2log (problem 3.3.6)

Percentage Accurate: 24.3% → 99.8%

Time: 9.2s

Alternatives: 7

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: 24.3% 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 23.0%

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

   1. sub-neg 23.0%

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

   2. +-commutative 23.0%

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

   3. log1p-undefine 23.0%

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

4. Applied egg-rr 23.0%

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

   1. sub-neg 23.0%

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

   2. rem-log-exp 23.0%

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

   3. exp-diff 22.9%

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

   4. log1p-undefine 23.0%

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

   5. +-commutative 23.0%

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

   6. rem-exp-log 25.5%

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

   7. rem-exp-log 25.9%

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

   8. *-lft-identity 25.9%

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

   9. associate-*l/ 25.7%

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

   10. distribute-lft-in 25.6%

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

   11. lft-mult-inverse 25.8%

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

   12. *-rgt-identity 25.8%

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

   13. 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, 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 23.0%

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

   1. sub-neg 23.0%

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

   2. +-commutative 23.0%

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

   3. log1p-undefine 23.0%

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

4. Applied egg-rr 23.0%

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

   1. sub-neg 23.0%

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

   2. rem-log-exp 23.0%

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

   3. exp-diff 22.9%

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

   4. log1p-undefine 23.0%

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

   5. +-commutative 23.0%

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

   6. rem-exp-log 25.5%

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

   7. rem-exp-log 25.9%

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

   8. *-lft-identity 25.9%

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

   9. associate-*l/ 25.7%

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

   10. distribute-lft-in 25.6%

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

   11. lft-mult-inverse 25.8%

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

   12. *-rgt-identity 25.8%

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

   13. 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 97.1%

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

9. Simplified 97.1%

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

10. Final simplification 97.1%

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

Alternative 3: 95.4% 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 23.0%

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

3. Taylor expanded in N around inf 96.3%

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

   1. associate--l+ 96.4%

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

   2. unpow2 96.4%

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

   3. associate-/r* 96.4%

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

   4. metadata-eval 96.4%

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

   5. associate-*r/ 96.4%

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

   6. associate-*r/ 96.4%

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

   7. metadata-eval 96.4%

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

   8. div-sub 96.4%

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

   9. sub-neg 96.4%

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

   10. metadata-eval 96.4%

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

   11. +-commutative 96.4%

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

   12. associate-*r/ 96.4%

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

   13. metadata-eval 96.4%

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

5. Simplified 96.4%

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

   1. clear-num 96.4%

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

7. Applied egg-rr 96.4%

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

8. Taylor expanded in N around -inf 96.7%

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

   1. mul-1-neg 96.7%

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

   2. *-commutative 96.7%

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

   3. distribute-rgt-neg-in 96.7%

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

   4. sub-neg 96.7%

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

   5. mul-1-neg 96.7%

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

   6. distribute-neg-frac2 96.7%

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

   7. sub-neg 96.7%

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

   8. associate-*r/ 96.7%

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

   9. metadata-eval 96.7%

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

   10. distribute-neg-frac 96.7%

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

   11. metadata-eval 96.7%

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

   12. metadata-eval 96.7%

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

10. Simplified 96.7%

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

11. Taylor expanded in N around inf 96.7%

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

    1. unpow2 96.7%

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

    2. associate-/r* 96.7%

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

    3. metadata-eval 96.7%

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

    4. associate-*r/ 96.7%

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

    5. associate-+r- 96.7%

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

    6. associate-*r/ 96.7%

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

    7. metadata-eval 96.7%

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

    8. div-sub 96.7%

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

    9. cancel-sign-sub-inv 96.7%

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

    10. metadata-eval 96.7%

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

    11. associate-*r/ 96.7%

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

    12. metadata-eval 96.7%

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

13. Simplified 96.7%

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

Alternative 4: 95.4% accurate, 18.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 23.0%

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

3. Taylor expanded in N around inf 96.3%

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

   1. associate--l+ 96.4%

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

   2. unpow2 96.4%

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

   3. associate-/r* 96.4%

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

   4. metadata-eval 96.4%

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

   5. associate-*r/ 96.4%

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

   6. associate-*r/ 96.4%

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

   7. metadata-eval 96.4%

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

   8. div-sub 96.4%

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

   9. sub-neg 96.4%

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

   10. metadata-eval 96.4%

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

   11. +-commutative 96.4%

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

   12. associate-*r/ 96.4%

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

   13. metadata-eval 96.4%

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

5. Simplified 96.4%

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

   1. clear-num 96.4%

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

7. Applied egg-rr 96.4%

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

8. Taylor expanded in N around -inf 96.7%

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

   1. mul-1-neg 96.7%

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

   2. *-commutative 96.7%

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

   3. distribute-rgt-neg-in 96.7%

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

   4. sub-neg 96.7%

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

   5. mul-1-neg 96.7%

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

   6. distribute-neg-frac2 96.7%

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

   7. sub-neg 96.7%

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

   8. associate-*r/ 96.7%

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

   9. metadata-eval 96.7%

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

   10. distribute-neg-frac 96.7%

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

   11. metadata-eval 96.7%

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

   12. metadata-eval 96.7%

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

10. Simplified 96.7%

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

11. Taylor expanded in N around 0 96.7%

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

12. Final simplification 96.7%

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

Alternative 5: 92.9% accurate, 41.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 23.0%

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

3. Taylor expanded in N around inf 96.3%

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

   1. associate--l+ 96.4%

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

   2. unpow2 96.4%

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

   3. associate-/r* 96.4%

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

   4. metadata-eval 96.4%

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

   5. associate-*r/ 96.4%

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

   6. associate-*r/ 96.4%

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

   7. metadata-eval 96.4%

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

   8. div-sub 96.4%

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

   9. sub-neg 96.4%

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

   10. metadata-eval 96.4%

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

   11. +-commutative 96.4%

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

   12. associate-*r/ 96.4%

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

   13. metadata-eval 96.4%

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

5. Simplified 96.4%

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

   1. clear-num 96.4%

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

7. Applied egg-rr 96.4%

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

8. Taylor expanded in N around inf 94.7%

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

   1. associate-*r/ 94.7%

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

   2. metadata-eval 94.7%

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

   3. metadata-eval 94.7%

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

   4. distribute-neg-frac 94.7%

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

   5. sub-neg 94.7%

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

10. Simplified 94.7%

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

11. Taylor expanded in N around 0 94.8%

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

    1. +-commutative 94.8%

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

13. Simplified 94.8%

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

Alternative 6: 84.1% accurate, 68.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(N):
	return 1.0 / N

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 23.0%

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

3. Taylor expanded in N around inf 85.4%

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

Alternative 7: 9.9% accurate, 205.0× speedup

Math

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

FPCore

(FPCore (N) :precision binary64 2.0)

C

double code(double N) {
	return 2.0;
}

Fortran

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

Java

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

Python

def code(N):
	return 2.0

Julia

function code(N)
	return 2.0
end

MATLAB

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

Wolfram

code[N_] := 2.0

Derivation

1. Initial program 23.0%

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

3. Taylor expanded in N around inf 96.3%

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

   1. associate--l+ 96.4%

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

   2. unpow2 96.4%

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

   3. associate-/r* 96.4%

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

   4. metadata-eval 96.4%

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

   5. associate-*r/ 96.4%

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

   6. associate-*r/ 96.4%

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

   7. metadata-eval 96.4%

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

   8. div-sub 96.4%

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

   9. sub-neg 96.4%

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

   10. metadata-eval 96.4%

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

   11. +-commutative 96.4%

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

   12. associate-*r/ 96.4%

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

   13. metadata-eval 96.4%

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

5. Simplified 96.4%

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

   1. clear-num 96.4%

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

7. Applied egg-rr 96.4%

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

8. Taylor expanded in N around inf 94.7%

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

   1. associate-*r/ 94.7%

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

   2. metadata-eval 94.7%

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

   3. metadata-eval 94.7%

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

   4. distribute-neg-frac 94.7%

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

   5. sub-neg 94.7%

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

10. Simplified 94.7%

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

11. Taylor expanded in N around 0 9.8%

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

Developer target: 99.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Reproduce

herbie shell --seed 2024096 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  :pre (and (> N 1.0) (< N 1e+40))

  :alt
  (log1p (/ 1.0 N))

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