2log (problem 3.3.6)

Percentage Accurate: 23.7% → 99.8%

Time: 7.1s

Alternatives: 8

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: 23.7% 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 28.3%

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

   1. diff-log 31.5%

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

4. Applied egg-rr 31.5%

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

   1. *-lft-identity 31.5%

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

   2. associate-*l/ 31.2%

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

   3. distribute-lft-in 31.2%

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

   4. lft-mult-inverse 31.5%

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

   5. *-rgt-identity 31.5%

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

   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: 96.3% accurate, 12.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 28.3%

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

3. Taylor expanded in N around -inf 94.9%

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

   1. mul-1-neg 94.9%

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

   2. distribute-neg-frac2 94.9%

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

5. Simplified 94.9%

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

   1. clear-num 94.9%

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

   2. inv-pow 94.9%

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

7. Applied egg-rr 94.9%

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

   1. unpow-1 94.9%

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

   2. +-commutative 94.9%

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

9. Simplified 94.9%

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

10. Final simplification 94.9%

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

Alternative 3: 96.3% 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 28.3%

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

   1. diff-log 31.5%

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

4. Applied egg-rr 31.5%

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

   1. *-lft-identity 31.5%

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

   2. associate-*l/ 31.2%

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

   3. distribute-lft-in 31.2%

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

   4. lft-mult-inverse 31.5%

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

   5. *-rgt-identity 31.5%

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

   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. pow2 99.2%

      \[\leadsto \mathsf{log1p}\left(\color{blue}{{\left(\sqrt{\frac{1}{N}}\right)}^{2}}\right) \]

   3. inv-pow 99.2%

      \[\leadsto \mathsf{log1p}\left({\left(\sqrt{\color{blue}{{N}^{-1}}}\right)}^{2}\right) \]

   4. sqrt-pow1 99.3%

      \[\leadsto \mathsf{log1p}\left({\color{blue}{\left({N}^{\left(\frac{-1}{2}\right)}\right)}}^{2}\right) \]

   5. metadata-eval 99.3%

      \[\leadsto \mathsf{log1p}\left({\left({N}^{\color{blue}{-0.5}}\right)}^{2}\right) \]

8. Applied egg-rr 99.3%

   \[\leadsto \mathsf{log1p}\left(\color{blue}{{\left({N}^{-0.5}\right)}^{2}}\right) \]

9. Taylor expanded in N around inf 94.8%

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

11. Simplified 94.9%

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

Alternative 4: 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 28.3%

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

3. Taylor expanded in N around inf 93.3%

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

   1. associate--l+ 93.4%

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

   2. unpow2 93.4%

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

   3. associate-/r* 93.4%

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

   4. metadata-eval 93.4%

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

   5. associate-*r/ 93.4%

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

   6. associate-*r/ 93.4%

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

   7. metadata-eval 93.4%

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

   8. div-sub 93.4%

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

   9. sub-neg 93.4%

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

   10. metadata-eval 93.4%

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

   11. +-commutative 93.4%

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

   12. associate-*r/ 93.4%

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

   13. metadata-eval 93.4%

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

5. Simplified 93.4%

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

   1. add-cube-cbrt 91.6%

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

   2. pow3 91.6%

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

   3. *-un-lft-identity 91.6%

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

   4. *-un-lft-identity 91.6%

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

   5. +-commutative 91.6%

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

7. Applied egg-rr 91.6%

   \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{1 + \frac{\frac{0.3333333333333333}{N} + -0.5}{N}}{N}}\right)}^{3}} \]
8. Step-by-step derivation

   1. rem-cube-cbrt 93.4%

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

   2. clear-num 93.4%

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

   3. +-commutative 93.4%

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

9. Applied egg-rr 93.4%

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

10. Taylor expanded in N around inf 93.9%

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

    1. associate-+r- 93.9%

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

    2. unpow2 93.9%

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

    3. associate-/r* 93.9%

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

    4. metadata-eval 93.9%

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

    5. associate-*r/ 93.9%

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

    6. associate-*r/ 93.9%

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

    7. metadata-eval 93.9%

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

    8. div-sub 93.9%

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

    9. sub-neg 93.9%

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

    10. distribute-lft-neg-in 93.9%

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

    11. associate-*r/ 93.9%

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

    12. metadata-eval 93.9%

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

    13. metadata-eval 93.9%

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

12. Simplified 93.9%

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

Alternative 5: 95.0% 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 28.3%

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

3. Taylor expanded in N around inf 93.3%

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

   1. associate--l+ 93.4%

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

   2. unpow2 93.4%

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

   3. associate-/r* 93.4%

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

   4. metadata-eval 93.4%

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

   5. associate-*r/ 93.4%

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

   6. associate-*r/ 93.4%

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

   7. metadata-eval 93.4%

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

   8. div-sub 93.4%

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

   9. sub-neg 93.4%

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

   10. metadata-eval 93.4%

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

   11. +-commutative 93.4%

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

   12. associate-*r/ 93.4%

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

   13. metadata-eval 93.4%

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

5. Simplified 93.4%

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

Alternative 6: 92.9% accurate, 22.8× speedup

Math

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

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 28.3%

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

3. Taylor expanded in N around inf 93.3%

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

   1. associate--l+ 93.4%

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

   2. unpow2 93.4%

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

   3. associate-/r* 93.4%

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

   4. metadata-eval 93.4%

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

   5. associate-*r/ 93.4%

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

   6. associate-*r/ 93.4%

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

   7. metadata-eval 93.4%

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

   8. div-sub 93.4%

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

   9. sub-neg 93.4%

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

   10. metadata-eval 93.4%

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

   11. +-commutative 93.4%

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

   12. associate-*r/ 93.4%

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

   13. metadata-eval 93.4%

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

5. Simplified 93.4%

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

   1. add-cube-cbrt 91.6%

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

   2. pow3 91.6%

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

   3. *-un-lft-identity 91.6%

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

   4. *-un-lft-identity 91.6%

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

   5. +-commutative 91.6%

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

7. Applied egg-rr 91.6%

   \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{1 + \frac{\frac{0.3333333333333333}{N} + -0.5}{N}}{N}}\right)}^{3}} \]
8. Step-by-step derivation

   1. rem-cube-cbrt 93.4%

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

   2. clear-num 93.4%

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

   3. +-commutative 93.4%

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

9. Applied egg-rr 93.4%

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

10. Taylor expanded in N around inf 91.2%

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

    1. associate-*r/ 91.2%

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

    2. metadata-eval 91.2%

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

12. Simplified 91.2%

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

13. Final simplification 91.2%

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

Alternative 7: 92.3% 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 28.3%

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

3. Taylor expanded in N around inf 90.6%

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

   1. associate-*r/ 90.6%

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

   2. metadata-eval 90.6%

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

5. Simplified 90.6%

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

Alternative 8: 84.5% 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 28.3%

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

3. Taylor expanded in N around inf 81.1%

   \[\leadsto \color{blue}{\frac{1}{N}} \]
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 2024121 
(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)))