2log (problem 3.3.6)

Percentage Accurate: 23.5% → 99.8%

Time: 10.4s

Alternatives: 9

Speedup: 68.3×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 23.5% 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 25.3%

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

   1. +-commutative 25.3%

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

   2. log1p-define 25.3%

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

3. Simplified 25.3%

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

   1. add-cbrt-cube 25.3%

      \[\leadsto \color{blue}{\sqrt[3]{\left(\left(\mathsf{log1p}\left(N\right) - \log N\right) \cdot \left(\mathsf{log1p}\left(N\right) - \log N\right)\right) \cdot \left(\mathsf{log1p}\left(N\right) - \log N\right)}} \]

   2. pow1/3 25.3%

      \[\leadsto \color{blue}{{\left(\left(\left(\mathsf{log1p}\left(N\right) - \log N\right) \cdot \left(\mathsf{log1p}\left(N\right) - \log N\right)\right) \cdot \left(\mathsf{log1p}\left(N\right) - \log N\right)\right)}^{0.3333333333333333}} \]

   3. pow3 25.3%

      \[\leadsto {\color{blue}{\left({\left(\mathsf{log1p}\left(N\right) - \log N\right)}^{3}\right)}}^{0.3333333333333333} \]

6. Applied egg-rr 25.3%

   \[\leadsto \color{blue}{{\left({\left(\mathsf{log1p}\left(N\right) - \log N\right)}^{3}\right)}^{0.3333333333333333}} \]
7. Step-by-step derivation

   1. unpow1/3 25.3%

      \[\leadsto \color{blue}{\sqrt[3]{{\left(\mathsf{log1p}\left(N\right) - \log N\right)}^{3}}} \]

   2. rem-cbrt-cube 25.3%

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

   3. log1p-undefine 25.3%

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

   4. +-commutative 25.3%

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

   5. diff-log 28.1%

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

8. Applied egg-rr 28.1%

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

   1. *-lft-identity 28.1%

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

   2. associate-*l/ 27.9%

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

   3. distribute-lft-in 27.8%

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

   4. lft-mult-inverse 28.1%

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

   5. *-rgt-identity 28.1%

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

10. Simplified 28.1%

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

    1. *-un-lft-identity 28.1%

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

    2. log-prod 28.1%

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

    3. metadata-eval 28.1%

       \[\leadsto \color{blue}{0} + \log \left(1 + \frac{1}{N}\right) \]

    4. log1p-define 99.8%

       \[\leadsto 0 + \color{blue}{\mathsf{log1p}\left(\frac{1}{N}\right)} \]

12. Applied egg-rr 99.8%

    \[\leadsto \color{blue}{0 + \mathsf{log1p}\left(\frac{1}{N}\right)} \]
13. Step-by-step derivation

    1. +-lft-identity 99.8%

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

14. Simplified 99.8%

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

Alternative 2: 96.7% accurate, 12.1× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    code = 1.0d0 / (n * (1.0d0 + (((((0.041666666666666664d0 / n) + (-0.08333333333333333d0)) / n) + 0.5d0) / n)))
end function

Java

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

Python

def code(N):
	return 1.0 / (N * (1.0 + (((((0.041666666666666664 / N) + -0.08333333333333333) / N) + 0.5) / N)))

Julia

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

MATLAB

function tmp = code(N)
	tmp = 1.0 / (N * (1.0 + (((((0.041666666666666664 / N) + -0.08333333333333333) / N) + 0.5) / N)));
end

Wolfram

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

Derivation

1. Initial program 25.3%

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

   1. +-commutative 25.3%

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

   2. log1p-define 25.3%

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

3. Simplified 25.3%

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

5. Taylor expanded in N around -inf 96.0%

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

   1. associate-*r/ 96.0%

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

7. Simplified 96.0%

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

   1. clear-num 96.1%

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

   2. inv-pow 96.1%

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

   3. +-commutative 96.1%

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

9. Applied egg-rr 96.1%

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

    1. unpow-1 96.1%

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

    2. +-commutative 96.1%

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

11. Simplified 96.1%

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

12. Taylor expanded in N around -inf 96.4%

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

13. Taylor expanded in N around -inf 96.4%

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

15. Simplified 96.4%

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

16. Final simplification 96.4%

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

Alternative 3: 96.4% accurate, 12.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 25.3%

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

   1. +-commutative 25.3%

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

   2. log1p-define 25.3%

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

3. Simplified 25.3%

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

5. Taylor expanded in N around -inf 96.0%

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

   1. associate-*r/ 96.0%

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

7. Simplified 96.0%

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

   1. clear-num 96.1%

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

   2. inv-pow 96.1%

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

   3. +-commutative 96.1%

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

9. Applied egg-rr 96.1%

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

    1. unpow-1 96.1%

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

    2. +-commutative 96.1%

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

11. Simplified 96.1%

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

12. Final simplification 96.1%

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

Alternative 4: 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 25.3%

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

   1. +-commutative 25.3%

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

   2. log1p-define 25.3%

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

3. Simplified 25.3%

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

5. Taylor expanded in N around -inf 96.0%

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

   1. associate-*r/ 96.0%

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

7. Simplified 96.0%

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

Alternative 5: 95.5% 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 25.3%

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

   1. +-commutative 25.3%

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

   2. log1p-define 25.3%

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

3. Simplified 25.3%

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

5. Taylor expanded in N around -inf 96.0%

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

   1. associate-*r/ 96.0%

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

7. Simplified 96.0%

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

   1. clear-num 96.1%

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

   2. inv-pow 96.1%

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

   3. +-commutative 96.1%

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

9. Applied egg-rr 96.1%

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

    1. unpow-1 96.1%

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

    2. +-commutative 96.1%

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

11. Simplified 96.1%

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

12. Taylor expanded in N around inf 95.1%

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

    1. associate--l+ 95.1%

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

    2. associate-*r/ 95.1%

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

    3. metadata-eval 95.1%

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

    4. unpow2 95.1%

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

    5. associate-/r* 95.1%

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

    6. metadata-eval 95.1%

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

    7. associate-*r/ 95.1%

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

    8. div-sub 95.1%

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

    9. sub-neg 95.1%

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

    10. associate-*r/ 95.1%

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

    11. metadata-eval 95.1%

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

    12. distribute-neg-frac 95.1%

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

    13. metadata-eval 95.1%

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

14. Simplified 95.1%

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

15. Final simplification 95.1%

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

Alternative 6: 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 25.3%

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

   1. +-commutative 25.3%

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

   2. log1p-define 25.3%

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

3. Simplified 25.3%

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

5. Taylor expanded in N around inf 94.6%

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

   1. sub-neg 94.6%

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

   2. associate-+l+ 94.6%

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

   3. sub-neg 94.6%

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

   4. unpow2 94.6%

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

   5. associate-/r* 94.6%

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

   6. metadata-eval 94.6%

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

   7. associate-*r/ 94.6%

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

   8. associate-*r/ 94.6%

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

   9. metadata-eval 94.6%

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

   10. div-sub 94.6%

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

   11. sub-neg 94.6%

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

   12. metadata-eval 94.6%

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

   13. +-commutative 94.6%

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

   14. associate-*r/ 94.6%

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

   15. metadata-eval 94.6%

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

7. Simplified 94.6%

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

Alternative 7: 93.0% 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 25.3%

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

   1. +-commutative 25.3%

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

   2. log1p-define 25.3%

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

3. Simplified 25.3%

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

5. Taylor expanded in N around -inf 96.0%

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

   1. associate-*r/ 96.0%

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

7. Simplified 96.0%

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

   1. clear-num 96.1%

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

   2. inv-pow 96.1%

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

   3. +-commutative 96.1%

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

9. Applied egg-rr 96.1%

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

    1. unpow-1 96.1%

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

    2. +-commutative 96.1%

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

11. Simplified 96.1%

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

12. Taylor expanded in N around inf 92.2%

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

    1. associate-*r/ 92.2%

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

    2. metadata-eval 92.2%

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

14. Simplified 92.2%

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

Alternative 8: 92.4% 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 25.3%

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

   1. +-commutative 25.3%

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

   2. log1p-define 25.3%

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

3. Simplified 25.3%

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

5. Taylor expanded in N around inf 91.6%

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

   1. associate-*r/ 91.6%

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

   2. metadata-eval 91.6%

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

7. Simplified 91.6%

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

Alternative 9: 84.7% 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 25.3%

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

   1. +-commutative 25.3%

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

   2. log1p-define 25.3%

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

3. Simplified 25.3%

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

5. Taylor expanded in N around inf 83.4%

   \[\leadsto \color{blue}{\frac{1}{N}} \]
6. 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 2024088 
(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)))