2log (problem 3.3.6)

Percentage Accurate: 54.6% → 100.0%

Time: 9.3s

Alternatives: 8

Speedup: 2.0×

Specification

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: 54.6% 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: 100.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 0.0005:\\ \;\;\;\;\left(\frac{0.3333333333333333}{{N}^{3}} + \frac{1 + \frac{-0.5}{N}}{N}\right) + \frac{-0.25}{{N}^{4}}\\ \mathbf{else}:\\ \;\;\;\;-\log \left(\frac{N}{N + 1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= (- (log (+ N 1.0)) (log N)) 0.0005)
   (+
    (+ (/ 0.3333333333333333 (pow N 3.0)) (/ (+ 1.0 (/ -0.5 N)) N))
    (/ -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)) <= 0.0005) {
		tmp = ((0.3333333333333333 / pow(N, 3.0)) + ((1.0 + (-0.5 / N)) / N)) + (-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)) <= 0.0005d0) then
        tmp = ((0.3333333333333333d0 / (n ** 3.0d0)) + ((1.0d0 + ((-0.5d0) / n)) / n)) + ((-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)) <= 0.0005) {
		tmp = ((0.3333333333333333 / Math.pow(N, 3.0)) + ((1.0 + (-0.5 / N)) / N)) + (-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)) <= 0.0005:
		tmp = ((0.3333333333333333 / math.pow(N, 3.0)) + ((1.0 + (-0.5 / N)) / N)) + (-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)) <= 0.0005)
		tmp = Float64(Float64(Float64(0.3333333333333333 / (N ^ 3.0)) + Float64(Float64(1.0 + Float64(-0.5 / N)) / N)) + 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)) <= 0.0005)
		tmp = ((0.3333333333333333 / (N ^ 3.0)) + ((1.0 + (-0.5 / N)) / N)) + (-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], 0.0005], N[(N[(N[(0.3333333333333333 / N[Power[N, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 + N[(-0.5 / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision]), $MachinePrecision] + N[(-0.25 / N[Power[N, 4.0], $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)) < 5.0000000000000001e-4

   1. Initial program 9.5%

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

      1. +-commutative 9.5%

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

      2. log1p-def 9.5%

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

   3. Simplified 9.5%

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

   4. Taylor expanded in N around inf 99.9%

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

      1. associate-*r/ 99.9%

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

      2. metadata-eval 99.9%

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

      3. +-commutative 99.9%

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

      4. associate-*r/ 99.9%

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

      5. metadata-eval 99.9%

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

      6. unpow2 99.9%

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

      7. associate-*r/ 99.9%

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

      8. metadata-eval 99.9%

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

   6. Simplified 99.9%

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

   7. Taylor expanded in N around 0 99.9%

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

      1. sub-neg 99.9%

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

      2. associate-*r/ 99.9%

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

      3. metadata-eval 99.9%

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

      4. +-commutative 99.9%

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

      5. associate-*r/ 99.9%

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

      6. metadata-eval 99.9%

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

      7. unpow2 99.9%

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

      8. distribute-neg-in 99.9%

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

      9. metadata-eval 99.9%

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

      10. associate-*r/ 99.9%

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

      11. associate-+r+ 99.9%

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

   9. Simplified 99.9%

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

   if 5.0000000000000001e-4 < (-.f64 (log.f64 (+.f64 N 1)) (log.f64 N))

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. log1p-def 99.9%

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

   3. Simplified 99.9%

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

      1. log1p-udef 99.9%

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

      2. diff-log 99.9%

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

      3. +-commutative 99.9%

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

   5. Applied egg-rr 99.9%

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

      1. clear-num 99.9%

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

      2. log-div 99.9%

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

      3. metadata-eval 99.9%

         \[\leadsto \color{blue}{0} - \log \left(\frac{N}{N + 1}\right) \]

   7. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{0 - \log \left(\frac{N}{N + 1}\right)} \]
   8. Step-by-step derivation

      1. neg-sub0 99.9%

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

   9. Simplified 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 0.0005:\\ \;\;\;\;\left(\frac{0.3333333333333333}{{N}^{3}} + \frac{1 + \frac{-0.5}{N}}{N}\right) + \frac{-0.25}{{N}^{4}}\\ \mathbf{else}:\\ \;\;\;\;-\log \left(\frac{N}{N + 1}\right)\\ \end{array} \]

Alternative 2: 99.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\frac{0.3333333333333333}{{N}^{3}} + \frac{1 + \frac{-0.5}{N}}{N}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= (- (log (+ N 1.0)) (log N)) 5e-6)
   (+ (/ 0.3333333333333333 (pow N 3.0)) (/ (+ 1.0 (/ -0.5 N)) N))
   (log (/ (+ N 1.0) N))))

C

double code(double N) {
	double tmp;
	if ((log((N + 1.0)) - log(N)) <= 5e-6) {
		tmp = (0.3333333333333333 / pow(N, 3.0)) + ((1.0 + (-0.5 / N)) / N);
	} else {
		tmp = log(((N + 1.0) / N));
	}
	return tmp;
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((log((n + 1.0d0)) - log(n)) <= 5d-6) then
        tmp = (0.3333333333333333d0 / (n ** 3.0d0)) + ((1.0d0 + ((-0.5d0) / n)) / n)
    else
        tmp = log(((n + 1.0d0) / n))
    end if
    code = tmp
end function

Java

public static double code(double N) {
	double tmp;
	if ((Math.log((N + 1.0)) - Math.log(N)) <= 5e-6) {
		tmp = (0.3333333333333333 / Math.pow(N, 3.0)) + ((1.0 + (-0.5 / N)) / N);
	} else {
		tmp = Math.log(((N + 1.0) / N));
	}
	return tmp;
}

Python

def code(N):
	tmp = 0
	if (math.log((N + 1.0)) - math.log(N)) <= 5e-6:
		tmp = (0.3333333333333333 / math.pow(N, 3.0)) + ((1.0 + (-0.5 / N)) / N)
	else:
		tmp = math.log(((N + 1.0) / N))
	return tmp

Julia

function code(N)
	tmp = 0.0
	if (Float64(log(Float64(N + 1.0)) - log(N)) <= 5e-6)
		tmp = Float64(Float64(0.3333333333333333 / (N ^ 3.0)) + Float64(Float64(1.0 + Float64(-0.5 / N)) / N));
	else
		tmp = log(Float64(Float64(N + 1.0) / N));
	end
	return tmp
end

MATLAB

function tmp_2 = code(N)
	tmp = 0.0;
	if ((log((N + 1.0)) - log(N)) <= 5e-6)
		tmp = (0.3333333333333333 / (N ^ 3.0)) + ((1.0 + (-0.5 / N)) / N);
	else
		tmp = log(((N + 1.0) / N));
	end
	tmp_2 = tmp;
end

Wolfram

code[N_] := If[LessEqual[N[(N[Log[N[(N + 1.0), $MachinePrecision]], $MachinePrecision] - N[Log[N], $MachinePrecision]), $MachinePrecision], 5e-6], N[(N[(0.3333333333333333 / N[Power[N, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 + N[(-0.5 / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision]), $MachinePrecision], N[Log[N[(N[(N + 1.0), $MachinePrecision] / N), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (log.f64 (+.f64 N 1)) (log.f64 N)) < 5.00000000000000041e-6

   1. Initial program 8.9%

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

      1. +-commutative 8.9%

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

      2. log1p-def 8.9%

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

   3. Simplified 8.9%

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

   4. Taylor expanded in N around inf 100.0%

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

      1. associate-*r/ 100.0%

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

      2. metadata-eval 100.0%

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

      3. +-commutative 100.0%

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

      4. associate-*r/ 100.0%

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

      5. metadata-eval 100.0%

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

      6. unpow2 100.0%

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

      7. associate-*r/ 100.0%

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

      8. metadata-eval 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in N around inf 100.0%

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

      1. +-commutative 100.0%

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

      2. associate-*r/ 100.0%

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

      3. metadata-eval 100.0%

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

      4. unpow2 100.0%

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

      5. associate--l+ 100.0%

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

      6. associate-*r/ 100.0%

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

      7. metadata-eval 100.0%

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

      8. associate-/r* 100.0%

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

      9. div-sub 100.0%

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

      10. sub-neg 100.0%

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

      11. distribute-neg-frac 100.0%

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

      12. metadata-eval 100.0%

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

   9. Simplified 100.0%

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

   if 5.00000000000000041e-6 < (-.f64 (log.f64 (+.f64 N 1)) (log.f64 N))

   1. Initial program 99.7%

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

      1. +-commutative 99.7%

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

      2. log1p-def 99.7%

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

   3. Simplified 99.7%

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

      1. log1p-udef 99.7%

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

      2. diff-log 99.8%

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

      3. +-commutative 99.8%

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

   5. Applied egg-rr 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\frac{0.3333333333333333}{{N}^{3}} + \frac{1 + \frac{-0.5}{N}}{N}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \end{array} \]

Alternative 3: 99.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;N \leq 330000:\\ \;\;\;\;-\log \left(\frac{N}{N + 1}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{0.5}{N}}{N}\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= N 330000.0) (- (log (/ N (+ N 1.0)))) (/ (- 1.0 (/ 0.5 N)) N)))

C

double code(double N) {
	double tmp;
	if (N <= 330000.0) {
		tmp = -log((N / (N + 1.0)));
	} else {
		tmp = (1.0 - (0.5 / N)) / N;
	}
	return tmp;
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= 330000.0d0) then
        tmp = -log((n / (n + 1.0d0)))
    else
        tmp = (1.0d0 - (0.5d0 / n)) / n
    end if
    code = tmp
end function

Java

public static double code(double N) {
	double tmp;
	if (N <= 330000.0) {
		tmp = -Math.log((N / (N + 1.0)));
	} else {
		tmp = (1.0 - (0.5 / N)) / N;
	}
	return tmp;
}

Python

def code(N):
	tmp = 0
	if N <= 330000.0:
		tmp = -math.log((N / (N + 1.0)))
	else:
		tmp = (1.0 - (0.5 / N)) / N
	return tmp

Julia

function code(N)
	tmp = 0.0
	if (N <= 330000.0)
		tmp = Float64(-log(Float64(N / Float64(N + 1.0))));
	else
		tmp = Float64(Float64(1.0 - Float64(0.5 / N)) / N);
	end
	return tmp
end

MATLAB

function tmp_2 = code(N)
	tmp = 0.0;
	if (N <= 330000.0)
		tmp = -log((N / (N + 1.0)));
	else
		tmp = (1.0 - (0.5 / N)) / N;
	end
	tmp_2 = tmp;
end

Wolfram

code[N_] := If[LessEqual[N, 330000.0], (-N[Log[N[(N / N[(N + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), N[(N[(1.0 - N[(0.5 / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if N < 3.3e5

   1. Initial program 99.0%

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

      1. +-commutative 99.0%

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

      2. log1p-def 99.0%

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

   3. Simplified 99.0%

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

      1. log1p-udef 99.0%

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

      2. diff-log 99.1%

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

      3. +-commutative 99.1%

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

   5. Applied egg-rr 99.1%

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

      1. clear-num 99.1%

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

      2. log-div 99.2%

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

      3. metadata-eval 99.2%

         \[\leadsto \color{blue}{0} - \log \left(\frac{N}{N + 1}\right) \]

   7. Applied egg-rr 99.2%

      \[\leadsto \color{blue}{0 - \log \left(\frac{N}{N + 1}\right)} \]
   8. Step-by-step derivation

      1. neg-sub0 99.2%

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

   9. Simplified 99.2%

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

   if 3.3e5 < N

   1. Initial program 7.6%

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

      1. +-commutative 7.6%

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

      2. log1p-def 7.6%

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

   3. Simplified 7.6%

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

   4. Taylor expanded in N around inf 100.0%

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

      1. associate-*r/ 100.0%

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

      2. metadata-eval 100.0%

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

      3. +-commutative 100.0%

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

      4. associate-*r/ 100.0%

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

      5. metadata-eval 100.0%

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

      6. unpow2 100.0%

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

      7. associate-*r/ 100.0%

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

      8. metadata-eval 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in N around inf 99.8%

      \[\leadsto \color{blue}{\frac{1}{N} - 0.5 \cdot \frac{1}{{N}^{2}}} \]
   8. Step-by-step derivation

      1. associate-*r/ 99.8%

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

      2. metadata-eval 99.8%

         \[\leadsto \frac{1}{N} - \frac{\color{blue}{0.5}}{{N}^{2}} \]

      3. unpow2 99.8%

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

      4. associate-/r* 99.8%

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

      5. div-sub 99.9%

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

   9. Simplified 99.9%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;N \leq 330000:\\ \;\;\;\;-\log \left(\frac{N}{N + 1}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{0.5}{N}}{N}\\ \end{array} \]

Alternative 4: 99.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;N \leq 190000:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{0.5}{N}}{N}\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= N 190000.0) (log (/ (+ N 1.0) N)) (/ (- 1.0 (/ 0.5 N)) N)))

C

double code(double N) {
	double tmp;
	if (N <= 190000.0) {
		tmp = log(((N + 1.0) / N));
	} else {
		tmp = (1.0 - (0.5 / N)) / N;
	}
	return tmp;
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= 190000.0d0) then
        tmp = log(((n + 1.0d0) / n))
    else
        tmp = (1.0d0 - (0.5d0 / n)) / n
    end if
    code = tmp
end function

Java

public static double code(double N) {
	double tmp;
	if (N <= 190000.0) {
		tmp = Math.log(((N + 1.0) / N));
	} else {
		tmp = (1.0 - (0.5 / N)) / N;
	}
	return tmp;
}

Python

def code(N):
	tmp = 0
	if N <= 190000.0:
		tmp = math.log(((N + 1.0) / N))
	else:
		tmp = (1.0 - (0.5 / N)) / N
	return tmp

Julia

function code(N)
	tmp = 0.0
	if (N <= 190000.0)
		tmp = log(Float64(Float64(N + 1.0) / N));
	else
		tmp = Float64(Float64(1.0 - Float64(0.5 / N)) / N);
	end
	return tmp
end

MATLAB

function tmp_2 = code(N)
	tmp = 0.0;
	if (N <= 190000.0)
		tmp = log(((N + 1.0) / N));
	else
		tmp = (1.0 - (0.5 / N)) / N;
	end
	tmp_2 = tmp;
end

Wolfram

code[N_] := If[LessEqual[N, 190000.0], N[Log[N[(N[(N + 1.0), $MachinePrecision] / N), $MachinePrecision]], $MachinePrecision], N[(N[(1.0 - N[(0.5 / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if N < 1.9e5

   1. Initial program 99.7%

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

      1. +-commutative 99.7%

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

      2. log1p-def 99.7%

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

   3. Simplified 99.7%

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

      1. log1p-udef 99.7%

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

      2. diff-log 99.8%

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

      3. +-commutative 99.8%

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

   5. Applied egg-rr 99.8%

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

   if 1.9e5 < N

   1. Initial program 8.9%

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

      1. +-commutative 8.9%

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

      2. log1p-def 8.9%

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

   3. Simplified 8.9%

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

   4. Taylor expanded in N around inf 100.0%

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

      1. associate-*r/ 100.0%

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

      2. metadata-eval 100.0%

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

      3. +-commutative 100.0%

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

      4. associate-*r/ 100.0%

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

      5. metadata-eval 100.0%

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

      6. unpow2 100.0%

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

      7. associate-*r/ 100.0%

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

      8. metadata-eval 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in N around inf 99.4%

      \[\leadsto \color{blue}{\frac{1}{N} - 0.5 \cdot \frac{1}{{N}^{2}}} \]
   8. Step-by-step derivation

      1. associate-*r/ 99.4%

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

      2. metadata-eval 99.4%

         \[\leadsto \frac{1}{N} - \frac{\color{blue}{0.5}}{{N}^{2}} \]

      3. unpow2 99.4%

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

      4. associate-/r* 99.4%

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

      5. div-sub 99.4%

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

   9. Simplified 99.4%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;N \leq 190000:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{0.5}{N}}{N}\\ \end{array} \]

Alternative 5: 98.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;N \leq 0.9:\\ \;\;\;\;N - \log N\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{0.5}{N}}{N}\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= N 0.9) (- N (log N)) (/ (- 1.0 (/ 0.5 N)) N)))

C

double code(double N) {
	double tmp;
	if (N <= 0.9) {
		tmp = N - log(N);
	} else {
		tmp = (1.0 - (0.5 / N)) / N;
	}
	return tmp;
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= 0.9d0) then
        tmp = n - log(n)
    else
        tmp = (1.0d0 - (0.5d0 / n)) / n
    end if
    code = tmp
end function

Java

public static double code(double N) {
	double tmp;
	if (N <= 0.9) {
		tmp = N - Math.log(N);
	} else {
		tmp = (1.0 - (0.5 / N)) / N;
	}
	return tmp;
}

Python

def code(N):
	tmp = 0
	if N <= 0.9:
		tmp = N - math.log(N)
	else:
		tmp = (1.0 - (0.5 / N)) / N
	return tmp

Julia

function code(N)
	tmp = 0.0
	if (N <= 0.9)
		tmp = Float64(N - log(N));
	else
		tmp = Float64(Float64(1.0 - Float64(0.5 / N)) / N);
	end
	return tmp
end

MATLAB

function tmp_2 = code(N)
	tmp = 0.0;
	if (N <= 0.9)
		tmp = N - log(N);
	else
		tmp = (1.0 - (0.5 / N)) / N;
	end
	tmp_2 = tmp;
end

Wolfram

code[N_] := If[LessEqual[N, 0.9], N[(N - N[Log[N], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(0.5 / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if N < 0.900000000000000022

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. log1p-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in N around 0 98.9%

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

      1. neg-mul-1 98.9%

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

      2. unsub-neg 98.9%

         \[\leadsto \color{blue}{N - \log N} \]

   6. Simplified 98.9%

      \[\leadsto \color{blue}{N - \log N} \]

   if 0.900000000000000022 < N

   1. Initial program 10.0%

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

      1. +-commutative 10.0%

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

      2. log1p-def 10.0%

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

   3. Simplified 10.0%

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

   4. Taylor expanded in N around inf 99.8%

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

      1. associate-*r/ 99.8%

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

      2. metadata-eval 99.8%

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

      3. +-commutative 99.8%

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

      4. associate-*r/ 99.8%

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

      5. metadata-eval 99.8%

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

      6. unpow2 99.8%

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

      7. associate-*r/ 99.8%

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

      8. metadata-eval 99.8%

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

   6. Simplified 99.8%

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

   7. Taylor expanded in N around inf 98.7%

      \[\leadsto \color{blue}{\frac{1}{N} - 0.5 \cdot \frac{1}{{N}^{2}}} \]
   8. Step-by-step derivation

      1. associate-*r/ 98.7%

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

      2. metadata-eval 98.7%

         \[\leadsto \frac{1}{N} - \frac{\color{blue}{0.5}}{{N}^{2}} \]

      3. unpow2 98.7%

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

      4. associate-/r* 98.7%

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

      5. div-sub 98.7%

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

   9. Simplified 98.7%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;N \leq 0.9:\\ \;\;\;\;N - \log N\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{0.5}{N}}{N}\\ \end{array} \]

Alternative 6: 98.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;N \leq 0.68:\\ \;\;\;\;-\log N\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{0.5}{N}}{N}\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= N 0.68) (- (log N)) (/ (- 1.0 (/ 0.5 N)) N)))

C

double code(double N) {
	double tmp;
	if (N <= 0.68) {
		tmp = -log(N);
	} else {
		tmp = (1.0 - (0.5 / N)) / N;
	}
	return tmp;
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= 0.68d0) then
        tmp = -log(n)
    else
        tmp = (1.0d0 - (0.5d0 / n)) / n
    end if
    code = tmp
end function

Java

public static double code(double N) {
	double tmp;
	if (N <= 0.68) {
		tmp = -Math.log(N);
	} else {
		tmp = (1.0 - (0.5 / N)) / N;
	}
	return tmp;
}

Python

def code(N):
	tmp = 0
	if N <= 0.68:
		tmp = -math.log(N)
	else:
		tmp = (1.0 - (0.5 / N)) / N
	return tmp

Julia

function code(N)
	tmp = 0.0
	if (N <= 0.68)
		tmp = Float64(-log(N));
	else
		tmp = Float64(Float64(1.0 - Float64(0.5 / N)) / N);
	end
	return tmp
end

MATLAB

function tmp_2 = code(N)
	tmp = 0.0;
	if (N <= 0.68)
		tmp = -log(N);
	else
		tmp = (1.0 - (0.5 / N)) / N;
	end
	tmp_2 = tmp;
end

Wolfram

code[N_] := If[LessEqual[N, 0.68], (-N[Log[N], $MachinePrecision]), N[(N[(1.0 - N[(0.5 / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if N < 0.680000000000000049

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. log1p-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in N around 0 98.0%

      \[\leadsto \color{blue}{-1 \cdot \log N} \]
   5. Step-by-step derivation

      1. neg-mul-1 98.0%

         \[\leadsto \color{blue}{-\log N} \]

   6. Simplified 98.0%

      \[\leadsto \color{blue}{-\log N} \]

   if 0.680000000000000049 < N

   1. Initial program 10.0%

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

      1. +-commutative 10.0%

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

      2. log1p-def 10.0%

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

   3. Simplified 10.0%

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

   4. Taylor expanded in N around inf 99.8%

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

      1. associate-*r/ 99.8%

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

      2. metadata-eval 99.8%

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

      3. +-commutative 99.8%

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

      4. associate-*r/ 99.8%

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

      5. metadata-eval 99.8%

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

      6. unpow2 99.8%

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

      7. associate-*r/ 99.8%

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

      8. metadata-eval 99.8%

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

   6. Simplified 99.8%

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

   7. Taylor expanded in N around inf 98.7%

      \[\leadsto \color{blue}{\frac{1}{N} - 0.5 \cdot \frac{1}{{N}^{2}}} \]
   8. Step-by-step derivation

      1. associate-*r/ 98.7%

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

      2. metadata-eval 98.7%

         \[\leadsto \frac{1}{N} - \frac{\color{blue}{0.5}}{{N}^{2}} \]

      3. unpow2 98.7%

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

      4. associate-/r* 98.7%

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

      5. div-sub 98.7%

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

   9. Simplified 98.7%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;N \leq 0.68:\\ \;\;\;\;-\log N\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{0.5}{N}}{N}\\ \end{array} \]

Alternative 7: 52.8% accurate, 12.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;N \leq 60000000:\\ \;\;\;\;\frac{\frac{\frac{1}{N}}{N}}{\frac{1}{N} + \frac{0.5}{N \cdot N}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{0.5}{N}}{N}\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= N 60000000.0)
   (/ (/ (/ 1.0 N) N) (+ (/ 1.0 N) (/ 0.5 (* N N))))
   (/ (- 1.0 (/ 0.5 N)) N)))

C

double code(double N) {
	double tmp;
	if (N <= 60000000.0) {
		tmp = ((1.0 / N) / N) / ((1.0 / N) + (0.5 / (N * N)));
	} else {
		tmp = (1.0 - (0.5 / N)) / N;
	}
	return tmp;
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= 60000000.0d0) then
        tmp = ((1.0d0 / n) / n) / ((1.0d0 / n) + (0.5d0 / (n * n)))
    else
        tmp = (1.0d0 - (0.5d0 / n)) / n
    end if
    code = tmp
end function

Java

public static double code(double N) {
	double tmp;
	if (N <= 60000000.0) {
		tmp = ((1.0 / N) / N) / ((1.0 / N) + (0.5 / (N * N)));
	} else {
		tmp = (1.0 - (0.5 / N)) / N;
	}
	return tmp;
}

Python

def code(N):
	tmp = 0
	if N <= 60000000.0:
		tmp = ((1.0 / N) / N) / ((1.0 / N) + (0.5 / (N * N)))
	else:
		tmp = (1.0 - (0.5 / N)) / N
	return tmp

Julia

function code(N)
	tmp = 0.0
	if (N <= 60000000.0)
		tmp = Float64(Float64(Float64(1.0 / N) / N) / Float64(Float64(1.0 / N) + Float64(0.5 / Float64(N * N))));
	else
		tmp = Float64(Float64(1.0 - Float64(0.5 / N)) / N);
	end
	return tmp
end

MATLAB

function tmp_2 = code(N)
	tmp = 0.0;
	if (N <= 60000000.0)
		tmp = ((1.0 / N) / N) / ((1.0 / N) + (0.5 / (N * N)));
	else
		tmp = (1.0 - (0.5 / N)) / N;
	end
	tmp_2 = tmp;
end

Wolfram

code[N_] := If[LessEqual[N, 60000000.0], N[(N[(N[(1.0 / N), $MachinePrecision] / N), $MachinePrecision] / N[(N[(1.0 / N), $MachinePrecision] + N[(0.5 / N[(N * N), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(0.5 / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if N < 6e7

   1. Initial program 98.7%

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

      1. +-commutative 98.7%

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

      2. log1p-def 98.7%

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

   3. Simplified 98.7%

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

   4. Taylor expanded in N around inf 4.3%

      \[\leadsto \color{blue}{\frac{1}{N} - 0.5 \cdot \frac{1}{{N}^{2}}} \]
   5. Step-by-step derivation

      1. associate-*r/ 4.3%

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

      2. metadata-eval 4.3%

         \[\leadsto \frac{1}{N} - \frac{\color{blue}{0.5}}{{N}^{2}} \]

      3. unpow2 4.3%

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

      4. associate-/r* 4.3%

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

   6. Simplified 4.3%

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

      1. flip-- 4.0%

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

      2. associate-/r* 4.0%

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

      3. associate-/r* 4.0%

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

      4. associate-/r* 4.0%

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

   8. Applied egg-rr 4.0%

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

      1. difference-of-squares 4.0%

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

      2. associate-/r* 4.0%

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

      3. div-sub 4.0%

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

   10. Simplified 4.0%

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

   11. Taylor expanded in N around inf 10.5%

       \[\leadsto \frac{\color{blue}{\frac{1}{{N}^{2}}}}{\frac{1}{N} + \frac{0.5}{N \cdot N}} \]
   12. Step-by-step derivation

       1. unpow2 10.5%

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

       2. associate-/r* 10.5%

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

   13. Simplified 10.5%

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

   if 6e7 < N

   1. Initial program 7.2%

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

      1. +-commutative 7.2%

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

      2. log1p-def 7.2%

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

   3. Simplified 7.2%

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

   4. Taylor expanded in N around inf 100.0%

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

      1. associate-*r/ 100.0%

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

      2. metadata-eval 100.0%

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

      3. +-commutative 100.0%

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

      4. associate-*r/ 100.0%

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

      5. metadata-eval 100.0%

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

      6. unpow2 100.0%

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

      7. associate-*r/ 100.0%

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

      8. metadata-eval 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in N around inf 100.0%

      \[\leadsto \color{blue}{\frac{1}{N} - 0.5 \cdot \frac{1}{{N}^{2}}} \]
   8. Step-by-step derivation

      1. associate-*r/ 100.0%

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

      2. metadata-eval 100.0%

         \[\leadsto \frac{1}{N} - \frac{\color{blue}{0.5}}{{N}^{2}} \]

      3. unpow2 100.0%

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

      4. associate-/r* 100.0%

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

      5. div-sub 100.0%

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

   9. Simplified 100.0%

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

4. Final simplification 58.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;N \leq 60000000:\\ \;\;\;\;\frac{\frac{\frac{1}{N}}{N}}{\frac{1}{N} + \frac{0.5}{N \cdot N}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{0.5}{N}}{N}\\ \end{array} \]

Alternative 8: 51.2% 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 49.4%

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

   1. +-commutative 49.4%

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

   2. log1p-def 49.4%

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

3. Simplified 49.4%

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

4. Taylor expanded in N around inf 56.7%

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

5. Final simplification 56.7%

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

Reproduce

herbie shell --seed 2023187 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1.0)) (log N)))