2log (problem 3.3.6)

Percentage Accurate: 23.5% → 99.8%

Time: 7.4s

Alternatives: 10

Speedup: 68.3×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 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 24.6%

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

   1. diff-log 28.0%

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

4. Applied egg-rr 28.0%

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

   1. *-lft-identity 28.0%

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

   2. associate-*l/ 27.6%

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

   3. distribute-lft-in 27.6%

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

   4. lft-mult-inverse 27.9%

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

   5. *-rgt-identity 27.9%

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

   6. log1p-define 99.8%

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

6. Simplified 99.8%

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

Alternative 2: 96.7% accurate, 10.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 24.6%

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

   1. diff-log 28.0%

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

4. Applied egg-rr 28.0%

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

   1. *-lft-identity 28.0%

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

   2. associate-*l/ 27.6%

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

   3. distribute-lft-in 27.6%

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

   4. lft-mult-inverse 27.9%

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

   5. *-rgt-identity 27.9%

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

   6. log1p-define 99.8%

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

6. Simplified 99.8%

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

7. Taylor expanded in N around inf 96.6%

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

9. Simplified 96.6%

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

    1. clear-num 96.7%

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

    2. inv-pow 96.7%

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

11. Applied egg-rr 96.7%

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

13. Simplified 96.7%

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

14. Taylor expanded in N around -inf 97.0%

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

15. Final simplification 97.0%

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

Alternative 3: 96.4% accurate, 10.8× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{N}{1 - \frac{-1 + \left(1.5 + \frac{\frac{0.25}{N} + -0.3333333333333333}{N}\right)}{N}}} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (/
  1.0
  (/
   N
   (- 1.0 (/ (+ -1.0 (+ 1.5 (/ (+ (/ 0.25 N) -0.3333333333333333) N))) N)))))

C

double code(double N) {
	return 1.0 / (N / (1.0 - ((-1.0 + (1.5 + (((0.25 / N) + -0.3333333333333333) / N))) / N)));
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    code = 1.0d0 / (n / (1.0d0 - (((-1.0d0) + (1.5d0 + (((0.25d0 / n) + (-0.3333333333333333d0)) / n))) / n)))
end function

Java

public static double code(double N) {
	return 1.0 / (N / (1.0 - ((-1.0 + (1.5 + (((0.25 / N) + -0.3333333333333333) / N))) / N)));
}

Python

def code(N):
	return 1.0 / (N / (1.0 - ((-1.0 + (1.5 + (((0.25 / N) + -0.3333333333333333) / N))) / N)))

Julia

function code(N)
	return Float64(1.0 / Float64(N / Float64(1.0 - Float64(Float64(-1.0 + Float64(1.5 + Float64(Float64(Float64(0.25 / N) + -0.3333333333333333) / N))) / N))))
end

MATLAB

function tmp = code(N)
	tmp = 1.0 / (N / (1.0 - ((-1.0 + (1.5 + (((0.25 / N) + -0.3333333333333333) / N))) / N)));
end

Wolfram

code[N_] := N[(1.0 / N[(N / N[(1.0 - N[(N[(-1.0 + N[(1.5 + N[(N[(N[(0.25 / N), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] / N), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 24.6%

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

   1. diff-log 28.0%

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

4. Applied egg-rr 28.0%

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

   1. *-lft-identity 28.0%

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

   2. associate-*l/ 27.6%

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

   3. distribute-lft-in 27.6%

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

   4. lft-mult-inverse 27.9%

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

   5. *-rgt-identity 27.9%

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

   6. log1p-define 99.8%

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

6. Simplified 99.8%

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

7. Taylor expanded in N around inf 96.6%

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

9. Simplified 96.6%

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

    1. clear-num 96.7%

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

    2. inv-pow 96.7%

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

11. Applied egg-rr 96.7%

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

13. Simplified 96.7%

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

    1. expm1-log1p-u 96.7%

       \[\leadsto \frac{1}{\frac{N}{1 - \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.5 + \frac{\frac{0.25}{N} + -0.3333333333333333}{N}\right)\right)}}{N}}} \]

    2. expm1-undefine 96.7%

       \[\leadsto \frac{1}{\frac{N}{1 - \frac{\color{blue}{e^{\mathsf{log1p}\left(0.5 + \frac{\frac{0.25}{N} + -0.3333333333333333}{N}\right)} - 1}}{N}}} \]

15. Applied egg-rr 96.7%

    \[\leadsto \frac{1}{\frac{N}{1 - \frac{\color{blue}{e^{\mathsf{log1p}\left(0.5 + \frac{\frac{0.25}{N} + -0.3333333333333333}{N}\right)} - 1}}{N}}} \]
16. Step-by-step derivation

    1. sub-neg 96.7%

       \[\leadsto \frac{1}{\frac{N}{1 - \frac{\color{blue}{e^{\mathsf{log1p}\left(0.5 + \frac{\frac{0.25}{N} + -0.3333333333333333}{N}\right)} + \left(-1\right)}}{N}}} \]

    2. log1p-undefine 96.7%

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

    3. rem-exp-log 96.7%

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

    4. associate-+r+ 96.7%

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

    5. metadata-eval 96.7%

       \[\leadsto \frac{1}{\frac{N}{1 - \frac{\left(\color{blue}{1.5} + \frac{\frac{0.25}{N} + -0.3333333333333333}{N}\right) + \left(-1\right)}{N}}} \]

    6. metadata-eval 96.7%

       \[\leadsto \frac{1}{\frac{N}{1 - \frac{\left(1.5 + \frac{\frac{0.25}{N} + -0.3333333333333333}{N}\right) + \color{blue}{-1}}{N}}} \]

17. Simplified 96.7%

    \[\leadsto \frac{1}{\frac{N}{1 - \frac{\color{blue}{\left(1.5 + \frac{\frac{0.25}{N} + -0.3333333333333333}{N}\right) + -1}}{N}}} \]

18. Final simplification 96.7%

    \[\leadsto \frac{1}{\frac{N}{1 - \frac{-1 + \left(1.5 + \frac{\frac{0.25}{N} + -0.3333333333333333}{N}\right)}{N}}} \]
19. Add Preprocessing

Alternative 4: 96.4% accurate, 12.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 24.6%

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

   1. diff-log 28.0%

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

4. Applied egg-rr 28.0%

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

   1. *-lft-identity 28.0%

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

   2. associate-*l/ 27.6%

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

   3. distribute-lft-in 27.6%

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

   4. lft-mult-inverse 27.9%

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

   5. *-rgt-identity 27.9%

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

   6. log1p-define 99.8%

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

6. Simplified 99.8%

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

7. Taylor expanded in N around inf 96.6%

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

9. Simplified 96.6%

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

    1. clear-num 96.7%

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

    2. inv-pow 96.7%

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

11. Applied egg-rr 96.7%

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

13. Simplified 96.7%

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

Alternative 5: 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 24.6%

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

   1. diff-log 28.0%

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

4. Applied egg-rr 28.0%

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

   1. *-lft-identity 28.0%

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

   2. associate-*l/ 27.6%

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

   3. distribute-lft-in 27.6%

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

   4. lft-mult-inverse 27.9%

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

   5. *-rgt-identity 27.9%

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

   6. log1p-define 99.8%

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

6. Simplified 99.8%

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

7. Taylor expanded in N around inf 96.6%

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

9. Simplified 96.6%

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

Alternative 6: 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 24.6%

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

   1. diff-log 28.0%

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

4. Applied egg-rr 28.0%

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

   1. *-lft-identity 28.0%

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

   2. associate-*l/ 27.6%

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

   3. distribute-lft-in 27.6%

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

   4. lft-mult-inverse 27.9%

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

   5. *-rgt-identity 27.9%

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

   6. log1p-define 99.8%

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

6. Simplified 99.8%

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

7. Taylor expanded in N around inf 96.6%

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

9. Simplified 96.6%

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

    1. clear-num 96.7%

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

    2. inv-pow 96.7%

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

11. Applied egg-rr 96.7%

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

13. Simplified 96.7%

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

14. Taylor expanded in N around inf 95.6%

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

    1. associate--l+ 95.6%

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

    2. associate-*r/ 95.6%

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

    3. metadata-eval 95.6%

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

    4. unpow2 95.6%

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

    5. associate-/r* 95.6%

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

    6. metadata-eval 95.6%

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

    7. associate-*r/ 95.6%

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

    8. div-sub 95.6%

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

    9. associate-*r/ 95.6%

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

    10. metadata-eval 95.6%

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

16. Simplified 95.6%

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

Alternative 7: 95.1% 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 24.6%

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

3. Taylor expanded in N around inf 95.1%

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

   1. associate--l+ 95.1%

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

   2. unpow2 95.1%

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

   3. associate-/r* 95.1%

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

   4. metadata-eval 95.1%

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

   5. associate-*r/ 95.1%

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

   6. associate-*r/ 95.1%

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

   7. metadata-eval 95.1%

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

   8. div-sub 95.1%

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

   9. sub-neg 95.1%

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

   10. metadata-eval 95.1%

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

   11. +-commutative 95.1%

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

   12. associate-*r/ 95.1%

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

   13. metadata-eval 95.1%

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

5. Simplified 95.1%

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

Alternative 8: 93.1% 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 24.6%

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

   1. diff-log 28.0%

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

4. Applied egg-rr 28.0%

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

   1. *-lft-identity 28.0%

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

   2. associate-*l/ 27.6%

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

   3. distribute-lft-in 27.6%

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

   4. lft-mult-inverse 27.9%

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

   5. *-rgt-identity 27.9%

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

   6. log1p-define 99.8%

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

6. Simplified 99.8%

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

7. Taylor expanded in N around inf 96.6%

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

9. Simplified 96.6%

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

    1. clear-num 96.7%

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

    2. inv-pow 96.7%

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

11. Applied egg-rr 96.7%

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

13. Simplified 96.7%

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

14. Taylor expanded in N around inf 93.1%

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

    1. associate-*r/ 93.1%

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

    2. metadata-eval 93.1%

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

16. Simplified 93.1%

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

17. Final simplification 93.1%

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

Alternative 9: 92.6% 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 24.6%

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

3. Taylor expanded in N around inf 92.5%

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

   1. associate-*r/ 92.5%

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

   2. metadata-eval 92.5%

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

5. Simplified 92.5%

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

Alternative 10: 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 24.6%

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

3. Taylor expanded in N around inf 84.1%

   \[\leadsto \color{blue}{\frac{1}{N}} \]
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 2024163 
(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)))