2log (problem 3.3.6)

Percentage Accurate: 23.9% → 99.8%

Time: 8.8s

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: 23.9% 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 24.7%

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

   1. diff-log 27.7%

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

4. Applied egg-rr 27.7%

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

   1. *-lft-identity 27.7%

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

   2. associate-*l/ 27.5%

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

   3. distribute-lft-in 27.6%

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

   4. *-rgt-identity 27.6%

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

   5. lft-mult-inverse 27.7%

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

   6. log1p-define 99.9%

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

6. Simplified 99.9%

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

7. Final simplification 99.9%

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

Alternative 2: 96.2% accurate, 12.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 24.7%

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

3. Taylor expanded in N around inf 95.8%

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

4. Taylor expanded in N around -inf 95.8%

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

   1. mul-1-neg 95.8%

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

   2. unsub-neg 95.8%

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

   3. mul-1-neg 95.8%

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

   4. unsub-neg 95.8%

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

   5. associate-*r/ 95.8%

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

   6. metadata-eval 95.8%

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

6. Simplified 95.8%

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

   1. div-sub 95.8%

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

8. Applied egg-rr 95.8%

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

9. Final simplification 95.8%

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

Alternative 3: 96.3% accurate, 12.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 24.7%

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

3. Taylor expanded in N around inf 95.8%

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

4. Taylor expanded in N around -inf 95.8%

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

   1. mul-1-neg 95.8%

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

   2. unsub-neg 95.8%

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

   3. mul-1-neg 95.8%

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

   4. unsub-neg 95.8%

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

   5. associate-*r/ 95.8%

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

   6. metadata-eval 95.8%

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

6. Simplified 95.8%

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

   1. clear-num 95.8%

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

   2. inv-pow 95.8%

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

   3. sub-neg 95.8%

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

   4. distribute-neg-frac 95.8%

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

   5. sub-neg 95.8%

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

   6. distribute-neg-in 95.8%

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

   7. metadata-eval 95.8%

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

   8. distribute-neg-frac2 95.8%

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

   9. distribute-frac-neg 95.8%

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

   10. frac-2neg 95.8%

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

8. Applied egg-rr 95.8%

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

   1. unpow-1 95.8%

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

10. Simplified 95.8%

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

11. Final simplification 95.8%

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

Alternative 4: 96.2% accurate, 13.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 24.7%

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

3. Taylor expanded in N around inf 95.8%

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

4. Taylor expanded in N around -inf 95.8%

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

   1. mul-1-neg 95.8%

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

   2. unsub-neg 95.8%

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

   3. mul-1-neg 95.8%

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

   4. unsub-neg 95.8%

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

   5. associate-*r/ 95.8%

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

   6. metadata-eval 95.8%

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

6. Simplified 95.8%

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

7. Final simplification 95.8%

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

Alternative 5: 95.0% accurate, 15.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 24.7%

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

3. Taylor expanded in N around inf 94.4%

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

   1. associate--l+ 94.4%

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

   2. unpow2 94.4%

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

   3. associate-/r* 94.4%

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

   4. metadata-eval 94.4%

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

   5. associate-*r/ 94.4%

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

   6. associate-*r/ 94.4%

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

   7. metadata-eval 94.4%

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

   8. div-sub 94.4%

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

   9. sub-neg 94.4%

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

   10. metadata-eval 94.4%

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

   11. +-commutative 94.4%

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

   12. associate-*r/ 94.4%

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

   13. metadata-eval 94.4%

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

5. Simplified 94.4%

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

   1. clear-num 94.4%

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

   2. inv-pow 94.4%

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

   3. +-commutative 94.4%

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

7. Applied egg-rr 94.4%

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

   1. unpow-1 94.4%

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

   2. +-commutative 94.4%

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

   3. metadata-eval 94.4%

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

   4. remove-double-neg 94.4%

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

   5. distribute-neg-in 94.4%

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

   6. distribute-neg-frac 94.4%

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

   7. unsub-neg 94.4%

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

   8. distribute-neg-frac 94.4%

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

   9. metadata-eval 94.4%

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

9. Simplified 94.4%

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

10. Final simplification 94.4%

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

Alternative 6: 95.0% 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 24.7%

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

3. Taylor expanded in N around inf 94.4%

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

   1. associate--l+ 94.4%

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

   2. unpow2 94.4%

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

   3. associate-/r* 94.4%

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

   4. metadata-eval 94.4%

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

   5. associate-*r/ 94.4%

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

   6. associate-*r/ 94.4%

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

   7. metadata-eval 94.4%

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

   8. div-sub 94.4%

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

   9. sub-neg 94.4%

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

   10. metadata-eval 94.4%

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

   11. +-commutative 94.4%

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

   12. associate-*r/ 94.4%

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

   13. metadata-eval 94.4%

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

5. Simplified 94.4%

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

6. Final simplification 94.4%

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

Alternative 7: 92.4% accurate, 22.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 24.7%

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

3. Taylor expanded in N around inf 91.4%

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

   1. associate-*r/ 91.4%

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

   2. metadata-eval 91.4%

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

5. Simplified 91.4%

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

   1. div-sub 91.5%

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

7. Applied egg-rr 91.5%

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

8. Final simplification 91.5%

   \[\leadsto \frac{1}{N} - \frac{\frac{0.5}{N}}{N} \]
9. Add Preprocessing

Alternative 8: 92.4% accurate, 22.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 24.7%

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

3. Taylor expanded in N around inf 91.4%

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

   1. associate-*r/ 91.4%

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

   2. metadata-eval 91.4%

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

5. Simplified 91.4%

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

6. Taylor expanded in N around 0 91.4%

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

   1. clear-num 91.5%

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

   2. inv-pow 91.5%

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

   3. div-sub 91.5%

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

   4. *-inverses 91.5%

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

8. Applied egg-rr 91.5%

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

   1. unpow-1 91.5%

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

10. Simplified 91.5%

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

11. Final simplification 91.5%

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

Alternative 9: 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 24.7%

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

3. Taylor expanded in N around inf 91.4%

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

   1. associate-*r/ 91.4%

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

   2. metadata-eval 91.4%

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

5. Simplified 91.4%

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

6. Final simplification 91.4%

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

Alternative 10: 84.5% accurate, 68.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(N):
	return 1.0 / N

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 24.7%

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

3. Taylor expanded in N around inf 83.7%

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

4. Final simplification 83.7%

   \[\leadsto \frac{1}{N} \]
5. 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 2024072 
(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)))