2log (problem 3.3.6)

Percentage Accurate: 24.2% → 99.8%

Time: 5.9s

Alternatives: 7

Speedup: 10.4×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 24.2% 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, 1.0× speedup

Math

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

FPCore

(FPCore (N) :precision binary64 (log1p (pow N -1.0)))

C

double code(double N) {
	return log1p(pow(N, -1.0));
}

Java

public static double code(double N) {
	return Math.log1p(Math.pow(N, -1.0));
}

Python

def code(N):
	return math.log1p(math.pow(N, -1.0))

Julia

function code(N)
	return log1p((N ^ -1.0))
end

Wolfram

code[N_] := N[Log[1 + N[Power[N, -1.0], $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 23.0%

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

   1. lift--.f64 N/A

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

   2. lift-log.f64 N/A

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

   3. lift-log.f64 N/A

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

   4. diff-log N/A

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

   5. lift-+.f64 N/A

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

   6. div-add N/A

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

   7. *-inverses N/A

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

   8. lower-log1p.f64 N/A

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

   9. inv-pow N/A

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

   10. lower-pow.f64 99.8

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

4. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\mathsf{log1p}\left({N}^{-1}\right)} \]
5. Step-by-step derivation

   1. lift-pow.f64 N/A

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

   2. inv-pow N/A

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

   3. lift-/.f64 99.9

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

6. Applied rewrites 99.9%

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

7. Final simplification 99.8%

   \[\leadsto \mathsf{log1p}\left({N}^{-1}\right) \]
8. Add Preprocessing

Alternative 2: 84.2% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (N) :precision binary64 (pow N -1.0))

C

double code(double N) {
	return pow(N, -1.0);
}

Fortran

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

Java

public static double code(double N) {
	return Math.pow(N, -1.0);
}

Python

def code(N):
	return math.pow(N, -1.0)

Julia

function code(N)
	return N ^ -1.0
end

MATLAB

function tmp = code(N)
	tmp = N ^ -1.0;
end

Wolfram

code[N_] := N[Power[N, -1.0], $MachinePrecision]

Derivation

1. Initial program 23.0%

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

3. Taylor expanded in N around inf

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

   1. lower-/.f64 85.1

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

5. Applied rewrites 85.1%

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

6. Final simplification 85.1%

   \[\leadsto {N}^{-1} \]
7. Add Preprocessing

Alternative 3: 96.3% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 23.0%

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

3. Taylor expanded in N around inf

   \[\leadsto \color{blue}{\frac{\left(1 + \frac{\frac{1}{3}}{{N}^{2}}\right) - \left(\frac{1}{2} \cdot \frac{1}{N} + \frac{1}{4} \cdot \frac{1}{{N}^{3}}\right)}{N}} \]

5. Applied rewrites 95.8%

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

Alternative 4: 95.0% accurate, 5.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 23.0%

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

3. Taylor expanded in N around inf

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

5. Applied rewrites 94.7%

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

Alternative 5: 92.7% accurate, 6.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 23.0%

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

3. Taylor expanded in N around -inf

   \[\leadsto \color{blue}{-1 \cdot \frac{\frac{1}{2} \cdot \frac{1}{N} - 1}{N}} \]
4. Step-by-step derivation

   1. metadata-eval N/A

      \[\leadsto \color{blue}{\frac{1}{-1}} \cdot \frac{\frac{1}{2} \cdot \frac{1}{N} - 1}{N} \]

   2. times-frac N/A

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

   3. *-lft-identity N/A

      \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot \frac{1}{N} - 1}}{-1 \cdot N} \]

   4. associate-/r* N/A

      \[\leadsto \color{blue}{\frac{\frac{\frac{1}{2} \cdot \frac{1}{N} - 1}{-1}}{N}} \]

   5. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{\frac{1}{2} \cdot \frac{1}{N} - 1}{-1}}{N}} \]

   6. div-sub N/A

      \[\leadsto \frac{\color{blue}{\frac{\frac{1}{2} \cdot \frac{1}{N}}{-1} - \frac{1}{-1}}}{N} \]

   7. metadata-eval N/A

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

   8. distribute-neg-frac2 N/A

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

   9. distribute-neg-frac N/A

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

   10. distribute-lft-neg-in N/A

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

   11. /-rgt-identity N/A

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

   12. metadata-eval N/A

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

   13. lower--.f64 N/A

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

   14. metadata-eval N/A

       \[\leadsto \frac{\color{blue}{\frac{-1}{2}} \cdot \frac{1}{N} - -1}{N} \]

   15. associate-*r/ N/A

       \[\leadsto \frac{\color{blue}{\frac{\frac{-1}{2} \cdot 1}{N}} - -1}{N} \]

   16. metadata-eval N/A

       \[\leadsto \frac{\frac{\color{blue}{\frac{-1}{2}}}{N} - -1}{N} \]

   17. lower-/.f64 92.3

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

5. Applied rewrites 92.3%

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

6. Taylor expanded in N around 0

   \[\leadsto \frac{N - \frac{1}{2}}{\color{blue}{{N}^{2}}} \]
7. Step-by-step derivation

8. Applied rewrites 92.1%

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

10. Applied rewrites 92.0%

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

11. Taylor expanded in N around inf

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

13. Applied rewrites 92.6%

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

Alternative 6: 92.3% accurate, 8.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 23.0%

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

3. Taylor expanded in N around -inf

   \[\leadsto \color{blue}{-1 \cdot \frac{\frac{1}{2} \cdot \frac{1}{N} - 1}{N}} \]
4. Step-by-step derivation

   1. metadata-eval N/A

      \[\leadsto \color{blue}{\frac{1}{-1}} \cdot \frac{\frac{1}{2} \cdot \frac{1}{N} - 1}{N} \]

   2. times-frac N/A

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

   3. *-lft-identity N/A

      \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot \frac{1}{N} - 1}}{-1 \cdot N} \]

   4. associate-/r* N/A

      \[\leadsto \color{blue}{\frac{\frac{\frac{1}{2} \cdot \frac{1}{N} - 1}{-1}}{N}} \]

   5. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{\frac{1}{2} \cdot \frac{1}{N} - 1}{-1}}{N}} \]

   6. div-sub N/A

      \[\leadsto \frac{\color{blue}{\frac{\frac{1}{2} \cdot \frac{1}{N}}{-1} - \frac{1}{-1}}}{N} \]

   7. metadata-eval N/A

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

   8. distribute-neg-frac2 N/A

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

   9. distribute-neg-frac N/A

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

   10. distribute-lft-neg-in N/A

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

   11. /-rgt-identity N/A

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

   12. metadata-eval N/A

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

   13. lower--.f64 N/A

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

   14. metadata-eval N/A

       \[\leadsto \frac{\color{blue}{\frac{-1}{2}} \cdot \frac{1}{N} - -1}{N} \]

   15. associate-*r/ N/A

       \[\leadsto \frac{\color{blue}{\frac{\frac{-1}{2} \cdot 1}{N}} - -1}{N} \]

   16. metadata-eval N/A

       \[\leadsto \frac{\frac{\color{blue}{\frac{-1}{2}}}{N} - -1}{N} \]

   17. lower-/.f64 92.3

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

5. Applied rewrites 92.3%

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

Alternative 7: 92.1% accurate, 10.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 23.0%

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

3. Taylor expanded in N around -inf

   \[\leadsto \color{blue}{-1 \cdot \frac{\frac{1}{2} \cdot \frac{1}{N} - 1}{N}} \]
4. Step-by-step derivation

   1. metadata-eval N/A

      \[\leadsto \color{blue}{\frac{1}{-1}} \cdot \frac{\frac{1}{2} \cdot \frac{1}{N} - 1}{N} \]

   2. times-frac N/A

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

   3. *-lft-identity N/A

      \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot \frac{1}{N} - 1}}{-1 \cdot N} \]

   4. associate-/r* N/A

      \[\leadsto \color{blue}{\frac{\frac{\frac{1}{2} \cdot \frac{1}{N} - 1}{-1}}{N}} \]

   5. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{\frac{1}{2} \cdot \frac{1}{N} - 1}{-1}}{N}} \]

   6. div-sub N/A

      \[\leadsto \frac{\color{blue}{\frac{\frac{1}{2} \cdot \frac{1}{N}}{-1} - \frac{1}{-1}}}{N} \]

   7. metadata-eval N/A

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

   8. distribute-neg-frac2 N/A

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

   9. distribute-neg-frac N/A

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

   10. distribute-lft-neg-in N/A

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

   11. /-rgt-identity N/A

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

   12. metadata-eval N/A

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

   13. lower--.f64 N/A

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

   14. metadata-eval N/A

       \[\leadsto \frac{\color{blue}{\frac{-1}{2}} \cdot \frac{1}{N} - -1}{N} \]

   15. associate-*r/ N/A

       \[\leadsto \frac{\color{blue}{\frac{\frac{-1}{2} \cdot 1}{N}} - -1}{N} \]

   16. metadata-eval N/A

       \[\leadsto \frac{\frac{\color{blue}{\frac{-1}{2}}}{N} - -1}{N} \]

   17. lower-/.f64 92.3

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

5. Applied rewrites 92.3%

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

6. Taylor expanded in N around 0

   \[\leadsto \frac{N - \frac{1}{2}}{\color{blue}{{N}^{2}}} \]
7. Step-by-step derivation

8. Applied rewrites 92.1%

   \[\leadsto \frac{-0.5 + N}{\color{blue}{N \cdot N}} \]
9. Add Preprocessing

Developer Target 1: 96.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \left(\left(\frac{1}{N} + \frac{-1}{2 \cdot {N}^{2}}\right) + \frac{1}{3 \cdot {N}^{3}}\right) + \frac{-1}{4 \cdot {N}^{4}} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (+
  (+ (+ (/ 1.0 N) (/ -1.0 (* 2.0 (pow N 2.0)))) (/ 1.0 (* 3.0 (pow N 3.0))))
  (/ -1.0 (* 4.0 (pow N 4.0)))))

C

double code(double N) {
	return (((1.0 / N) + (-1.0 / (2.0 * pow(N, 2.0)))) + (1.0 / (3.0 * pow(N, 3.0)))) + (-1.0 / (4.0 * pow(N, 4.0)));
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    code = (((1.0d0 / n) + ((-1.0d0) / (2.0d0 * (n ** 2.0d0)))) + (1.0d0 / (3.0d0 * (n ** 3.0d0)))) + ((-1.0d0) / (4.0d0 * (n ** 4.0d0)))
end function

Java

public static double code(double N) {
	return (((1.0 / N) + (-1.0 / (2.0 * Math.pow(N, 2.0)))) + (1.0 / (3.0 * Math.pow(N, 3.0)))) + (-1.0 / (4.0 * Math.pow(N, 4.0)));
}

Python

def code(N):
	return (((1.0 / N) + (-1.0 / (2.0 * math.pow(N, 2.0)))) + (1.0 / (3.0 * math.pow(N, 3.0)))) + (-1.0 / (4.0 * math.pow(N, 4.0)))

Julia

function code(N)
	return Float64(Float64(Float64(Float64(1.0 / N) + Float64(-1.0 / Float64(2.0 * (N ^ 2.0)))) + Float64(1.0 / Float64(3.0 * (N ^ 3.0)))) + Float64(-1.0 / Float64(4.0 * (N ^ 4.0))))
end

MATLAB

function tmp = code(N)
	tmp = (((1.0 / N) + (-1.0 / (2.0 * (N ^ 2.0)))) + (1.0 / (3.0 * (N ^ 3.0)))) + (-1.0 / (4.0 * (N ^ 4.0)));
end

Wolfram

code[N_] := N[(N[(N[(N[(1.0 / N), $MachinePrecision] + N[(-1.0 / N[(2.0 * N[Power[N, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(3.0 * N[Power[N, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[(4.0 * N[Power[N, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024339 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  :pre (and (> N 1.0) (< N 1e+40))

  :alt
  (! :herbie-platform default (+ (/ 1 N) (/ -1 (* 2 (pow N 2))) (/ 1 (* 3 (pow N 3))) (/ -1 (* 4 (pow N 4)))))

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