2log (problem 3.3.6)

Percentage Accurate: 24.0% → 99.8%

Time: 7.5s

Alternatives: 10

Speedup: 68.3×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 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, 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 25.9%

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

   1. diff-log 28.4%

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

4. Applied egg-rr 28.4%

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

   1. *-lft-identity 28.4%

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

   2. associate-*l/ 28.0%

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

   3. distribute-lft-in 28.1%

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

   4. lft-mult-inverse 28.3%

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

   5. *-rgt-identity 28.3%

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

   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: 96.7% accurate, 12.1× speedup

Math

\[\begin{array}{l} \\ \frac{1 + \frac{1}{N \cdot \left(\frac{\frac{0.1111111111111111}{N} + -1.3333333333333333}{N} - 2\right)}}{N} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (/
  (+
   1.0
   (/
    1.0
    (* N (- (/ (+ (/ 0.1111111111111111 N) -1.3333333333333333) N) 2.0))))
  N))

C

double code(double N) {
	return (1.0 + (1.0 / (N * ((((0.1111111111111111 / N) + -1.3333333333333333) / N) - 2.0)))) / N;
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    code = (1.0d0 + (1.0d0 / (n * ((((0.1111111111111111d0 / n) + (-1.3333333333333333d0)) / n) - 2.0d0)))) / n
end function

Java

public static double code(double N) {
	return (1.0 + (1.0 / (N * ((((0.1111111111111111 / N) + -1.3333333333333333) / N) - 2.0)))) / N;
}

Python

def code(N):
	return (1.0 + (1.0 / (N * ((((0.1111111111111111 / N) + -1.3333333333333333) / N) - 2.0)))) / N

Julia

function code(N)
	return Float64(Float64(1.0 + Float64(1.0 / Float64(N * Float64(Float64(Float64(Float64(0.1111111111111111 / N) + -1.3333333333333333) / N) - 2.0)))) / N)
end

MATLAB

function tmp = code(N)
	tmp = (1.0 + (1.0 / (N * ((((0.1111111111111111 / N) + -1.3333333333333333) / N) - 2.0)))) / N;
end

Wolfram

code[N_] := N[(N[(1.0 + N[(1.0 / N[(N * N[(N[(N[(N[(0.1111111111111111 / N), $MachinePrecision] + -1.3333333333333333), $MachinePrecision] / N), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision]

Derivation

1. Initial program 25.9%

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

   1. diff-log 28.4%

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

4. Applied egg-rr 28.4%

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

   1. *-lft-identity 28.4%

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

   2. associate-*l/ 28.0%

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

   3. distribute-lft-in 28.1%

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

   4. lft-mult-inverse 28.3%

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

   5. *-rgt-identity 28.3%

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

   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. Taylor expanded in N around inf 95.8%

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

9. Simplified 95.8%

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

    1. clear-num 95.8%

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

    2. inv-pow 95.8%

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

11. Applied egg-rr 95.8%

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

    1. unpow-1 95.8%

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

    2. metadata-eval 95.8%

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

    3. distribute-neg-frac 95.8%

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

    4. metadata-eval 95.8%

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

    5. associate-*r/ 95.8%

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

    6. sub-neg 95.8%

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

    7. associate-*r/ 95.8%

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

    8. metadata-eval 95.8%

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

13. Simplified 95.8%

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

14. Taylor expanded in N around inf 96.4%

    \[\leadsto \frac{1 + \frac{1}{\color{blue}{N \cdot \left(\frac{0.1111111111111111}{{N}^{2}} - \left(2 + 1.3333333333333333 \cdot \frac{1}{N}\right)\right)}}}{N} \]
15. Step-by-step derivation

    1. +-commutative 96.4%

       \[\leadsto \frac{1 + \frac{1}{N \cdot \left(\frac{0.1111111111111111}{{N}^{2}} - \color{blue}{\left(1.3333333333333333 \cdot \frac{1}{N} + 2\right)}\right)}}{N} \]

    2. associate--r+ 96.4%

       \[\leadsto \frac{1 + \frac{1}{N \cdot \color{blue}{\left(\left(\frac{0.1111111111111111}{{N}^{2}} - 1.3333333333333333 \cdot \frac{1}{N}\right) - 2\right)}}}{N} \]

    3. unpow2 96.4%

       \[\leadsto \frac{1 + \frac{1}{N \cdot \left(\left(\frac{0.1111111111111111}{\color{blue}{N \cdot N}} - 1.3333333333333333 \cdot \frac{1}{N}\right) - 2\right)}}{N} \]

    4. associate-/r* 96.4%

       \[\leadsto \frac{1 + \frac{1}{N \cdot \left(\left(\color{blue}{\frac{\frac{0.1111111111111111}{N}}{N}} - 1.3333333333333333 \cdot \frac{1}{N}\right) - 2\right)}}{N} \]

    5. metadata-eval 96.4%

       \[\leadsto \frac{1 + \frac{1}{N \cdot \left(\left(\frac{\frac{\color{blue}{0.1111111111111111 \cdot 1}}{N}}{N} - 1.3333333333333333 \cdot \frac{1}{N}\right) - 2\right)}}{N} \]

    6. associate-*r/ 96.4%

       \[\leadsto \frac{1 + \frac{1}{N \cdot \left(\left(\frac{\color{blue}{0.1111111111111111 \cdot \frac{1}{N}}}{N} - 1.3333333333333333 \cdot \frac{1}{N}\right) - 2\right)}}{N} \]

    7. associate-*r/ 96.4%

       \[\leadsto \frac{1 + \frac{1}{N \cdot \left(\left(\frac{0.1111111111111111 \cdot \frac{1}{N}}{N} - \color{blue}{\frac{1.3333333333333333 \cdot 1}{N}}\right) - 2\right)}}{N} \]

    8. metadata-eval 96.4%

       \[\leadsto \frac{1 + \frac{1}{N \cdot \left(\left(\frac{0.1111111111111111 \cdot \frac{1}{N}}{N} - \frac{\color{blue}{1.3333333333333333}}{N}\right) - 2\right)}}{N} \]

    9. div-sub 96.4%

       \[\leadsto \frac{1 + \frac{1}{N \cdot \left(\color{blue}{\frac{0.1111111111111111 \cdot \frac{1}{N} - 1.3333333333333333}{N}} - 2\right)}}{N} \]

    10. sub-neg 96.4%

        \[\leadsto \frac{1 + \frac{1}{N \cdot \left(\frac{\color{blue}{0.1111111111111111 \cdot \frac{1}{N} + \left(-1.3333333333333333\right)}}{N} - 2\right)}}{N} \]

    11. associate-*r/ 96.4%

        \[\leadsto \frac{1 + \frac{1}{N \cdot \left(\frac{\color{blue}{\frac{0.1111111111111111 \cdot 1}{N}} + \left(-1.3333333333333333\right)}{N} - 2\right)}}{N} \]

    12. metadata-eval 96.4%

        \[\leadsto \frac{1 + \frac{1}{N \cdot \left(\frac{\frac{\color{blue}{0.1111111111111111}}{N} + \left(-1.3333333333333333\right)}{N} - 2\right)}}{N} \]

    13. metadata-eval 96.4%

        \[\leadsto \frac{1 + \frac{1}{N \cdot \left(\frac{\frac{0.1111111111111111}{N} + \color{blue}{-1.3333333333333333}}{N} - 2\right)}}{N} \]

16. Simplified 96.4%

    \[\leadsto \frac{1 + \frac{1}{\color{blue}{N \cdot \left(\frac{\frac{0.1111111111111111}{N} + -1.3333333333333333}{N} - 2\right)}}}{N} \]
17. Add Preprocessing

Alternative 3: 96.1% accurate, 12.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 25.9%

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

   1. diff-log 28.4%

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

4. Applied egg-rr 28.4%

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

   1. *-lft-identity 28.4%

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

   2. associate-*l/ 28.0%

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

   3. distribute-lft-in 28.1%

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

   4. lft-mult-inverse 28.3%

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

   5. *-rgt-identity 28.3%

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

   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. Taylor expanded in N around inf 95.8%

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

9. Simplified 95.8%

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

    1. clear-num 95.8%

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

    2. inv-pow 95.8%

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

11. Applied egg-rr 95.8%

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

    1. unpow-1 95.8%

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

    2. metadata-eval 95.8%

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

    3. *-lft-identity 95.8%

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

    4. remove-double-neg 95.8%

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

    5. neg-mul-1 95.8%

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

    6. times-frac 95.8%

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

    7. metadata-eval 95.8%

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

    8. metadata-eval 95.8%

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

    9. distribute-neg-frac 95.8%

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

    10. metadata-eval 95.8%

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

    11. associate-*r/ 95.8%

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

    12. sub-neg 95.8%

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

    13. distribute-neg-frac2 95.8%

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

    14. mul-1-neg 95.8%

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

    15. distribute-lft-in 95.8%

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

13. Simplified 95.8%

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

14. Final simplification 95.8%

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

Alternative 4: 96.1% accurate, 13.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 25.9%

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

   1. diff-log 28.4%

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

4. Applied egg-rr 28.4%

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

   1. *-lft-identity 28.4%

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

   2. associate-*l/ 28.0%

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

   3. distribute-lft-in 28.1%

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

   4. lft-mult-inverse 28.3%

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

   5. *-rgt-identity 28.3%

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

   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. Taylor expanded in N around inf 95.8%

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

9. Simplified 95.8%

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

Alternative 5: 95.4% accurate, 15.8× speedup

Math

\[\begin{array}{l} \\ \frac{1 + \frac{-1}{N \cdot \left(2 + \frac{1.3333333333333333}{N}\right)}}{N} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (/ (+ 1.0 (/ -1.0 (* N (+ 2.0 (/ 1.3333333333333333 N))))) N))

C

double code(double N) {
	return (1.0 + (-1.0 / (N * (2.0 + (1.3333333333333333 / N))))) / N;
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    code = (1.0d0 + ((-1.0d0) / (n * (2.0d0 + (1.3333333333333333d0 / n))))) / n
end function

Java

public static double code(double N) {
	return (1.0 + (-1.0 / (N * (2.0 + (1.3333333333333333 / N))))) / N;
}

Python

def code(N):
	return (1.0 + (-1.0 / (N * (2.0 + (1.3333333333333333 / N))))) / N

Julia

function code(N)
	return Float64(Float64(1.0 + Float64(-1.0 / Float64(N * Float64(2.0 + Float64(1.3333333333333333 / N))))) / N)
end

MATLAB

function tmp = code(N)
	tmp = (1.0 + (-1.0 / (N * (2.0 + (1.3333333333333333 / N))))) / N;
end

Wolfram

code[N_] := N[(N[(1.0 + N[(-1.0 / N[(N * N[(2.0 + N[(1.3333333333333333 / N), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision]

Derivation

1. Initial program 25.9%

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

   1. diff-log 28.4%

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

4. Applied egg-rr 28.4%

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

   1. *-lft-identity 28.4%

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

   2. associate-*l/ 28.0%

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

   3. distribute-lft-in 28.1%

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

   4. lft-mult-inverse 28.3%

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

   5. *-rgt-identity 28.3%

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

   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. Taylor expanded in N around inf 95.8%

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

9. Simplified 95.8%

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

    1. clear-num 95.8%

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

    2. inv-pow 95.8%

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

11. Applied egg-rr 95.8%

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

    1. unpow-1 95.8%

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

    2. metadata-eval 95.8%

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

    3. distribute-neg-frac 95.8%

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

    4. metadata-eval 95.8%

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

    5. associate-*r/ 95.8%

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

    6. sub-neg 95.8%

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

    7. associate-*r/ 95.8%

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

    8. metadata-eval 95.8%

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

13. Simplified 95.8%

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

14. Taylor expanded in N around inf 95.2%

    \[\leadsto \frac{1 + \frac{1}{\color{blue}{-1 \cdot \left(N \cdot \left(2 + 1.3333333333333333 \cdot \frac{1}{N}\right)\right)}}}{N} \]
15. Step-by-step derivation

    1. associate-*r* 95.2%

       \[\leadsto \frac{1 + \frac{1}{\color{blue}{\left(-1 \cdot N\right) \cdot \left(2 + 1.3333333333333333 \cdot \frac{1}{N}\right)}}}{N} \]

    2. neg-mul-1 95.2%

       \[\leadsto \frac{1 + \frac{1}{\color{blue}{\left(-N\right)} \cdot \left(2 + 1.3333333333333333 \cdot \frac{1}{N}\right)}}{N} \]

    3. associate-*r/ 95.2%

       \[\leadsto \frac{1 + \frac{1}{\left(-N\right) \cdot \left(2 + \color{blue}{\frac{1.3333333333333333 \cdot 1}{N}}\right)}}{N} \]

    4. metadata-eval 95.2%

       \[\leadsto \frac{1 + \frac{1}{\left(-N\right) \cdot \left(2 + \frac{\color{blue}{1.3333333333333333}}{N}\right)}}{N} \]

16. Simplified 95.2%

    \[\leadsto \frac{1 + \frac{1}{\color{blue}{\left(-N\right) \cdot \left(2 + \frac{1.3333333333333333}{N}\right)}}}{N} \]

17. Final simplification 95.2%

    \[\leadsto \frac{1 + \frac{-1}{N \cdot \left(2 + \frac{1.3333333333333333}{N}\right)}}{N} \]
18. Add Preprocessing

Alternative 6: 94.7% accurate, 15.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 25.9%

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

3. Taylor expanded in N around inf 94.4%

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

   1. associate--l+ 94.5%

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

   2. unpow2 94.5%

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

   3. associate-/r* 94.5%

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

   4. metadata-eval 94.5%

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

   5. associate-*r/ 94.5%

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

   6. associate-*r/ 94.5%

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

   7. metadata-eval 94.5%

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

   8. div-sub 94.5%

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

   9. sub-neg 94.5%

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

   10. metadata-eval 94.5%

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

   11. +-commutative 94.5%

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

   12. associate-*r/ 94.5%

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

   13. metadata-eval 94.5%

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

5. Simplified 94.5%

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

   1. clear-num 94.6%

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

   2. inv-pow 94.6%

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

7. Applied egg-rr 94.6%

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

   1. unpow-1 94.6%

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

9. Applied egg-rr 94.6%

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

Alternative 7: 94.7% accurate, 18.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 25.9%

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

3. Taylor expanded in N around inf 94.4%

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

   1. associate--l+ 94.5%

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

   2. unpow2 94.5%

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

   3. associate-/r* 94.5%

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

   4. metadata-eval 94.5%

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

   5. associate-*r/ 94.5%

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

   6. associate-*r/ 94.5%

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

   7. metadata-eval 94.5%

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

   8. div-sub 94.5%

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

   9. sub-neg 94.5%

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

   10. metadata-eval 94.5%

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

   11. +-commutative 94.5%

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

   12. associate-*r/ 94.5%

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

   13. metadata-eval 94.5%

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

5. Simplified 94.5%

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

Alternative 8: 92.1% accurate, 22.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 25.9%

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

3. Taylor expanded in N around inf 91.5%

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

   1. associate-*r/ 91.5%

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

   2. metadata-eval 91.5%

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

5. Simplified 91.5%

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

   1. expm1-log1p-u 91.5%

      \[\leadsto \frac{1 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.5}{N}\right)\right)}}{N} \]

   2. expm1-undefine 91.5%

      \[\leadsto \frac{1 - \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{0.5}{N}\right)} - 1\right)}}{N} \]

   3. log1p-undefine 91.5%

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

   4. add-exp-log 91.5%

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

7. Applied egg-rr 91.5%

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

   1. clear-num 91.5%

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

   2. inv-pow 91.5%

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

   3. add-exp-log 91.5%

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

   4. expm1-define 91.5%

      \[\leadsto {\left(\frac{N}{1 - \color{blue}{\mathsf{expm1}\left(\log \left(1 + \frac{0.5}{N}\right)\right)}}\right)}^{-1} \]

   5. log1p-define 91.6%

      \[\leadsto {\left(\frac{N}{1 - \mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(\frac{0.5}{N}\right)}\right)}\right)}^{-1} \]

   6. expm1-log1p-u 91.6%

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

9. Applied egg-rr 91.6%

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

    1. unpow-1 91.6%

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

    2. sub-neg 91.6%

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

    3. distribute-neg-frac 91.6%

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

    4. metadata-eval 91.6%

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

11. Simplified 91.6%

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

12. Final simplification 91.6%

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

Alternative 9: 92.1% accurate, 29.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 25.9%

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

3. Taylor expanded in N around inf 91.5%

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

   1. associate-*r/ 91.5%

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

   2. metadata-eval 91.5%

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

5. Simplified 91.5%

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

Alternative 10: 84.3% 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 25.9%

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

3. Taylor expanded in N around inf 82.9%

   \[\leadsto \color{blue}{\frac{1}{N}} \]
4. 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 2024165 
(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)))