadata-eval 94.2%

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

   13. unsub-neg 94.2%

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

   14. unsub-neg 94.2%

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

   15. unsub-neg 94.2%

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

9. Simplified 94.2%

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

10. Taylor expanded in N around inf 91.7%

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

    1. distribute-rgt-in 91.8%

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

    2. *-lft-identity 91.8%

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

    3. associate-*l* 91.8%

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

    4. unpow-1 91.8%

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

    5. pow-plus 91.8%

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

    6. metadata-eval 91.8%

       \[\leadsto \frac{1}{N + 0.5 \cdot {N}^{\color{blue}{0}}} \]

    7. metadata-eval 91.8%

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

    8. metadata-eval 91.8%

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

12. Simplified 91.8%

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

13. Taylor expanded in N around 0 10.0%

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

Developer Target 1: 99.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Reproduce

herbie shell --seed 2024129 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  :pre (and (> N 1.0) (< N 1e+40))

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

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