2log (problem 3.3.6)

Percentage Accurate: 54.0% → 99.9%

Time: 6.4s

Alternatives: 7

Speedup: 41.0×

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: 54.0% 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: 99.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 10^{-6}:\\ \;\;\;\;\frac{1}{N} + \left(\frac{0.3333333333333333}{{N}^{3}} + \frac{-0.5}{{N}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= (- (log (+ N 1.0)) (log N)) 1e-6)
   (+ (/ 1.0 N) (+ (/ 0.3333333333333333 (pow N 3.0)) (/ -0.5 (pow N 2.0))))
   (log (/ (+ N 1.0) N))))

C

double code(double N) {
	double tmp;
	if ((log((N + 1.0)) - log(N)) <= 1e-6) {
		tmp = (1.0 / N) + ((0.3333333333333333 / pow(N, 3.0)) + (-0.5 / pow(N, 2.0)));
	} 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)) <= 1d-6) then
        tmp = (1.0d0 / n) + ((0.3333333333333333d0 / (n ** 3.0d0)) + ((-0.5d0) / (n ** 2.0d0)))
    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)) <= 1e-6) {
		tmp = (1.0 / N) + ((0.3333333333333333 / Math.pow(N, 3.0)) + (-0.5 / Math.pow(N, 2.0)));
	} 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)) <= 1e-6:
		tmp = (1.0 / N) + ((0.3333333333333333 / math.pow(N, 3.0)) + (-0.5 / math.pow(N, 2.0)))
	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)) <= 1e-6)
		tmp = Float64(Float64(1.0 / N) + Float64(Float64(0.3333333333333333 / (N ^ 3.0)) + Float64(-0.5 / (N ^ 2.0))));
	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)) <= 1e-6)
		tmp = (1.0 / N) + ((0.3333333333333333 / (N ^ 3.0)) + (-0.5 / (N ^ 2.0)));
	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], 1e-6], 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 + 1.0), $MachinePrecision] / N), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (log.f64 (+.f64 N 1)) (log.f64 N)) < 9.99999999999999955e-7

   1. Initial program 6.4%

      \[\log \left(N + 1\right) - \log N \]

   2. Taylor expanded in N around inf 100.0%

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

      1. sub-neg 100.0%

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

      2. +-commutative 100.0%

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

      3. associate-+l+ 100.0%

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

      4. associate-*r/ 100.0%

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

      5. metadata-eval 100.0%

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

      6. associate-*r/ 100.0%

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

      7. metadata-eval 100.0%

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

      8. distribute-neg-frac 100.0%

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

      9. metadata-eval 100.0%

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

   4. Simplified 100.0%

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

   if 9.99999999999999955e-7 < (-.f64 (log.f64 (+.f64 N 1)) (log.f64 N))

   1. Initial program 100.0%

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

      1. diff-log 100.0%

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

   3. Applied egg-rr 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 10^{-6}:\\ \;\;\;\;\frac{1}{N} + \left(\frac{0.3333333333333333}{{N}^{3}} + \frac{-0.5}{{N}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \end{array} \]

Alternative 2: 99.8% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (N)
 :precision binary64
 (if (<= (- (log (+ N 1.0)) (log N)) 1e-6)
   (/ 1.0 (+ N 0.5))
   (log (/ (+ N 1.0) N))))

C

double code(double N) {
	double tmp;
	if ((log((N + 1.0)) - log(N)) <= 1e-6) {
		tmp = 1.0 / (N + 0.5);
	} 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)) <= 1d-6) then
        tmp = 1.0d0 / (n + 0.5d0)
    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)) <= 1e-6) {
		tmp = 1.0 / (N + 0.5);
	} 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)) <= 1e-6:
		tmp = 1.0 / (N + 0.5)
	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)) <= 1e-6)
		tmp = Float64(1.0 / Float64(N + 0.5));
	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)) <= 1e-6)
		tmp = 1.0 / (N + 0.5);
	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], 1e-6], N[(1.0 / N[(N + 0.5), $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)) < 9.99999999999999955e-7

   1. Initial program 6.4%

      \[\log \left(N + 1\right) - \log N \]

   2. Taylor expanded in N around inf 99.9%

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

      1. associate-*r/ 99.9%

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

      2. metadata-eval 99.9%

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

   4. Simplified 99.9%

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

      1. frac-sub 30.6%

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

      2. *-un-lft-identity 30.6%

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

      3. unpow2 30.6%

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

      4. distribute-lft-out-- 30.6%

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

      5. unpow2 30.6%

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

      6. cube-mult 30.5%

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

   6. Applied egg-rr 30.5%

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

      1. associate-/l* 34.2%

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

      2. sub-neg 34.2%

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

      3. metadata-eval 34.2%

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

   8. Simplified 34.2%

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

   9. Taylor expanded in N around inf 50.0%

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

       1. unpow2 50.0%

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

       2. distribute-rgt-out 50.0%

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

       3. +-commutative 50.0%

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

   11. Simplified 50.0%

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

       1. add-log-exp 6.4%

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

       2. *-un-lft-identity 6.4%

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

       3. log-prod 6.4%

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

       4. metadata-eval 6.4%

          \[\leadsto \color{blue}{0} + \log \left(e^{\frac{N}{N \cdot \left(N + 0.5\right)}}\right) \]

       5. add-log-exp 50.0%

          \[\leadsto 0 + \color{blue}{\frac{N}{N \cdot \left(N + 0.5\right)}} \]

       6. associate-/r* 99.9%

          \[\leadsto 0 + \color{blue}{\frac{\frac{N}{N}}{N + 0.5}} \]

       7. *-inverses 99.9%

          \[\leadsto 0 + \frac{\color{blue}{1}}{N + 0.5} \]

   13. Applied egg-rr 99.9%

       \[\leadsto \color{blue}{0 + \frac{1}{N + 0.5}} \]
   14. Step-by-step derivation

       1. +-lft-identity 99.9%

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

   15. Simplified 99.9%

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

   if 9.99999999999999955e-7 < (-.f64 (log.f64 (+.f64 N 1)) (log.f64 N))

   1. Initial program 100.0%

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

      1. diff-log 100.0%

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

   3. Applied egg-rr 100.0%

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

4. Final simplification 100.0%

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

Alternative 3: 99.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;N \leq 0.6:\\ \;\;\;\;N - \log N\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N + 0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= N 0.6) (- N (log N)) (/ 1.0 (+ N 0.5))))

C

double code(double N) {
	double tmp;
	if (N <= 0.6) {
		tmp = N - log(N);
	} else {
		tmp = 1.0 / (N + 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= 0.6d0) then
        tmp = n - log(n)
    else
        tmp = 1.0d0 / (n + 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double N) {
	double tmp;
	if (N <= 0.6) {
		tmp = N - Math.log(N);
	} else {
		tmp = 1.0 / (N + 0.5);
	}
	return tmp;
}

Python

def code(N):
	tmp = 0
	if N <= 0.6:
		tmp = N - math.log(N)
	else:
		tmp = 1.0 / (N + 0.5)
	return tmp

Julia

function code(N)
	tmp = 0.0
	if (N <= 0.6)
		tmp = Float64(N - log(N));
	else
		tmp = Float64(1.0 / Float64(N + 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(N)
	tmp = 0.0;
	if (N <= 0.6)
		tmp = N - log(N);
	else
		tmp = 1.0 / (N + 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[N_] := If[LessEqual[N, 0.6], N[(N - N[Log[N], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N + 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if N < 0.599999999999999978

   1. Initial program 100.0%

      \[\log \left(N + 1\right) - \log N \]

   2. Taylor expanded in N around 0 99.5%

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

      1. neg-mul-1 99.5%

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

      2. unsub-neg 99.5%

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

   4. Simplified 99.5%

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

   if 0.599999999999999978 < N

   1. Initial program 7.1%

      \[\log \left(N + 1\right) - \log N \]

   2. Taylor expanded in N around inf 99.3%

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

      1. associate-*r/ 99.3%

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

      2. metadata-eval 99.3%

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

   4. Simplified 99.3%

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

      1. frac-sub 30.5%

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

      2. *-un-lft-identity 30.5%

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

      3. unpow2 30.5%

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

      4. distribute-lft-out-- 30.5%

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

      5. unpow2 30.5%

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

      6. cube-mult 30.5%

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

   6. Applied egg-rr 30.5%

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

      1. associate-/l* 34.1%

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

      2. sub-neg 34.1%

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

      3. metadata-eval 34.1%

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

   8. Simplified 34.1%

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

   9. Taylor expanded in N around inf 49.8%

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

       1. unpow2 49.8%

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

       2. distribute-rgt-out 49.8%

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

       3. +-commutative 49.8%

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

   11. Simplified 49.8%

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

       1. add-log-exp 6.6%

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

       2. *-un-lft-identity 6.6%

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

       3. log-prod 6.6%

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

       4. metadata-eval 6.6%

          \[\leadsto \color{blue}{0} + \log \left(e^{\frac{N}{N \cdot \left(N + 0.5\right)}}\right) \]

       5. add-log-exp 49.8%

          \[\leadsto 0 + \color{blue}{\frac{N}{N \cdot \left(N + 0.5\right)}} \]

       6. associate-/r* 99.4%

          \[\leadsto 0 + \color{blue}{\frac{\frac{N}{N}}{N + 0.5}} \]

       7. *-inverses 99.4%

          \[\leadsto 0 + \frac{\color{blue}{1}}{N + 0.5} \]

   13. Applied egg-rr 99.4%

       \[\leadsto \color{blue}{0 + \frac{1}{N + 0.5}} \]
   14. Step-by-step derivation

       1. +-lft-identity 99.4%

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

   15. Simplified 99.4%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;N \leq 0.6:\\ \;\;\;\;N - \log N\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N + 0.5}\\ \end{array} \]

Alternative 4: 98.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;N \leq 0.27:\\ \;\;\;\;-\log N\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N + 0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (N) :precision binary64 (if (<= N 0.27) (- (log N)) (/ 1.0 (+ N 0.5))))

C

double code(double N) {
	double tmp;
	if (N <= 0.27) {
		tmp = -log(N);
	} else {
		tmp = 1.0 / (N + 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= 0.27d0) then
        tmp = -log(n)
    else
        tmp = 1.0d0 / (n + 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double N) {
	double tmp;
	if (N <= 0.27) {
		tmp = -Math.log(N);
	} else {
		tmp = 1.0 / (N + 0.5);
	}
	return tmp;
}

Python

def code(N):
	tmp = 0
	if N <= 0.27:
		tmp = -math.log(N)
	else:
		tmp = 1.0 / (N + 0.5)
	return tmp

Julia

function code(N)
	tmp = 0.0
	if (N <= 0.27)
		tmp = Float64(-log(N));
	else
		tmp = Float64(1.0 / Float64(N + 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(N)
	tmp = 0.0;
	if (N <= 0.27)
		tmp = -log(N);
	else
		tmp = 1.0 / (N + 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[N_] := If[LessEqual[N, 0.27], (-N[Log[N], $MachinePrecision]), N[(1.0 / N[(N + 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if N < 0.27000000000000002

   1. Initial program 100.0%

      \[\log \left(N + 1\right) - \log N \]

   2. Taylor expanded in N around 0 98.3%

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

      1. neg-mul-1 98.3%

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

   4. Simplified 98.3%

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

   if 0.27000000000000002 < N

   1. Initial program 7.1%

      \[\log \left(N + 1\right) - \log N \]

   2. Taylor expanded in N around inf 99.3%

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

      1. associate-*r/ 99.3%

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

      2. metadata-eval 99.3%

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

   4. Simplified 99.3%

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

      1. frac-sub 30.5%

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

      2. *-un-lft-identity 30.5%

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

      3. unpow2 30.5%

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

      4. distribute-lft-out-- 30.5%

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

      5. unpow2 30.5%

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

      6. cube-mult 30.5%

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

   6. Applied egg-rr 30.5%

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

      1. associate-/l* 34.1%

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

      2. sub-neg 34.1%

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

      3. metadata-eval 34.1%

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

   8. Simplified 34.1%

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

   9. Taylor expanded in N around inf 49.8%

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

       1. unpow2 49.8%

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

       2. distribute-rgt-out 49.8%

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

       3. +-commutative 49.8%

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

   11. Simplified 49.8%

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

       1. add-log-exp 6.6%

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

       2. *-un-lft-identity 6.6%

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

       3. log-prod 6.6%

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

       4. metadata-eval 6.6%

          \[\leadsto \color{blue}{0} + \log \left(e^{\frac{N}{N \cdot \left(N + 0.5\right)}}\right) \]

       5. add-log-exp 49.8%

          \[\leadsto 0 + \color{blue}{\frac{N}{N \cdot \left(N + 0.5\right)}} \]

       6. associate-/r* 99.4%

          \[\leadsto 0 + \color{blue}{\frac{\frac{N}{N}}{N + 0.5}} \]

       7. *-inverses 99.4%

          \[\leadsto 0 + \frac{\color{blue}{1}}{N + 0.5} \]

   13. Applied egg-rr 99.4%

       \[\leadsto \color{blue}{0 + \frac{1}{N + 0.5}} \]
   14. Step-by-step derivation

       1. +-lft-identity 99.4%

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

   15. Simplified 99.4%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;N \leq 0.27:\\ \;\;\;\;-\log N\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N + 0.5}\\ \end{array} \]

Alternative 5: 56.2% accurate, 40.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;N \leq 0.5:\\ \;\;\;\;2\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N}\\ \end{array} \end{array} \]

FPCore

(FPCore (N) :precision binary64 (if (<= N 0.5) 2.0 (/ 1.0 N)))

C

double code(double N) {
	double tmp;
	if (N <= 0.5) {
		tmp = 2.0;
	} else {
		tmp = 1.0 / N;
	}
	return tmp;
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= 0.5d0) then
        tmp = 2.0d0
    else
        tmp = 1.0d0 / n
    end if
    code = tmp
end function

Java

public static double code(double N) {
	double tmp;
	if (N <= 0.5) {
		tmp = 2.0;
	} else {
		tmp = 1.0 / N;
	}
	return tmp;
}

Python

def code(N):
	tmp = 0
	if N <= 0.5:
		tmp = 2.0
	else:
		tmp = 1.0 / N
	return tmp

Julia

function code(N)
	tmp = 0.0
	if (N <= 0.5)
		tmp = 2.0;
	else
		tmp = Float64(1.0 / N);
	end
	return tmp
end

MATLAB

function tmp_2 = code(N)
	tmp = 0.0;
	if (N <= 0.5)
		tmp = 2.0;
	else
		tmp = 1.0 / N;
	end
	tmp_2 = tmp;
end

Wolfram

code[N_] := If[LessEqual[N, 0.5], 2.0, N[(1.0 / N), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if N < 0.5

   1. Initial program 100.0%

      \[\log \left(N + 1\right) - \log N \]

   2. Taylor expanded in N around inf 0.9%

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

      1. associate-*r/ 0.9%

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

      2. metadata-eval 0.9%

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

   4. Simplified 0.9%

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

      1. frac-sub 0.9%

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

      2. *-un-lft-identity 0.9%

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

      3. unpow2 0.9%

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

      4. distribute-lft-out-- 0.9%

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

      5. unpow2 0.9%

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

      6. cube-mult 0.9%

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

   6. Applied egg-rr 0.9%

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

      1. associate-/l* 0.9%

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

      2. sub-neg 0.9%

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

      3. metadata-eval 0.9%

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

   8. Simplified 0.9%

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

   9. Taylor expanded in N around inf 14.4%

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

       1. unpow2 14.4%

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

       2. distribute-rgt-out 14.4%

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

       3. +-commutative 14.4%

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

   11. Simplified 14.4%

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

   12. Taylor expanded in N around 0 14.4%

       \[\leadsto \color{blue}{2} \]

   if 0.5 < N

   1. Initial program 7.1%

      \[\log \left(N + 1\right) - \log N \]

   2. Taylor expanded in N around inf 98.8%

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

4. Final simplification 58.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;N \leq 0.5:\\ \;\;\;\;2\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N}\\ \end{array} \]

Alternative 6: 56.9% accurate, 41.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.4%

   \[\log \left(N + 1\right) - \log N \]

2. Taylor expanded in N around inf 52.4%

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

   1. associate-*r/ 52.4%

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

   2. metadata-eval 52.4%

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

4. Simplified 52.4%

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

   1. frac-sub 16.4%

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

   2. *-un-lft-identity 16.4%

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

   3. unpow2 16.4%

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

   4. distribute-lft-out-- 16.4%

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

   5. unpow2 16.4%

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

   6. cube-mult 16.4%

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

6. Applied egg-rr 16.4%

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

   1. associate-/l* 18.3%

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

   2. sub-neg 18.3%

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

   3. metadata-eval 18.3%

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

8. Simplified 18.3%

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

9. Taylor expanded in N around inf 32.9%

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

    1. unpow2 32.9%

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

    2. distribute-rgt-out 32.9%

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

    3. +-commutative 32.9%

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

11. Simplified 32.9%

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

    1. add-log-exp 10.3%

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

    2. *-un-lft-identity 10.3%

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

    3. log-prod 10.3%

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

    4. metadata-eval 10.3%

       \[\leadsto \color{blue}{0} + \log \left(e^{\frac{N}{N \cdot \left(N + 0.5\right)}}\right) \]

    5. add-log-exp 32.9%

       \[\leadsto 0 + \color{blue}{\frac{N}{N \cdot \left(N + 0.5\right)}} \]

    6. associate-/r* 58.9%

       \[\leadsto 0 + \color{blue}{\frac{\frac{N}{N}}{N + 0.5}} \]

    7. *-inverses 58.9%

       \[\leadsto 0 + \frac{\color{blue}{1}}{N + 0.5} \]

13. Applied egg-rr 58.9%

    \[\leadsto \color{blue}{0 + \frac{1}{N + 0.5}} \]
14. Step-by-step derivation

    1. +-lft-identity 58.9%

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

15. Simplified 58.9%

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

16. Final simplification 58.9%

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

Alternative 7: 9.9% accurate, 205.0× speedup

Math

\[\begin{array}{l} \\ 2 \end{array} \]

FPCore

(FPCore (N) :precision binary64 2.0)

C

double code(double N) {
	return 2.0;
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    code = 2.0d0
end function

Java

public static double code(double N) {
	return 2.0;
}

Python

def code(N):
	return 2.0

Julia

function code(N)
	return 2.0
end

MATLAB

function tmp = code(N)
	tmp = 2.0;
end

Wolfram

code[N_] := 2.0

Derivation

1. Initial program 51.4%

   \[\log \left(N + 1\right) - \log N \]

2. Taylor expanded in N around inf 52.4%

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

   1. associate-*r/ 52.4%

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

   2. metadata-eval 52.4%

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

4. Simplified 52.4%

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

   1. frac-sub 16.4%

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

   2. *-un-lft-identity 16.4%

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

   3. unpow2 16.4%

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

   4. distribute-lft-out-- 16.4%

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

   5. unpow2 16.4%

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

   6. cube-mult 16.4%

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

6. Applied egg-rr 16.4%

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

   1. associate-/l* 18.3%

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

   2. sub-neg 18.3%

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

   3. metadata-eval 18.3%

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

8. Simplified 18.3%

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

9. Taylor expanded in N around inf 32.9%

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

    1. unpow2 32.9%

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

    2. distribute-rgt-out 32.9%

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

    3. +-commutative 32.9%

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

11. Simplified 32.9%

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

12. Taylor expanded in N around 0 9.5%

    \[\leadsto \color{blue}{2} \]

13. Final simplification 9.5%

    \[\leadsto 2 \]

Reproduce

herbie shell --seed 2023333 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1.0)) (log N)))