2log (problem 3.3.6)

Percentage Accurate: 23.9% → 99.8%

Time: 8.8s

Alternatives: 8

Speedup: 29.3×

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: 23.9% 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 25.5%

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

   1. diff-log 28.1%

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

4. Applied egg-rr 28.1%

   \[\leadsto \color{blue}{\log \left(\frac{N + 1}{N}\right)} \]
5. 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 27.9%

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

   4. lft-mult-inverse 28.2%

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

   5. *-rgt-identity 28.2%

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

   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: 97.1% accurate, 10.8× speedup

Math

\[\begin{array}{l} \\ \frac{1 - \frac{\frac{\frac{0.25 + \frac{-0.2}{N}}{N} - 0.3333333333333333}{N} - -0.5}{N}}{N} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (/
  (- 1.0 (/ (- (/ (- (/ (+ 0.25 (/ -0.2 N)) N) 0.3333333333333333) N) -0.5) N))
  N))

C

double code(double N) {
	return (1.0 - ((((((0.25 + (-0.2 / N)) / N) - 0.3333333333333333) / N) - -0.5) / N)) / N;
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    code = (1.0d0 - ((((((0.25d0 + ((-0.2d0) / n)) / n) - 0.3333333333333333d0) / n) - (-0.5d0)) / n)) / n
end function

Java

public static double code(double N) {
	return (1.0 - ((((((0.25 + (-0.2 / N)) / N) - 0.3333333333333333) / N) - -0.5) / N)) / N;
}

Python

def code(N):
	return (1.0 - ((((((0.25 + (-0.2 / N)) / N) - 0.3333333333333333) / N) - -0.5) / N)) / N

Julia

function code(N)
	return Float64(Float64(1.0 - Float64(Float64(Float64(Float64(Float64(Float64(0.25 + Float64(-0.2 / N)) / N) - 0.3333333333333333) / N) - -0.5) / N)) / N)
end

MATLAB

function tmp = code(N)
	tmp = (1.0 - ((((((0.25 + (-0.2 / N)) / N) - 0.3333333333333333) / N) - -0.5) / N)) / N;
end

Wolfram

code[N_] := N[(N[(1.0 - N[(N[(N[(N[(N[(N[(0.25 + N[(-0.2 / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] / N), $MachinePrecision] - -0.5), $MachinePrecision] / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision]

Derivation

1. Initial program 25.5%

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

3. Taylor expanded in N around -inf 97.0%

   \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{0.2 \cdot \frac{1}{N} - 0.25}{N} - 0.3333333333333333}{N} - 0.5}{N} - 1}{N}} \]
4. Step-by-step derivation

   1. mul-1-neg 97.0%

      \[\leadsto \color{blue}{-\frac{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{0.2 \cdot \frac{1}{N} - 0.25}{N} - 0.3333333333333333}{N} - 0.5}{N} - 1}{N}} \]

   2. distribute-neg-frac2 97.0%

      \[\leadsto \color{blue}{\frac{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{0.2 \cdot \frac{1}{N} - 0.25}{N} - 0.3333333333333333}{N} - 0.5}{N} - 1}{-N}} \]

5. Simplified 97.0%

   \[\leadsto \color{blue}{\frac{-1 - \frac{-0.5 - \frac{-0.3333333333333333 - \frac{-0.25 + \frac{0.2}{N}}{N}}{N}}{N}}{-N}} \]

6. Taylor expanded in N around -inf 97.0%

   \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{0.2 \cdot \frac{1}{N} - 0.25}{N} - 0.3333333333333333}{N} - 0.5}{N} - 1}{N}} \]

8. Simplified 97.0%

   \[\leadsto \color{blue}{\frac{1 + \frac{-0.5 + \frac{0.3333333333333333 - \frac{0.25 + \frac{-0.2}{N}}{N}}{N}}{N}}{N}} \]

9. Final simplification 97.0%

   \[\leadsto \frac{1 - \frac{\frac{\frac{0.25 + \frac{-0.2}{N}}{N} - 0.3333333333333333}{N} - -0.5}{N}}{N} \]
10. Add Preprocessing

Alternative 3: 96.3% accurate, 13.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 25.5%

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

3. Taylor expanded in N around -inf 96.2%

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

   1. mul-1-neg 96.2%

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

   2. distribute-neg-frac2 96.2%

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

5. Simplified 96.2%

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

6. Final simplification 96.2%

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

Alternative 4: 95.1% accurate, 15.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 25.5%

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

3. Taylor expanded in N around inf 94.7%

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

4. Taylor expanded in N around -inf 94.8%

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

   1. mul-1-neg 94.8%

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

   2. unsub-neg 94.8%

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

   3. sub-neg 94.8%

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

   4. associate-*r/ 94.8%

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

   5. metadata-eval 94.8%

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

   6. distribute-neg-frac 94.8%

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

   7. metadata-eval 94.8%

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

6. Simplified 94.8%

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

   1. clear-num 94.8%

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

   2. inv-pow 94.8%

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

8. Applied egg-rr 94.8%

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

   1. unpow-1 94.8%

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

10. Simplified 94.8%

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

11. Final simplification 94.8%

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

Alternative 5: 95.1% accurate, 18.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 25.5%

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

3. Taylor expanded in N around inf 94.7%

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

   1. associate--l+ 94.8%

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

   2. unpow2 94.8%

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

   3. associate-/r* 94.8%

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

   4. metadata-eval 94.8%

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

   5. associate-*r/ 94.8%

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

   6. associate-*r/ 94.8%

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

   7. metadata-eval 94.8%

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

   8. div-sub 94.8%

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

   9. sub-neg 94.8%

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

   10. metadata-eval 94.8%

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

   11. +-commutative 94.8%

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

   12. associate-*r/ 94.8%

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

   13. metadata-eval 94.8%

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

5. Simplified 94.8%

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

6. Final simplification 94.8%

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

Alternative 6: 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 25.5%

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

3. Taylor expanded in N around inf 91.8%

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

   1. associate-*r/ 91.8%

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

   2. metadata-eval 91.8%

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

5. Simplified 91.8%

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

   1. clear-num 91.9%

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

   2. inv-pow 91.9%

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

7. Applied egg-rr 91.9%

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

   1. unpow-1 91.9%

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

   2. sub-neg 91.9%

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

   3. distribute-neg-frac 91.9%

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

   4. metadata-eval 91.9%

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

9. Simplified 91.9%

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

10. Taylor expanded in N around inf 92.5%

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

    1. associate-*r/ 92.5%

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

    2. metadata-eval 92.5%

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

12. Simplified 92.5%

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

13. Final simplification 92.5%

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

Alternative 7: 92.5% 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 25.5%

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

3. Taylor expanded in N around inf 91.8%

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

   1. associate-*r/ 91.8%

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

   2. metadata-eval 91.8%

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

5. Simplified 91.8%

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

Alternative 8: 83.9% accurate, 29.3× 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 25.5%

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

3. Taylor expanded in N around inf 91.8%

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

   1. associate-*r/ 91.8%

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

   2. metadata-eval 91.8%

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

5. Simplified 91.8%

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

   1. clear-num 91.9%

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

   2. inv-pow 91.9%

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

7. Applied egg-rr 91.9%

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

   1. unpow-1 91.9%

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

   2. sub-neg 91.9%

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

   3. distribute-neg-frac 91.9%

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

   4. metadata-eval 91.9%

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

9. Simplified 91.9%

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

    1. associate-/r/ 91.8%

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

    2. frac-2neg 91.8%

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

    3. metadata-eval 91.8%

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

    4. add-sqr-sqrt 0.0%

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

    5. sqrt-unprod 82.5%

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

    6. sqr-neg 82.5%

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

    7. sqrt-unprod 82.5%

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

    8. add-sqr-sqrt 82.5%

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

11. Applied egg-rr 82.5%

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

    1. associate-*l/ 82.5%

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

    2. *-lft-identity 82.5%

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

13. Simplified 82.5%

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

14. Final simplification 82.5%

    \[\leadsto \frac{\frac{0.5}{N} + 1}{N} \]
15. 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]

Developer Target 2: 26.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Developer Target 3: 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 2024179 
(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)))

  :alt
  (! :herbie-platform default (log (+ 1 (/ 1 N))))

  :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)))