2log (problem 3.3.6)

Percentage Accurate: 24.0% → 99.3%

Time: 10.2s

Alternatives: 9

Speedup: 68.3×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 24.0% 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: 99.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 0.001:\\ \;\;\;\;\frac{1}{N} + \left(\frac{0.3333333333333333}{{N}^{3}} - \left(\frac{0.5}{{N}^{2}} + \frac{0.25}{{N}^{4}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-\log \left(\frac{N}{N + 1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= (- (log (+ N 1.0)) (log N)) 0.001)
   (+
    (/ 1.0 N)
    (-
     (/ 0.3333333333333333 (pow N 3.0))
     (+ (/ 0.5 (pow N 2.0)) (/ 0.25 (pow N 4.0)))))
   (- (log (/ N (+ N 1.0))))))

C

double code(double N) {
	double tmp;
	if ((log((N + 1.0)) - log(N)) <= 0.001) {
		tmp = (1.0 / N) + ((0.3333333333333333 / pow(N, 3.0)) - ((0.5 / pow(N, 2.0)) + (0.25 / pow(N, 4.0))));
	} else {
		tmp = -log((N / (N + 1.0)));
	}
	return tmp;
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((log((n + 1.0d0)) - log(n)) <= 0.001d0) then
        tmp = (1.0d0 / n) + ((0.3333333333333333d0 / (n ** 3.0d0)) - ((0.5d0 / (n ** 2.0d0)) + (0.25d0 / (n ** 4.0d0))))
    else
        tmp = -log((n / (n + 1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double N) {
	double tmp;
	if ((Math.log((N + 1.0)) - Math.log(N)) <= 0.001) {
		tmp = (1.0 / N) + ((0.3333333333333333 / Math.pow(N, 3.0)) - ((0.5 / Math.pow(N, 2.0)) + (0.25 / Math.pow(N, 4.0))));
	} else {
		tmp = -Math.log((N / (N + 1.0)));
	}
	return tmp;
}

Python

def code(N):
	tmp = 0
	if (math.log((N + 1.0)) - math.log(N)) <= 0.001:
		tmp = (1.0 / N) + ((0.3333333333333333 / math.pow(N, 3.0)) - ((0.5 / math.pow(N, 2.0)) + (0.25 / math.pow(N, 4.0))))
	else:
		tmp = -math.log((N / (N + 1.0)))
	return tmp

Julia

function code(N)
	tmp = 0.0
	if (Float64(log(Float64(N + 1.0)) - log(N)) <= 0.001)
		tmp = Float64(Float64(1.0 / N) + Float64(Float64(0.3333333333333333 / (N ^ 3.0)) - Float64(Float64(0.5 / (N ^ 2.0)) + Float64(0.25 / (N ^ 4.0)))));
	else
		tmp = Float64(-log(Float64(N / Float64(N + 1.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(N)
	tmp = 0.0;
	if ((log((N + 1.0)) - log(N)) <= 0.001)
		tmp = (1.0 / N) + ((0.3333333333333333 / (N ^ 3.0)) - ((0.5 / (N ^ 2.0)) + (0.25 / (N ^ 4.0))));
	else
		tmp = -log((N / (N + 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[N_] := If[LessEqual[N[(N[Log[N[(N + 1.0), $MachinePrecision]], $MachinePrecision] - N[Log[N], $MachinePrecision]), $MachinePrecision], 0.001], N[(N[(1.0 / N), $MachinePrecision] + N[(N[(0.3333333333333333 / N[Power[N, 3.0], $MachinePrecision]), $MachinePrecision] - N[(N[(0.5 / N[Power[N, 2.0], $MachinePrecision]), $MachinePrecision] + N[(0.25 / N[Power[N, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-N[Log[N[(N / N[(N + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (log.f64 (+.f64 N 1)) (log.f64 N)) < 1e-3

   1. Initial program 17.9%

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

      1. +-commutative 17.9%

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

      2. log1p-define 17.9%

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

   3. Simplified 17.9%

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

   5. Taylor expanded in N around inf 99.8%

      \[\leadsto \color{blue}{\left(0.3333333333333333 \cdot \frac{1}{{N}^{3}} + \frac{1}{N}\right) - \left(0.25 \cdot \frac{1}{{N}^{4}} + 0.5 \cdot \frac{1}{{N}^{2}}\right)} \]
   6. Step-by-step derivation

      1. +-commutative 99.8%

         \[\leadsto \color{blue}{\left(\frac{1}{N} + 0.3333333333333333 \cdot \frac{1}{{N}^{3}}\right)} - \left(0.25 \cdot \frac{1}{{N}^{4}} + 0.5 \cdot \frac{1}{{N}^{2}}\right) \]

      2. associate--l+ 99.8%

         \[\leadsto \color{blue}{\frac{1}{N} + \left(0.3333333333333333 \cdot \frac{1}{{N}^{3}} - \left(0.25 \cdot \frac{1}{{N}^{4}} + 0.5 \cdot \frac{1}{{N}^{2}}\right)\right)} \]

      3. associate-*r/ 99.8%

         \[\leadsto \frac{1}{N} + \left(\color{blue}{\frac{0.3333333333333333 \cdot 1}{{N}^{3}}} - \left(0.25 \cdot \frac{1}{{N}^{4}} + 0.5 \cdot \frac{1}{{N}^{2}}\right)\right) \]

      4. metadata-eval 99.8%

         \[\leadsto \frac{1}{N} + \left(\frac{\color{blue}{0.3333333333333333}}{{N}^{3}} - \left(0.25 \cdot \frac{1}{{N}^{4}} + 0.5 \cdot \frac{1}{{N}^{2}}\right)\right) \]

      5. +-commutative 99.8%

         \[\leadsto \frac{1}{N} + \left(\frac{0.3333333333333333}{{N}^{3}} - \color{blue}{\left(0.5 \cdot \frac{1}{{N}^{2}} + 0.25 \cdot \frac{1}{{N}^{4}}\right)}\right) \]

      6. associate-*r/ 99.8%

         \[\leadsto \frac{1}{N} + \left(\frac{0.3333333333333333}{{N}^{3}} - \left(\color{blue}{\frac{0.5 \cdot 1}{{N}^{2}}} + 0.25 \cdot \frac{1}{{N}^{4}}\right)\right) \]

      7. metadata-eval 99.8%

         \[\leadsto \frac{1}{N} + \left(\frac{0.3333333333333333}{{N}^{3}} - \left(\frac{\color{blue}{0.5}}{{N}^{2}} + 0.25 \cdot \frac{1}{{N}^{4}}\right)\right) \]

      8. associate-*r/ 99.8%

         \[\leadsto \frac{1}{N} + \left(\frac{0.3333333333333333}{{N}^{3}} - \left(\frac{0.5}{{N}^{2}} + \color{blue}{\frac{0.25 \cdot 1}{{N}^{4}}}\right)\right) \]

      9. metadata-eval 99.8%

         \[\leadsto \frac{1}{N} + \left(\frac{0.3333333333333333}{{N}^{3}} - \left(\frac{0.5}{{N}^{2}} + \frac{\color{blue}{0.25}}{{N}^{4}}\right)\right) \]

   7. Simplified 99.8%

      \[\leadsto \color{blue}{\frac{1}{N} + \left(\frac{0.3333333333333333}{{N}^{3}} - \left(\frac{0.5}{{N}^{2}} + \frac{0.25}{{N}^{4}}\right)\right)} \]

   if 1e-3 < (-.f64 (log.f64 (+.f64 N 1)) (log.f64 N))

   1. Initial program 93.4%

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

      1. +-commutative 93.4%

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

      2. log1p-define 93.6%

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

   3. Simplified 93.6%

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

      1. add-exp-log 93.7%

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

   6. Applied egg-rr 93.7%

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

      1. rem-exp-log 93.6%

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

      2. log1p-undefine 93.4%

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

      3. +-commutative 93.4%

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

      4. diff-log 94.9%

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

      5. clear-num 94.6%

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

      6. log-rec 95.4%

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

   8. Applied egg-rr 95.4%

      \[\leadsto \color{blue}{-\log \left(\frac{N}{N + 1}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 0.001:\\ \;\;\;\;\frac{1}{N} + \left(\frac{0.3333333333333333}{{N}^{3}} - \left(\frac{0.5}{{N}^{2}} + \frac{0.25}{{N}^{4}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-\log \left(\frac{N}{N + 1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 0.001:\\ \;\;\;\;\frac{0.3333333333333333}{{N}^{3}} + \left(\frac{1}{N} - \left(\frac{0.5}{{N}^{2}} + \frac{0.25}{{N}^{4}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-\log \left(\frac{N}{N + 1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= (- (log (+ N 1.0)) (log N)) 0.001)
   (+
    (/ 0.3333333333333333 (pow N 3.0))
    (- (/ 1.0 N) (+ (/ 0.5 (pow N 2.0)) (/ 0.25 (pow N 4.0)))))
   (- (log (/ N (+ N 1.0))))))

C

double code(double N) {
	double tmp;
	if ((log((N + 1.0)) - log(N)) <= 0.001) {
		tmp = (0.3333333333333333 / pow(N, 3.0)) + ((1.0 / N) - ((0.5 / pow(N, 2.0)) + (0.25 / pow(N, 4.0))));
	} else {
		tmp = -log((N / (N + 1.0)));
	}
	return tmp;
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((log((n + 1.0d0)) - log(n)) <= 0.001d0) then
        tmp = (0.3333333333333333d0 / (n ** 3.0d0)) + ((1.0d0 / n) - ((0.5d0 / (n ** 2.0d0)) + (0.25d0 / (n ** 4.0d0))))
    else
        tmp = -log((n / (n + 1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double N) {
	double tmp;
	if ((Math.log((N + 1.0)) - Math.log(N)) <= 0.001) {
		tmp = (0.3333333333333333 / Math.pow(N, 3.0)) + ((1.0 / N) - ((0.5 / Math.pow(N, 2.0)) + (0.25 / Math.pow(N, 4.0))));
	} else {
		tmp = -Math.log((N / (N + 1.0)));
	}
	return tmp;
}

Python

def code(N):
	tmp = 0
	if (math.log((N + 1.0)) - math.log(N)) <= 0.001:
		tmp = (0.3333333333333333 / math.pow(N, 3.0)) + ((1.0 / N) - ((0.5 / math.pow(N, 2.0)) + (0.25 / math.pow(N, 4.0))))
	else:
		tmp = -math.log((N / (N + 1.0)))
	return tmp

Julia

function code(N)
	tmp = 0.0
	if (Float64(log(Float64(N + 1.0)) - log(N)) <= 0.001)
		tmp = Float64(Float64(0.3333333333333333 / (N ^ 3.0)) + Float64(Float64(1.0 / N) - Float64(Float64(0.5 / (N ^ 2.0)) + Float64(0.25 / (N ^ 4.0)))));
	else
		tmp = Float64(-log(Float64(N / Float64(N + 1.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(N)
	tmp = 0.0;
	if ((log((N + 1.0)) - log(N)) <= 0.001)
		tmp = (0.3333333333333333 / (N ^ 3.0)) + ((1.0 / N) - ((0.5 / (N ^ 2.0)) + (0.25 / (N ^ 4.0))));
	else
		tmp = -log((N / (N + 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[N_] := If[LessEqual[N[(N[Log[N[(N + 1.0), $MachinePrecision]], $MachinePrecision] - N[Log[N], $MachinePrecision]), $MachinePrecision], 0.001], N[(N[(0.3333333333333333 / N[Power[N, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / N), $MachinePrecision] - N[(N[(0.5 / N[Power[N, 2.0], $MachinePrecision]), $MachinePrecision] + N[(0.25 / N[Power[N, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-N[Log[N[(N / N[(N + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (log.f64 (+.f64 N 1)) (log.f64 N)) < 1e-3

   1. Initial program 17.9%

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

      1. +-commutative 17.9%

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

      2. log1p-define 17.9%

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

   3. Simplified 17.9%

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

   5. Taylor expanded in N around inf 99.8%

      \[\leadsto \color{blue}{\left(0.3333333333333333 \cdot \frac{1}{{N}^{3}} + \frac{1}{N}\right) - \left(0.25 \cdot \frac{1}{{N}^{4}} + 0.5 \cdot \frac{1}{{N}^{2}}\right)} \]
   6. Step-by-step derivation

      1. associate--l+ 99.8%

         \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{1}{{N}^{3}} + \left(\frac{1}{N} - \left(0.25 \cdot \frac{1}{{N}^{4}} + 0.5 \cdot \frac{1}{{N}^{2}}\right)\right)} \]

      2. associate-*r/ 99.8%

         \[\leadsto \color{blue}{\frac{0.3333333333333333 \cdot 1}{{N}^{3}}} + \left(\frac{1}{N} - \left(0.25 \cdot \frac{1}{{N}^{4}} + 0.5 \cdot \frac{1}{{N}^{2}}\right)\right) \]

      3. metadata-eval 99.8%

         \[\leadsto \frac{\color{blue}{0.3333333333333333}}{{N}^{3}} + \left(\frac{1}{N} - \left(0.25 \cdot \frac{1}{{N}^{4}} + 0.5 \cdot \frac{1}{{N}^{2}}\right)\right) \]

      4. +-commutative 99.8%

         \[\leadsto \frac{0.3333333333333333}{{N}^{3}} + \left(\frac{1}{N} - \color{blue}{\left(0.5 \cdot \frac{1}{{N}^{2}} + 0.25 \cdot \frac{1}{{N}^{4}}\right)}\right) \]

      5. associate-*r/ 99.8%

         \[\leadsto \frac{0.3333333333333333}{{N}^{3}} + \left(\frac{1}{N} - \left(\color{blue}{\frac{0.5 \cdot 1}{{N}^{2}}} + 0.25 \cdot \frac{1}{{N}^{4}}\right)\right) \]

      6. metadata-eval 99.8%

         \[\leadsto \frac{0.3333333333333333}{{N}^{3}} + \left(\frac{1}{N} - \left(\frac{\color{blue}{0.5}}{{N}^{2}} + 0.25 \cdot \frac{1}{{N}^{4}}\right)\right) \]

      7. associate-*r/ 99.8%

         \[\leadsto \frac{0.3333333333333333}{{N}^{3}} + \left(\frac{1}{N} - \left(\frac{0.5}{{N}^{2}} + \color{blue}{\frac{0.25 \cdot 1}{{N}^{4}}}\right)\right) \]

      8. metadata-eval 99.8%

         \[\leadsto \frac{0.3333333333333333}{{N}^{3}} + \left(\frac{1}{N} - \left(\frac{0.5}{{N}^{2}} + \frac{\color{blue}{0.25}}{{N}^{4}}\right)\right) \]

   7. Simplified 99.8%

      \[\leadsto \color{blue}{\frac{0.3333333333333333}{{N}^{3}} + \left(\frac{1}{N} - \left(\frac{0.5}{{N}^{2}} + \frac{0.25}{{N}^{4}}\right)\right)} \]

   if 1e-3 < (-.f64 (log.f64 (+.f64 N 1)) (log.f64 N))

   1. Initial program 93.4%

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

      1. +-commutative 93.4%

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

      2. log1p-define 93.6%

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

   3. Simplified 93.6%

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

      1. add-exp-log 93.7%

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

   6. Applied egg-rr 93.7%

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

      1. rem-exp-log 93.6%

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

      2. log1p-undefine 93.4%

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

      3. +-commutative 93.4%

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

      4. diff-log 94.9%

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

      5. clear-num 94.6%

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

      6. log-rec 95.4%

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

   8. Applied egg-rr 95.4%

      \[\leadsto \color{blue}{-\log \left(\frac{N}{N + 1}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 0.001:\\ \;\;\;\;\frac{0.3333333333333333}{{N}^{3}} + \left(\frac{1}{N} - \left(\frac{0.5}{{N}^{2}} + \frac{0.25}{{N}^{4}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-\log \left(\frac{N}{N + 1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 8 \cdot 10^{-5}:\\ \;\;\;\;\frac{1}{N} + \left(\frac{0.3333333333333333}{{N}^{3}} + \frac{-0.5}{{N}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;-\log \left(\frac{N}{N + 1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= (- (log (+ N 1.0)) (log N)) 8e-5)
   (+ (/ 1.0 N) (+ (/ 0.3333333333333333 (pow N 3.0)) (/ -0.5 (pow N 2.0))))
   (- (log (/ N (+ N 1.0))))))

C

double code(double N) {
	double tmp;
	if ((log((N + 1.0)) - log(N)) <= 8e-5) {
		tmp = (1.0 / N) + ((0.3333333333333333 / pow(N, 3.0)) + (-0.5 / pow(N, 2.0)));
	} else {
		tmp = -log((N / (N + 1.0)));
	}
	return tmp;
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((log((n + 1.0d0)) - log(n)) <= 8d-5) then
        tmp = (1.0d0 / n) + ((0.3333333333333333d0 / (n ** 3.0d0)) + ((-0.5d0) / (n ** 2.0d0)))
    else
        tmp = -log((n / (n + 1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double N) {
	double tmp;
	if ((Math.log((N + 1.0)) - Math.log(N)) <= 8e-5) {
		tmp = (1.0 / N) + ((0.3333333333333333 / Math.pow(N, 3.0)) + (-0.5 / Math.pow(N, 2.0)));
	} else {
		tmp = -Math.log((N / (N + 1.0)));
	}
	return tmp;
}

Python

def code(N):
	tmp = 0
	if (math.log((N + 1.0)) - math.log(N)) <= 8e-5:
		tmp = (1.0 / N) + ((0.3333333333333333 / math.pow(N, 3.0)) + (-0.5 / math.pow(N, 2.0)))
	else:
		tmp = -math.log((N / (N + 1.0)))
	return tmp

Julia

function code(N)
	tmp = 0.0
	if (Float64(log(Float64(N + 1.0)) - log(N)) <= 8e-5)
		tmp = Float64(Float64(1.0 / N) + Float64(Float64(0.3333333333333333 / (N ^ 3.0)) + Float64(-0.5 / (N ^ 2.0))));
	else
		tmp = Float64(-log(Float64(N / Float64(N + 1.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(N)
	tmp = 0.0;
	if ((log((N + 1.0)) - log(N)) <= 8e-5)
		tmp = (1.0 / N) + ((0.3333333333333333 / (N ^ 3.0)) + (-0.5 / (N ^ 2.0)));
	else
		tmp = -log((N / (N + 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[N_] := If[LessEqual[N[(N[Log[N[(N + 1.0), $MachinePrecision]], $MachinePrecision] - N[Log[N], $MachinePrecision]), $MachinePrecision], 8e-5], N[(N[(1.0 / N), $MachinePrecision] + N[(N[(0.3333333333333333 / N[Power[N, 3.0], $MachinePrecision]), $MachinePrecision] + N[(-0.5 / N[Power[N, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-N[Log[N[(N / N[(N + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (log.f64 (+.f64 N 1)) (log.f64 N)) < 8.00000000000000065e-5

   1. Initial program 16.8%

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

      1. +-commutative 16.8%

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

      2. log1p-define 16.9%

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

   3. Simplified 16.9%

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

   5. Taylor expanded in N around inf 99.7%

      \[\leadsto \color{blue}{\left(0.3333333333333333 \cdot \frac{1}{{N}^{3}} + \frac{1}{N}\right) - 0.5 \cdot \frac{1}{{N}^{2}}} \]
   6. Step-by-step derivation

      1. sub-neg 99.7%

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

      2. +-commutative 99.7%

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

      3. associate-+l+ 99.7%

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

      4. associate-*r/ 99.7%

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

      5. metadata-eval 99.7%

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

      6. associate-*r/ 99.7%

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

      7. metadata-eval 99.7%

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

      8. distribute-neg-frac 99.7%

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

      9. metadata-eval 99.7%

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

   7. Simplified 99.7%

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

   if 8.00000000000000065e-5 < (-.f64 (log.f64 (+.f64 N 1)) (log.f64 N))

   1. Initial program 91.0%

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

      1. +-commutative 91.0%

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

      2. log1p-define 91.2%

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

   3. Simplified 91.2%

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

      1. add-exp-log 91.3%

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

   6. Applied egg-rr 91.3%

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

      1. rem-exp-log 91.2%

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

      2. log1p-undefine 91.0%

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

      3. +-commutative 91.0%

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

      4. diff-log 93.3%

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

      5. clear-num 93.0%

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

      6. log-rec 93.9%

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

   8. Applied egg-rr 93.9%

      \[\leadsto \color{blue}{-\log \left(\frac{N}{N + 1}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 8 \cdot 10^{-5}:\\ \;\;\;\;\frac{1}{N} + \left(\frac{0.3333333333333333}{{N}^{3}} + \frac{-0.5}{{N}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;-\log \left(\frac{N}{N + 1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 8 \cdot 10^{-5}:\\ \;\;\;\;\frac{0.3333333333333333}{{N}^{3}} + \frac{\frac{N + -0.5}{N}}{N}\\ \mathbf{else}:\\ \;\;\;\;-\log \left(\frac{N}{N + 1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= (- (log (+ N 1.0)) (log N)) 8e-5)
   (+ (/ 0.3333333333333333 (pow N 3.0)) (/ (/ (+ N -0.5) N) N))
   (- (log (/ N (+ N 1.0))))))

C

double code(double N) {
	double tmp;
	if ((log((N + 1.0)) - log(N)) <= 8e-5) {
		tmp = (0.3333333333333333 / pow(N, 3.0)) + (((N + -0.5) / N) / N);
	} else {
		tmp = -log((N / (N + 1.0)));
	}
	return tmp;
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((log((n + 1.0d0)) - log(n)) <= 8d-5) then
        tmp = (0.3333333333333333d0 / (n ** 3.0d0)) + (((n + (-0.5d0)) / n) / n)
    else
        tmp = -log((n / (n + 1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double N) {
	double tmp;
	if ((Math.log((N + 1.0)) - Math.log(N)) <= 8e-5) {
		tmp = (0.3333333333333333 / Math.pow(N, 3.0)) + (((N + -0.5) / N) / N);
	} else {
		tmp = -Math.log((N / (N + 1.0)));
	}
	return tmp;
}

Python

def code(N):
	tmp = 0
	if (math.log((N + 1.0)) - math.log(N)) <= 8e-5:
		tmp = (0.3333333333333333 / math.pow(N, 3.0)) + (((N + -0.5) / N) / N)
	else:
		tmp = -math.log((N / (N + 1.0)))
	return tmp

Julia

function code(N)
	tmp = 0.0
	if (Float64(log(Float64(N + 1.0)) - log(N)) <= 8e-5)
		tmp = Float64(Float64(0.3333333333333333 / (N ^ 3.0)) + Float64(Float64(Float64(N + -0.5) / N) / N));
	else
		tmp = Float64(-log(Float64(N / Float64(N + 1.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(N)
	tmp = 0.0;
	if ((log((N + 1.0)) - log(N)) <= 8e-5)
		tmp = (0.3333333333333333 / (N ^ 3.0)) + (((N + -0.5) / N) / N);
	else
		tmp = -log((N / (N + 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[N_] := If[LessEqual[N[(N[Log[N[(N + 1.0), $MachinePrecision]], $MachinePrecision] - N[Log[N], $MachinePrecision]), $MachinePrecision], 8e-5], N[(N[(0.3333333333333333 / N[Power[N, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N + -0.5), $MachinePrecision] / N), $MachinePrecision] / N), $MachinePrecision]), $MachinePrecision], (-N[Log[N[(N / N[(N + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (log.f64 (+.f64 N 1)) (log.f64 N)) < 8.00000000000000065e-5

   1. Initial program 16.8%

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

      1. +-commutative 16.8%

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

      2. log1p-define 16.9%

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

   3. Simplified 16.9%

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

   5. Taylor expanded in N around inf 99.7%

      \[\leadsto \color{blue}{\left(0.3333333333333333 \cdot \frac{1}{{N}^{3}} + \frac{1}{N}\right) - 0.5 \cdot \frac{1}{{N}^{2}}} \]
   6. Step-by-step derivation

      1. associate--l+ 99.7%

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

      2. associate-*r/ 99.7%

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

      3. metadata-eval 99.7%

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

      4. associate-*r/ 99.7%

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

      5. metadata-eval 99.7%

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

   7. Simplified 99.7%

      \[\leadsto \color{blue}{\frac{0.3333333333333333}{{N}^{3}} + \left(\frac{1}{N} - \frac{0.5}{{N}^{2}}\right)} \]
   8. Step-by-step derivation

      1. frac-sub 99.4%

         \[\leadsto \frac{0.3333333333333333}{{N}^{3}} + \color{blue}{\frac{1 \cdot {N}^{2} - N \cdot 0.5}{N \cdot {N}^{2}}} \]

      2. *-un-lft-identity 99.4%

         \[\leadsto \frac{0.3333333333333333}{{N}^{3}} + \frac{\color{blue}{{N}^{2}} - N \cdot 0.5}{N \cdot {N}^{2}} \]

      3. unpow2 99.4%

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

      4. cube-mult 99.3%

         \[\leadsto \frac{0.3333333333333333}{{N}^{3}} + \frac{{N}^{2} - N \cdot 0.5}{\color{blue}{{N}^{3}}} \]

   9. Applied egg-rr 99.3%

      \[\leadsto \frac{0.3333333333333333}{{N}^{3}} + \color{blue}{\frac{{N}^{2} - N \cdot 0.5}{{N}^{3}}} \]
   10. Step-by-step derivation

       1. unpow2 99.3%

          \[\leadsto \frac{0.3333333333333333}{{N}^{3}} + \frac{\color{blue}{N \cdot N} - N \cdot 0.5}{{N}^{3}} \]

       2. distribute-lft-out-- 99.3%

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

   11. Simplified 99.3%

       \[\leadsto \frac{0.3333333333333333}{{N}^{3}} + \color{blue}{\frac{N \cdot \left(N - 0.5\right)}{{N}^{3}}} \]
   12. Step-by-step derivation

       1. *-un-lft-identity 99.3%

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

       2. cube-mult 99.3%

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

       3. unpow2 99.3%

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

       4. times-frac 99.6%

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

       5. sub-neg 99.6%

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

       6. metadata-eval 99.6%

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

   13. Applied egg-rr 99.6%

       \[\leadsto \frac{0.3333333333333333}{{N}^{3}} + \color{blue}{\frac{1}{N} \cdot \frac{N \cdot \left(N + -0.5\right)}{{N}^{2}}} \]
   14. Step-by-step derivation

       1. associate-*l/ 99.6%

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

       2. *-lft-identity 99.6%

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

       3. unpow2 99.6%

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

       4. times-frac 99.7%

          \[\leadsto \frac{0.3333333333333333}{{N}^{3}} + \frac{\color{blue}{\frac{N}{N} \cdot \frac{N + -0.5}{N}}}{N} \]

       5. *-inverses 99.7%

          \[\leadsto \frac{0.3333333333333333}{{N}^{3}} + \frac{\color{blue}{1} \cdot \frac{N + -0.5}{N}}{N} \]

       6. +-commutative 99.7%

          \[\leadsto \frac{0.3333333333333333}{{N}^{3}} + \frac{1 \cdot \frac{\color{blue}{-0.5 + N}}{N}}{N} \]

   15. Simplified 99.7%

       \[\leadsto \frac{0.3333333333333333}{{N}^{3}} + \color{blue}{\frac{1 \cdot \frac{-0.5 + N}{N}}{N}} \]

   if 8.00000000000000065e-5 < (-.f64 (log.f64 (+.f64 N 1)) (log.f64 N))

   1. Initial program 91.0%

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

      1. +-commutative 91.0%

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

      2. log1p-define 91.2%

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

   3. Simplified 91.2%

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

      1. add-exp-log 91.3%

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

   6. Applied egg-rr 91.3%

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

      1. rem-exp-log 91.2%

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

      2. log1p-undefine 91.0%

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

      3. +-commutative 91.0%

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

      4. diff-log 93.3%

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

      5. clear-num 93.0%

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

      6. log-rec 93.9%

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

   8. Applied egg-rr 93.9%

      \[\leadsto \color{blue}{-\log \left(\frac{N}{N + 1}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 8 \cdot 10^{-5}:\\ \;\;\;\;\frac{0.3333333333333333}{{N}^{3}} + \frac{\frac{N + -0.5}{N}}{N}\\ \mathbf{else}:\\ \;\;\;\;-\log \left(\frac{N}{N + 1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;N \leq 250000:\\ \;\;\;\;-\log \left(\frac{N}{N + 1}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N} + \frac{\frac{-0.5}{N}}{N}\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= N 250000.0) (- (log (/ N (+ N 1.0)))) (+ (/ 1.0 N) (/ (/ -0.5 N) N))))

C

double code(double N) {
	double tmp;
	if (N <= 250000.0) {
		tmp = -log((N / (N + 1.0)));
	} else {
		tmp = (1.0 / N) + ((-0.5 / N) / N);
	}
	return tmp;
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= 250000.0d0) then
        tmp = -log((n / (n + 1.0d0)))
    else
        tmp = (1.0d0 / n) + (((-0.5d0) / n) / n)
    end if
    code = tmp
end function

Java

public static double code(double N) {
	double tmp;
	if (N <= 250000.0) {
		tmp = -Math.log((N / (N + 1.0)));
	} else {
		tmp = (1.0 / N) + ((-0.5 / N) / N);
	}
	return tmp;
}

Python

def code(N):
	tmp = 0
	if N <= 250000.0:
		tmp = -math.log((N / (N + 1.0)))
	else:
		tmp = (1.0 / N) + ((-0.5 / N) / N)
	return tmp

Julia

function code(N)
	tmp = 0.0
	if (N <= 250000.0)
		tmp = Float64(-log(Float64(N / Float64(N + 1.0))));
	else
		tmp = Float64(Float64(1.0 / N) + Float64(Float64(-0.5 / N) / N));
	end
	return tmp
end

MATLAB

function tmp_2 = code(N)
	tmp = 0.0;
	if (N <= 250000.0)
		tmp = -log((N / (N + 1.0)));
	else
		tmp = (1.0 / N) + ((-0.5 / N) / N);
	end
	tmp_2 = tmp;
end

Wolfram

code[N_] := If[LessEqual[N, 250000.0], (-N[Log[N[(N / N[(N + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), N[(N[(1.0 / N), $MachinePrecision] + N[(N[(-0.5 / N), $MachinePrecision] / N), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if N < 2.5e5

   1. Initial program 87.2%

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

      1. +-commutative 87.2%

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

      2. log1p-define 87.3%

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

   3. Simplified 87.3%

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

      1. add-exp-log 87.4%

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

   6. Applied egg-rr 87.4%

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

      1. rem-exp-log 87.3%

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

      2. log1p-undefine 87.2%

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

      3. +-commutative 87.2%

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

      4. diff-log 90.1%

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

      5. clear-num 89.9%

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

      6. log-rec 90.9%

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

   8. Applied egg-rr 90.9%

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

   if 2.5e5 < N

   1. Initial program 15.0%

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

      1. +-commutative 15.0%

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

      2. log1p-define 15.1%

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

   3. Simplified 15.1%

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

      1. add-exp-log 15.1%

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

   6. Applied egg-rr 15.1%

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

   7. Taylor expanded in N around inf 94.5%

      \[\leadsto e^{\color{blue}{\left(\log \left(\frac{1}{N}\right) + 0.20833333333333334 \cdot \frac{1}{{N}^{2}}\right) - \left(0.125 \cdot \frac{1}{{N}^{3}} + 0.5 \cdot \frac{1}{N}\right)}} \]
   8. Step-by-step derivation

      1. log-rec 94.5%

         \[\leadsto e^{\left(\color{blue}{\left(-\log N\right)} + 0.20833333333333334 \cdot \frac{1}{{N}^{2}}\right) - \left(0.125 \cdot \frac{1}{{N}^{3}} + 0.5 \cdot \frac{1}{N}\right)} \]

      2. +-commutative 94.5%

         \[\leadsto e^{\color{blue}{\left(0.20833333333333334 \cdot \frac{1}{{N}^{2}} + \left(-\log N\right)\right)} - \left(0.125 \cdot \frac{1}{{N}^{3}} + 0.5 \cdot \frac{1}{N}\right)} \]

      3. unsub-neg 94.5%

         \[\leadsto e^{\color{blue}{\left(0.20833333333333334 \cdot \frac{1}{{N}^{2}} - \log N\right)} - \left(0.125 \cdot \frac{1}{{N}^{3}} + 0.5 \cdot \frac{1}{N}\right)} \]

      4. associate-*r/ 94.5%

         \[\leadsto e^{\left(\color{blue}{\frac{0.20833333333333334 \cdot 1}{{N}^{2}}} - \log N\right) - \left(0.125 \cdot \frac{1}{{N}^{3}} + 0.5 \cdot \frac{1}{N}\right)} \]

      5. metadata-eval 94.5%

         \[\leadsto e^{\left(\frac{\color{blue}{0.20833333333333334}}{{N}^{2}} - \log N\right) - \left(0.125 \cdot \frac{1}{{N}^{3}} + 0.5 \cdot \frac{1}{N}\right)} \]

      6. +-commutative 94.5%

         \[\leadsto e^{\left(\frac{0.20833333333333334}{{N}^{2}} - \log N\right) - \color{blue}{\left(0.5 \cdot \frac{1}{N} + 0.125 \cdot \frac{1}{{N}^{3}}\right)}} \]

      7. associate-*r/ 94.5%

         \[\leadsto e^{\left(\frac{0.20833333333333334}{{N}^{2}} - \log N\right) - \left(\color{blue}{\frac{0.5 \cdot 1}{N}} + 0.125 \cdot \frac{1}{{N}^{3}}\right)} \]

      8. metadata-eval 94.5%

         \[\leadsto e^{\left(\frac{0.20833333333333334}{{N}^{2}} - \log N\right) - \left(\frac{\color{blue}{0.5}}{N} + 0.125 \cdot \frac{1}{{N}^{3}}\right)} \]

      9. associate-*r/ 94.5%

         \[\leadsto e^{\left(\frac{0.20833333333333334}{{N}^{2}} - \log N\right) - \left(\frac{0.5}{N} + \color{blue}{\frac{0.125 \cdot 1}{{N}^{3}}}\right)} \]

      10. metadata-eval 94.5%

          \[\leadsto e^{\left(\frac{0.20833333333333334}{{N}^{2}} - \log N\right) - \left(\frac{0.5}{N} + \frac{\color{blue}{0.125}}{{N}^{3}}\right)} \]

   9. Simplified 94.5%

      \[\leadsto e^{\color{blue}{\left(\frac{0.20833333333333334}{{N}^{2}} - \log N\right) - \left(\frac{0.5}{N} + \frac{0.125}{{N}^{3}}\right)}} \]

   10. Taylor expanded in N around inf 94.0%

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

       1. exp-neg 94.0%

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

       2. mul-1-neg 94.0%

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

       3. log-rec 94.0%

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

       4. remove-double-neg 94.0%

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

       5. rem-exp-log 98.9%

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

       6. associate-*r/ 98.9%

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

       7. metadata-eval 98.9%

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

       8. exp-neg 98.9%

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

       9. mul-1-neg 98.9%

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

       10. log-rec 98.9%

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

       11. remove-double-neg 98.9%

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

       12. rem-exp-log 98.9%

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

       13. distribute-lft-neg-in 98.9%

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

       14. associate-*r/ 98.9%

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

       15. metadata-eval 98.9%

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

       16. distribute-neg-frac 98.9%

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

       17. metadata-eval 98.9%

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

   12. Simplified 98.9%

       \[\leadsto \color{blue}{\frac{1}{N} + \frac{\frac{-0.5}{N}}{N}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;N \leq 250000:\\ \;\;\;\;-\log \left(\frac{N}{N + 1}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N} + \frac{\frac{-0.5}{N}}{N}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;N \leq 240000:\\ \;\;\;\;\log \left(1 + \frac{1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N} + \frac{\frac{-0.5}{N}}{N}\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= N 240000.0) (log (+ 1.0 (/ 1.0 N))) (+ (/ 1.0 N) (/ (/ -0.5 N) N))))

C

double code(double N) {
	double tmp;
	if (N <= 240000.0) {
		tmp = log((1.0 + (1.0 / N)));
	} else {
		tmp = (1.0 / N) + ((-0.5 / N) / N);
	}
	return tmp;
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= 240000.0d0) then
        tmp = log((1.0d0 + (1.0d0 / n)))
    else
        tmp = (1.0d0 / n) + (((-0.5d0) / n) / n)
    end if
    code = tmp
end function

Java

public static double code(double N) {
	double tmp;
	if (N <= 240000.0) {
		tmp = Math.log((1.0 + (1.0 / N)));
	} else {
		tmp = (1.0 / N) + ((-0.5 / N) / N);
	}
	return tmp;
}

Python

def code(N):
	tmp = 0
	if N <= 240000.0:
		tmp = math.log((1.0 + (1.0 / N)))
	else:
		tmp = (1.0 / N) + ((-0.5 / N) / N)
	return tmp

Julia

function code(N)
	tmp = 0.0
	if (N <= 240000.0)
		tmp = log(Float64(1.0 + Float64(1.0 / N)));
	else
		tmp = Float64(Float64(1.0 / N) + Float64(Float64(-0.5 / N) / N));
	end
	return tmp
end

MATLAB

function tmp_2 = code(N)
	tmp = 0.0;
	if (N <= 240000.0)
		tmp = log((1.0 + (1.0 / N)));
	else
		tmp = (1.0 / N) + ((-0.5 / N) / N);
	end
	tmp_2 = tmp;
end

Wolfram

code[N_] := If[LessEqual[N, 240000.0], N[Log[N[(1.0 + N[(1.0 / N), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(1.0 / N), $MachinePrecision] + N[(N[(-0.5 / N), $MachinePrecision] / N), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if N < 2.4e5

   1. Initial program 87.8%

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

      1. +-commutative 87.8%

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

      2. log1p-define 88.0%

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

   3. Simplified 88.0%

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

      1. add-log-exp 87.9%

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

      2. log1p-expm1-u 87.9%

         \[\leadsto \log \left(e^{\mathsf{log1p}\left(N\right)}\right) - \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log N\right)\right)} \]

      3. log1p-undefine 87.9%

         \[\leadsto \log \left(e^{\mathsf{log1p}\left(N\right)}\right) - \color{blue}{\log \left(1 + \mathsf{expm1}\left(\log N\right)\right)} \]

      4. diff-log 87.9%

         \[\leadsto \color{blue}{\log \left(\frac{e^{\mathsf{log1p}\left(N\right)}}{1 + \mathsf{expm1}\left(\log N\right)}\right)} \]

      5. log1p-undefine 87.9%

         \[\leadsto \log \left(\frac{e^{\color{blue}{\log \left(1 + N\right)}}}{1 + \mathsf{expm1}\left(\log N\right)}\right) \]

      6. rem-exp-log 87.6%

         \[\leadsto \log \left(\frac{\color{blue}{1 + N}}{1 + \mathsf{expm1}\left(\log N\right)}\right) \]

      7. +-commutative 87.6%

         \[\leadsto \log \left(\frac{\color{blue}{N + 1}}{1 + \mathsf{expm1}\left(\log N\right)}\right) \]

      8. add-exp-log 87.6%

         \[\leadsto \log \left(\frac{N + 1}{\color{blue}{e^{\log \left(1 + \mathsf{expm1}\left(\log N\right)\right)}}}\right) \]

      9. log1p-undefine 87.6%

         \[\leadsto \log \left(\frac{N + 1}{e^{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log N\right)\right)}}}\right) \]

      10. log1p-expm1-u 87.6%

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

      11. add-exp-log 90.6%

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

   6. Applied egg-rr 90.6%

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

   7. Taylor expanded in N around 0 90.7%

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

   if 2.4e5 < N

   1. Initial program 15.3%

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

      1. +-commutative 15.3%

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

      2. log1p-define 15.3%

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

   3. Simplified 15.3%

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

      1. add-exp-log 15.3%

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

   6. Applied egg-rr 15.3%

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

   7. Taylor expanded in N around inf 94.5%

      \[\leadsto e^{\color{blue}{\left(\log \left(\frac{1}{N}\right) + 0.20833333333333334 \cdot \frac{1}{{N}^{2}}\right) - \left(0.125 \cdot \frac{1}{{N}^{3}} + 0.5 \cdot \frac{1}{N}\right)}} \]
   8. Step-by-step derivation

      1. log-rec 94.5%

         \[\leadsto e^{\left(\color{blue}{\left(-\log N\right)} + 0.20833333333333334 \cdot \frac{1}{{N}^{2}}\right) - \left(0.125 \cdot \frac{1}{{N}^{3}} + 0.5 \cdot \frac{1}{N}\right)} \]

      2. +-commutative 94.5%

         \[\leadsto e^{\color{blue}{\left(0.20833333333333334 \cdot \frac{1}{{N}^{2}} + \left(-\log N\right)\right)} - \left(0.125 \cdot \frac{1}{{N}^{3}} + 0.5 \cdot \frac{1}{N}\right)} \]

      3. unsub-neg 94.5%

         \[\leadsto e^{\color{blue}{\left(0.20833333333333334 \cdot \frac{1}{{N}^{2}} - \log N\right)} - \left(0.125 \cdot \frac{1}{{N}^{3}} + 0.5 \cdot \frac{1}{N}\right)} \]

      4. associate-*r/ 94.5%

         \[\leadsto e^{\left(\color{blue}{\frac{0.20833333333333334 \cdot 1}{{N}^{2}}} - \log N\right) - \left(0.125 \cdot \frac{1}{{N}^{3}} + 0.5 \cdot \frac{1}{N}\right)} \]

      5. metadata-eval 94.5%

         \[\leadsto e^{\left(\frac{\color{blue}{0.20833333333333334}}{{N}^{2}} - \log N\right) - \left(0.125 \cdot \frac{1}{{N}^{3}} + 0.5 \cdot \frac{1}{N}\right)} \]

      6. +-commutative 94.5%

         \[\leadsto e^{\left(\frac{0.20833333333333334}{{N}^{2}} - \log N\right) - \color{blue}{\left(0.5 \cdot \frac{1}{N} + 0.125 \cdot \frac{1}{{N}^{3}}\right)}} \]

      7. associate-*r/ 94.5%

         \[\leadsto e^{\left(\frac{0.20833333333333334}{{N}^{2}} - \log N\right) - \left(\color{blue}{\frac{0.5 \cdot 1}{N}} + 0.125 \cdot \frac{1}{{N}^{3}}\right)} \]

      8. metadata-eval 94.5%

         \[\leadsto e^{\left(\frac{0.20833333333333334}{{N}^{2}} - \log N\right) - \left(\frac{\color{blue}{0.5}}{N} + 0.125 \cdot \frac{1}{{N}^{3}}\right)} \]

      9. associate-*r/ 94.5%

         \[\leadsto e^{\left(\frac{0.20833333333333334}{{N}^{2}} - \log N\right) - \left(\frac{0.5}{N} + \color{blue}{\frac{0.125 \cdot 1}{{N}^{3}}}\right)} \]

      10. metadata-eval 94.5%

          \[\leadsto e^{\left(\frac{0.20833333333333334}{{N}^{2}} - \log N\right) - \left(\frac{0.5}{N} + \frac{\color{blue}{0.125}}{{N}^{3}}\right)} \]

   9. Simplified 94.5%

      \[\leadsto e^{\color{blue}{\left(\frac{0.20833333333333334}{{N}^{2}} - \log N\right) - \left(\frac{0.5}{N} + \frac{0.125}{{N}^{3}}\right)}} \]

   10. Taylor expanded in N around inf 93.9%

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

       1. exp-neg 93.9%

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

       2. mul-1-neg 93.9%

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

       3. log-rec 93.9%

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

       4. remove-double-neg 93.9%

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

       5. rem-exp-log 98.8%

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

       6. associate-*r/ 98.8%

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

       7. metadata-eval 98.8%

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

       8. exp-neg 98.8%

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

       9. mul-1-neg 98.8%

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

       10. log-rec 98.8%

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

       11. remove-double-neg 98.8%

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

       12. rem-exp-log 98.8%

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

       13. distribute-lft-neg-in 98.8%

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

       14. associate-*r/ 98.8%

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

       15. metadata-eval 98.8%

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

       16. distribute-neg-frac 98.8%

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

       17. metadata-eval 98.8%

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

   12. Simplified 98.8%

       \[\leadsto \color{blue}{\frac{1}{N} + \frac{\frac{-0.5}{N}}{N}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;N \leq 240000:\\ \;\;\;\;\log \left(1 + \frac{1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N} + \frac{\frac{-0.5}{N}}{N}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 92.2% 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 24.1%

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

   1. +-commutative 24.1%

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

   2. log1p-define 24.1%

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

3. Simplified 24.1%

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

   1. add-exp-log 24.1%

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

6. Applied egg-rr 24.1%

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

7. Taylor expanded in N around inf 91.1%

   \[\leadsto e^{\color{blue}{\left(\log \left(\frac{1}{N}\right) + 0.20833333333333334 \cdot \frac{1}{{N}^{2}}\right) - \left(0.125 \cdot \frac{1}{{N}^{3}} + 0.5 \cdot \frac{1}{N}\right)}} \]
8. Step-by-step derivation

   1. log-rec 91.1%

      \[\leadsto e^{\left(\color{blue}{\left(-\log N\right)} + 0.20833333333333334 \cdot \frac{1}{{N}^{2}}\right) - \left(0.125 \cdot \frac{1}{{N}^{3}} + 0.5 \cdot \frac{1}{N}\right)} \]

   2. +-commutative 91.1%

      \[\leadsto e^{\color{blue}{\left(0.20833333333333334 \cdot \frac{1}{{N}^{2}} + \left(-\log N\right)\right)} - \left(0.125 \cdot \frac{1}{{N}^{3}} + 0.5 \cdot \frac{1}{N}\right)} \]

   3. unsub-neg 91.1%

      \[\leadsto e^{\color{blue}{\left(0.20833333333333334 \cdot \frac{1}{{N}^{2}} - \log N\right)} - \left(0.125 \cdot \frac{1}{{N}^{3}} + 0.5 \cdot \frac{1}{N}\right)} \]

   4. associate-*r/ 91.1%

      \[\leadsto e^{\left(\color{blue}{\frac{0.20833333333333334 \cdot 1}{{N}^{2}}} - \log N\right) - \left(0.125 \cdot \frac{1}{{N}^{3}} + 0.5 \cdot \frac{1}{N}\right)} \]

   5. metadata-eval 91.1%

      \[\leadsto e^{\left(\frac{\color{blue}{0.20833333333333334}}{{N}^{2}} - \log N\right) - \left(0.125 \cdot \frac{1}{{N}^{3}} + 0.5 \cdot \frac{1}{N}\right)} \]

   6. +-commutative 91.1%

      \[\leadsto e^{\left(\frac{0.20833333333333334}{{N}^{2}} - \log N\right) - \color{blue}{\left(0.5 \cdot \frac{1}{N} + 0.125 \cdot \frac{1}{{N}^{3}}\right)}} \]

   7. associate-*r/ 91.1%

      \[\leadsto e^{\left(\frac{0.20833333333333334}{{N}^{2}} - \log N\right) - \left(\color{blue}{\frac{0.5 \cdot 1}{N}} + 0.125 \cdot \frac{1}{{N}^{3}}\right)} \]

   8. metadata-eval 91.1%

      \[\leadsto e^{\left(\frac{0.20833333333333334}{{N}^{2}} - \log N\right) - \left(\frac{\color{blue}{0.5}}{N} + 0.125 \cdot \frac{1}{{N}^{3}}\right)} \]

   9. associate-*r/ 91.1%

      \[\leadsto e^{\left(\frac{0.20833333333333334}{{N}^{2}} - \log N\right) - \left(\frac{0.5}{N} + \color{blue}{\frac{0.125 \cdot 1}{{N}^{3}}}\right)} \]

   10. metadata-eval 91.1%

       \[\leadsto e^{\left(\frac{0.20833333333333334}{{N}^{2}} - \log N\right) - \left(\frac{0.5}{N} + \frac{\color{blue}{0.125}}{{N}^{3}}\right)} \]

9. Simplified 91.1%

   \[\leadsto e^{\color{blue}{\left(\frac{0.20833333333333334}{{N}^{2}} - \log N\right) - \left(\frac{0.5}{N} + \frac{0.125}{{N}^{3}}\right)}} \]

10. Taylor expanded in N around inf 87.9%

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

    1. exp-neg 87.9%

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

    2. mul-1-neg 87.9%

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

    3. log-rec 87.9%

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

    4. remove-double-neg 87.9%

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

    5. rem-exp-log 92.2%

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

    6. associate-*r/ 92.2%

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

    7. metadata-eval 92.2%

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

    8. exp-neg 92.2%

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

    9. mul-1-neg 92.2%

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

    10. log-rec 92.2%

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

    11. remove-double-neg 92.2%

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

    12. rem-exp-log 92.2%

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

    13. distribute-lft-neg-in 92.2%

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

    14. associate-*r/ 92.2%

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

    15. metadata-eval 92.2%

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

    16. distribute-neg-frac 92.2%

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

    17. metadata-eval 92.2%

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

12. Simplified 92.2%

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

13. Final simplification 92.2%

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

Alternative 8: 84.3% 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 24.1%

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

   1. +-commutative 24.1%

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

   2. log1p-define 24.1%

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

3. Simplified 24.1%

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

5. Taylor expanded in N around inf 84.2%

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

6. Final simplification 84.2%

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

Alternative 9: 3.3% accurate, 205.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (N) :precision binary64 0.0)

C

double code(double N) {
	return 0.0;
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    code = 0.0d0
end function

Java

public static double code(double N) {
	return 0.0;
}

Python

def code(N):
	return 0.0

Julia

function code(N)
	return 0.0
end

MATLAB

function tmp = code(N)
	tmp = 0.0;
end

Wolfram

code[N_] := 0.0

Derivation

1. Initial program 24.1%

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

   1. +-commutative 24.1%

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

   2. log1p-define 24.1%

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

3. Simplified 24.1%

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

   1. sub-neg 24.1%

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

   2. +-commutative 24.1%

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

   3. add-sqr-sqrt 24.2%

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

   4. distribute-rgt-neg-in 24.2%

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

   5. fma-define 25.6%

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

6. Applied egg-rr 25.6%

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

7. Taylor expanded in N around inf 3.3%

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

   1. log-rec 3.3%

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

   2. log-rec 3.3%

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

   3. distribute-rgt1-in 3.3%

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

   4. metadata-eval 3.3%

      \[\leadsto \color{blue}{0} \cdot \left(-\log N\right) \]

   5. mul0-lft 3.3%

      \[\leadsto \color{blue}{0} \]

9. Simplified 3.3%

   \[\leadsto \color{blue}{0} \]

10. Final simplification 3.3%

    \[\leadsto 0 \]
11. 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 2024044 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  :pre (and (> N 1.0) (< N 1e+40))

  :herbie-target
  (log1p (/ 1.0 N))

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