2log (problem 3.3.6)

Percentage Accurate: 23.7% → 99.5%

Time: 8.4s

Alternatives: 11

Speedup: 68.3×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 23.7% 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: 99.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 0.0006:\\ \;\;\;\;\frac{1}{\frac{N}{1 + \frac{\frac{0.3333333333333333 + \frac{-0.25}{N}}{N} - 0.5}{N}}}\\ \mathbf{else}:\\ \;\;\;\;-\log \left(\frac{N}{N + 1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= (- (log (+ N 1.0)) (log N)) 0.0006)
   (/ 1.0 (/ N (+ 1.0 (/ (- (/ (+ 0.3333333333333333 (/ -0.25 N)) N) 0.5) N))))
   (- (log (/ N (+ N 1.0))))))

C

double code(double N) {
	double tmp;
	if ((log((N + 1.0)) - log(N)) <= 0.0006) {
		tmp = 1.0 / (N / (1.0 + ((((0.3333333333333333 + (-0.25 / N)) / N) - 0.5) / 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)) <= 0.0006d0) then
        tmp = 1.0d0 / (n / (1.0d0 + ((((0.3333333333333333d0 + ((-0.25d0) / n)) / n) - 0.5d0) / 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)) <= 0.0006) {
		tmp = 1.0 / (N / (1.0 + ((((0.3333333333333333 + (-0.25 / N)) / N) - 0.5) / 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)) <= 0.0006:
		tmp = 1.0 / (N / (1.0 + ((((0.3333333333333333 + (-0.25 / N)) / N) - 0.5) / 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)) <= 0.0006)
		tmp = Float64(1.0 / Float64(N / Float64(1.0 + Float64(Float64(Float64(Float64(0.3333333333333333 + Float64(-0.25 / N)) / N) - 0.5) / 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)) <= 0.0006)
		tmp = 1.0 / (N / (1.0 + ((((0.3333333333333333 + (-0.25 / N)) / N) - 0.5) / 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], 0.0006], N[(1.0 / N[(N / N[(1.0 + N[(N[(N[(N[(0.3333333333333333 + N[(-0.25 / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision] - 0.5), $MachinePrecision] / N), $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 #s(literal 1 binary64))) (log.f64 N)) < 5.99999999999999947e-4

   1. Initial program 18.0%

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

      1. +-commutative 18.0%

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

      2. log1p-define 18.0%

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

   3. Simplified 18.0%

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

   5. Taylor expanded in N around -inf 99.8%

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

      1. mul-1-neg 99.8%

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

      2. distribute-neg-frac2 99.8%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in N around inf 99.7%

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

      1. associate-*r/ 99.7%

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

      2. metadata-eval 99.7%

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

      3. associate-*r/ 99.7%

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

      4. metadata-eval 99.7%

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

   10. Simplified 99.7%

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

       1. clear-num 99.8%

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

       2. inv-pow 99.8%

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

       3. +-commutative 99.8%

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

       4. div-inv 99.8%

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

       5. fma-define 99.8%

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

       6. pow-flip 99.8%

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

       7. metadata-eval 99.8%

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

       8. +-commutative 99.8%

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

       9. div-inv 99.8%

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

       10. fma-define 99.8%

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

       11. pow-flip 99.8%

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

       12. metadata-eval 99.8%

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

   12. Applied egg-rr 99.8%

       \[\leadsto \color{blue}{{\left(\frac{N}{\mathsf{fma}\left(0.3333333333333333, {N}^{-2}, 1\right) - \mathsf{fma}\left(0.25, {N}^{-3}, \frac{0.5}{N}\right)}\right)}^{-1}} \]
   13. Step-by-step derivation

       1. unpow-1 99.8%

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

   14. Simplified 99.8%

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

   15. Taylor expanded in N around -inf 99.8%

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

       1. mul-1-neg 99.8%

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

       2. unsub-neg 99.8%

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

       3. mul-1-neg 99.8%

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

       4. unsub-neg 99.8%

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

       5. sub-neg 99.8%

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

       6. associate-*r/ 99.8%

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

       7. metadata-eval 99.8%

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

       8. distribute-neg-frac 99.8%

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

       9. metadata-eval 99.8%

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

   17. Simplified 99.8%

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

   if 5.99999999999999947e-4 < (-.f64 (log.f64 (+.f64 N #s(literal 1 binary64))) (log.f64 N))

   1. Initial program 90.5%

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

      1. +-commutative 90.5%

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

      2. log1p-define 90.6%

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

   3. Simplified 90.6%

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

      1. log1p-expm1-u 90.6%

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

      2. expm1-undefine 90.8%

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

      3. exp-diff 90.3%

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

      4. log1p-undefine 90.0%

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

      5. rem-exp-log 91.7%

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

      6. add-exp-log 92.9%

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

      7. +-commutative 92.9%

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

   6. Applied egg-rr 92.9%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\frac{N + 1}{N} - 1\right)} \]
   7. Step-by-step derivation

      1. add-exp-log 92.9%

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

      2. expm1-define 92.9%

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

      3. log1p-expm1-u 93.0%

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

      4. clear-num 92.9%

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

      5. log-div 94.3%

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

      6. metadata-eval 94.3%

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

   8. Applied egg-rr 94.3%

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

      1. neg-sub0 94.3%

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

   10. Simplified 94.3%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 0.0006:\\ \;\;\;\;\frac{1}{\frac{N}{1 + \frac{\frac{0.3333333333333333 + \frac{-0.25}{N}}{N} - 0.5}{N}}}\\ \mathbf{else}:\\ \;\;\;\;-\log \left(\frac{N}{N + 1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;N \leq 1300:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{N}{1 + \frac{\frac{0.3333333333333333 + \frac{-0.25}{N}}{N} - 0.5}{N}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= N 1300.0)
   (log (/ (+ N 1.0) N))
   (/
    1.0
    (/ N (+ 1.0 (/ (- (/ (+ 0.3333333333333333 (/ -0.25 N)) N) 0.5) N))))))

C

double code(double N) {
	double tmp;
	if (N <= 1300.0) {
		tmp = log(((N + 1.0) / N));
	} else {
		tmp = 1.0 / (N / (1.0 + ((((0.3333333333333333 + (-0.25 / N)) / N) - 0.5) / N)));
	}
	return tmp;
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= 1300.0d0) then
        tmp = log(((n + 1.0d0) / n))
    else
        tmp = 1.0d0 / (n / (1.0d0 + ((((0.3333333333333333d0 + ((-0.25d0) / n)) / n) - 0.5d0) / n)))
    end if
    code = tmp
end function

Java

public static double code(double N) {
	double tmp;
	if (N <= 1300.0) {
		tmp = Math.log(((N + 1.0) / N));
	} else {
		tmp = 1.0 / (N / (1.0 + ((((0.3333333333333333 + (-0.25 / N)) / N) - 0.5) / N)));
	}
	return tmp;
}

Python

def code(N):
	tmp = 0
	if N <= 1300.0:
		tmp = math.log(((N + 1.0) / N))
	else:
		tmp = 1.0 / (N / (1.0 + ((((0.3333333333333333 + (-0.25 / N)) / N) - 0.5) / N)))
	return tmp

Julia

function code(N)
	tmp = 0.0
	if (N <= 1300.0)
		tmp = log(Float64(Float64(N + 1.0) / N));
	else
		tmp = Float64(1.0 / Float64(N / Float64(1.0 + Float64(Float64(Float64(Float64(0.3333333333333333 + Float64(-0.25 / N)) / N) - 0.5) / N))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(N)
	tmp = 0.0;
	if (N <= 1300.0)
		tmp = log(((N + 1.0) / N));
	else
		tmp = 1.0 / (N / (1.0 + ((((0.3333333333333333 + (-0.25 / N)) / N) - 0.5) / N)));
	end
	tmp_2 = tmp;
end

Wolfram

code[N_] := If[LessEqual[N, 1300.0], N[Log[N[(N[(N + 1.0), $MachinePrecision] / N), $MachinePrecision]], $MachinePrecision], N[(1.0 / N[(N / N[(1.0 + N[(N[(N[(N[(0.3333333333333333 + N[(-0.25 / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision] - 0.5), $MachinePrecision] / N), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if N < 1300

   1. Initial program 90.8%

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

      1. +-commutative 90.8%

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

      2. log1p-define 90.9%

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

   3. Simplified 90.9%

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

      1. add-log-exp 90.9%

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

      2. log1p-expm1-u 90.9%

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

      3. log1p-undefine 90.9%

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

      4. diff-log 90.9%

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

      5. log1p-undefine 90.6%

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

      6. rem-exp-log 91.8%

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

      7. +-commutative 91.8%

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

      8. add-exp-log 92.2%

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

      9. log1p-undefine 92.2%

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

      10. log1p-expm1-u 92.2%

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

      11. add-exp-log 93.4%

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

   6. Applied egg-rr 93.4%

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

   if 1300 < N

   1. Initial program 18.3%

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

      1. +-commutative 18.3%

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

      2. log1p-define 18.3%

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

   3. Simplified 18.3%

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

   5. Taylor expanded in N around -inf 99.7%

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

      1. mul-1-neg 99.7%

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

      2. distribute-neg-frac2 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in N around inf 99.7%

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

      1. associate-*r/ 99.7%

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

      2. metadata-eval 99.7%

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

      3. associate-*r/ 99.7%

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

      4. metadata-eval 99.7%

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

   10. Simplified 99.7%

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

       1. clear-num 99.7%

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

       2. inv-pow 99.7%

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

       3. +-commutative 99.7%

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

       4. div-inv 99.7%

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

       5. fma-define 99.7%

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

       6. pow-flip 99.7%

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

       7. metadata-eval 99.7%

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

       8. +-commutative 99.7%

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

       9. div-inv 99.7%

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

       10. fma-define 99.7%

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

       11. pow-flip 99.7%

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

       12. metadata-eval 99.7%

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

   12. Applied egg-rr 99.7%

       \[\leadsto \color{blue}{{\left(\frac{N}{\mathsf{fma}\left(0.3333333333333333, {N}^{-2}, 1\right) - \mathsf{fma}\left(0.25, {N}^{-3}, \frac{0.5}{N}\right)}\right)}^{-1}} \]
   13. Step-by-step derivation

       1. unpow-1 99.7%

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

   14. Simplified 99.7%

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

   15. Taylor expanded in N around -inf 99.8%

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

       1. mul-1-neg 99.8%

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

       2. unsub-neg 99.8%

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

       3. mul-1-neg 99.8%

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

       4. unsub-neg 99.8%

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

       5. sub-neg 99.8%

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

       6. associate-*r/ 99.8%

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

       7. metadata-eval 99.8%

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

       8. distribute-neg-frac 99.8%

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

       9. metadata-eval 99.8%

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

   17. Simplified 99.8%

       \[\leadsto \frac{1}{\frac{N}{\color{blue}{1 - \frac{0.5 - \frac{0.3333333333333333 + \frac{-0.25}{N}}{N}}{N}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;N \leq 1300:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{N}{1 + \frac{\frac{0.3333333333333333 + \frac{-0.25}{N}}{N} - 0.5}{N}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.6% accurate, 10.8× speedup

Math

\[\begin{array}{l} \\ \frac{-1}{N \cdot \left(-1 - \frac{0.5 + \frac{0.041666666666666664 \cdot \frac{1}{N} - 0.08333333333333333}{N}}{N}\right)} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (/
  -1.0
  (*
   N
   (-
    -1.0
    (/
     (+ 0.5 (/ (- (* 0.041666666666666664 (/ 1.0 N)) 0.08333333333333333) N))
     N)))))

C

double code(double N) {
	return -1.0 / (N * (-1.0 - ((0.5 + (((0.041666666666666664 * (1.0 / N)) - 0.08333333333333333) / N)) / N)));
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    code = (-1.0d0) / (n * ((-1.0d0) - ((0.5d0 + (((0.041666666666666664d0 * (1.0d0 / n)) - 0.08333333333333333d0) / n)) / n)))
end function

Java

public static double code(double N) {
	return -1.0 / (N * (-1.0 - ((0.5 + (((0.041666666666666664 * (1.0 / N)) - 0.08333333333333333) / N)) / N)));
}

Python

def code(N):
	return -1.0 / (N * (-1.0 - ((0.5 + (((0.041666666666666664 * (1.0 / N)) - 0.08333333333333333) / N)) / N)))

Julia

function code(N)
	return Float64(-1.0 / Float64(N * Float64(-1.0 - Float64(Float64(0.5 + Float64(Float64(Float64(0.041666666666666664 * Float64(1.0 / N)) - 0.08333333333333333) / N)) / N))))
end

MATLAB

function tmp = code(N)
	tmp = -1.0 / (N * (-1.0 - ((0.5 + (((0.041666666666666664 * (1.0 / N)) - 0.08333333333333333) / N)) / N)));
end

Wolfram

code[N_] := N[(-1.0 / N[(N * N[(-1.0 - N[(N[(0.5 + N[(N[(N[(0.041666666666666664 * N[(1.0 / N), $MachinePrecision]), $MachinePrecision] - 0.08333333333333333), $MachinePrecision] / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 23.4%

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

   1. +-commutative 23.4%

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

   2. log1p-define 23.4%

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

3. Simplified 23.4%

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

5. Taylor expanded in N around -inf 96.7%

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

   1. mul-1-neg 96.7%

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

   2. distribute-neg-frac2 96.7%

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

7. Simplified 96.7%

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

8. Taylor expanded in N around inf 96.7%

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

   1. associate-*r/ 96.7%

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

   2. metadata-eval 96.7%

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

   3. associate-*r/ 96.7%

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

   4. metadata-eval 96.7%

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

10. Simplified 96.7%

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

    1. clear-num 96.7%

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

    2. inv-pow 96.7%

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

    3. +-commutative 96.7%

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

    4. div-inv 96.7%

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

    5. fma-define 96.7%

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

    6. pow-flip 96.7%

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

    7. metadata-eval 96.7%

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

    8. +-commutative 96.7%

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

    9. div-inv 96.7%

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

    10. fma-define 96.7%

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

    11. pow-flip 96.7%

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

    12. metadata-eval 96.7%

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

12. Applied egg-rr 96.7%

    \[\leadsto \color{blue}{{\left(\frac{N}{\mathsf{fma}\left(0.3333333333333333, {N}^{-2}, 1\right) - \mathsf{fma}\left(0.25, {N}^{-3}, \frac{0.5}{N}\right)}\right)}^{-1}} \]
13. Step-by-step derivation

    1. unpow-1 96.7%

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

14. Simplified 96.7%

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

15. Taylor expanded in N around -inf 97.1%

    \[\leadsto \frac{1}{\color{blue}{-1 \cdot \left(N \cdot \left(-1 \cdot \frac{0.5 + -1 \cdot \frac{0.08333333333333333 - 0.041666666666666664 \cdot \frac{1}{N}}{N}}{N} - 1\right)\right)}} \]

16. Final simplification 97.1%

    \[\leadsto \frac{-1}{N \cdot \left(-1 - \frac{0.5 + \frac{0.041666666666666664 \cdot \frac{1}{N} - 0.08333333333333333}{N}}{N}\right)} \]
17. Add Preprocessing

Alternative 4: 96.2% accurate, 12.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 23.4%

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

   1. +-commutative 23.4%

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

   2. log1p-define 23.4%

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

3. Simplified 23.4%

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

5. Taylor expanded in N around -inf 96.7%

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

   1. mul-1-neg 96.7%

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

   2. distribute-neg-frac2 96.7%

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

7. Simplified 96.7%

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

8. Taylor expanded in N around inf 96.7%

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

   1. associate-*r/ 96.7%

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

   2. metadata-eval 96.7%

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

   3. associate-*r/ 96.7%

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

   4. metadata-eval 96.7%

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

10. Simplified 96.7%

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

    1. clear-num 96.7%

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

    2. inv-pow 96.7%

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

    3. +-commutative 96.7%

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

    4. div-inv 96.7%

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

    5. fma-define 96.7%

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

    6. pow-flip 96.7%

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

    7. metadata-eval 96.7%

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

    8. +-commutative 96.7%

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

    9. div-inv 96.7%

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

    10. fma-define 96.7%

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

    11. pow-flip 96.7%

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

    12. metadata-eval 96.7%

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

12. Applied egg-rr 96.7%

    \[\leadsto \color{blue}{{\left(\frac{N}{\mathsf{fma}\left(0.3333333333333333, {N}^{-2}, 1\right) - \mathsf{fma}\left(0.25, {N}^{-3}, \frac{0.5}{N}\right)}\right)}^{-1}} \]
13. Step-by-step derivation

    1. unpow-1 96.7%

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

14. Simplified 96.7%

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

15. Taylor expanded in N around -inf 96.7%

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

    1. mul-1-neg 96.7%

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

    2. unsub-neg 96.7%

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

    3. mul-1-neg 96.7%

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

    4. unsub-neg 96.7%

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

    5. sub-neg 96.7%

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

    6. associate-*r/ 96.7%

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

    7. metadata-eval 96.7%

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

    8. distribute-neg-frac 96.7%

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

    9. metadata-eval 96.7%

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

17. Simplified 96.7%

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

18. Final simplification 96.7%

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

Alternative 5: 96.2% accurate, 13.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 23.4%

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

   1. +-commutative 23.4%

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

   2. log1p-define 23.4%

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

3. Simplified 23.4%

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

5. Taylor expanded in N around -inf 96.7%

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

   1. mul-1-neg 96.7%

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

   2. distribute-neg-frac2 96.7%

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

7. Simplified 96.7%

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

8. Taylor expanded in N around inf 96.7%

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

   1. associate-*r/ 96.7%

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

   2. metadata-eval 96.7%

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

   3. associate-*r/ 96.7%

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

   4. metadata-eval 96.7%

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

10. Simplified 96.7%

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

11. Taylor expanded in N around -inf 96.7%

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

    1. mul-1-neg 96.7%

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

    2. unsub-neg 96.7%

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

    3. mul-1-neg 96.7%

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

    4. unsub-neg 96.7%

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

    5. sub-neg 96.7%

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

    6. associate-*r/ 96.7%

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

    7. metadata-eval 96.7%

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

    8. distribute-neg-frac 96.7%

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

    9. metadata-eval 96.7%

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

13. Simplified 96.7%

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

14. Final simplification 96.7%

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

Alternative 6: 95.4% accurate, 15.8× speedup

Math

\[\begin{array}{l} \\ \frac{1}{N \cdot \left(1 + \frac{0.5 + \frac{-0.08333333333333333}{N}}{N}\right)} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (/ 1.0 (* N (+ 1.0 (/ (+ 0.5 (/ -0.08333333333333333 N)) N)))))

C

double code(double N) {
	return 1.0 / (N * (1.0 + ((0.5 + (-0.08333333333333333 / N)) / N)));
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    code = 1.0d0 / (n * (1.0d0 + ((0.5d0 + ((-0.08333333333333333d0) / n)) / n)))
end function

Java

public static double code(double N) {
	return 1.0 / (N * (1.0 + ((0.5 + (-0.08333333333333333 / N)) / N)));
}

Python

def code(N):
	return 1.0 / (N * (1.0 + ((0.5 + (-0.08333333333333333 / N)) / N)))

Julia

function code(N)
	return Float64(1.0 / Float64(N * Float64(1.0 + Float64(Float64(0.5 + Float64(-0.08333333333333333 / N)) / N))))
end

MATLAB

function tmp = code(N)
	tmp = 1.0 / (N * (1.0 + ((0.5 + (-0.08333333333333333 / N)) / N)));
end

Wolfram

code[N_] := N[(1.0 / N[(N * N[(1.0 + N[(N[(0.5 + N[(-0.08333333333333333 / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 23.4%

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

   1. +-commutative 23.4%

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

   2. log1p-define 23.4%

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

3. Simplified 23.4%

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

5. Taylor expanded in N around -inf 96.7%

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

   1. mul-1-neg 96.7%

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

   2. distribute-neg-frac2 96.7%

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

7. Simplified 96.7%

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

8. Taylor expanded in N around inf 96.7%

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

   1. associate-*r/ 96.7%

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

   2. metadata-eval 96.7%

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

   3. associate-*r/ 96.7%

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

   4. metadata-eval 96.7%

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

10. Simplified 96.7%

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

    1. clear-num 96.7%

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

    2. inv-pow 96.7%

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

    3. +-commutative 96.7%

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

    4. div-inv 96.7%

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

    5. fma-define 96.7%

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

    6. pow-flip 96.7%

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

    7. metadata-eval 96.7%

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

    8. +-commutative 96.7%

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

    9. div-inv 96.7%

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

    10. fma-define 96.7%

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

    11. pow-flip 96.7%

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

    12. metadata-eval 96.7%

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

12. Applied egg-rr 96.7%

    \[\leadsto \color{blue}{{\left(\frac{N}{\mathsf{fma}\left(0.3333333333333333, {N}^{-2}, 1\right) - \mathsf{fma}\left(0.25, {N}^{-3}, \frac{0.5}{N}\right)}\right)}^{-1}} \]
13. Step-by-step derivation

    1. unpow-1 96.7%

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

14. Simplified 96.7%

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

15. Taylor expanded in N around inf 95.9%

    \[\leadsto \frac{1}{\color{blue}{N \cdot \left(\left(1 + 0.5 \cdot \frac{1}{N}\right) - \frac{0.08333333333333333}{{N}^{2}}\right)}} \]
16. Step-by-step derivation

    1. associate--l+ 95.9%

       \[\leadsto \frac{1}{N \cdot \color{blue}{\left(1 + \left(0.5 \cdot \frac{1}{N} - \frac{0.08333333333333333}{{N}^{2}}\right)\right)}} \]

    2. associate-*r/ 95.9%

       \[\leadsto \frac{1}{N \cdot \left(1 + \left(\color{blue}{\frac{0.5 \cdot 1}{N}} - \frac{0.08333333333333333}{{N}^{2}}\right)\right)} \]

    3. metadata-eval 95.9%

       \[\leadsto \frac{1}{N \cdot \left(1 + \left(\frac{\color{blue}{0.5}}{N} - \frac{0.08333333333333333}{{N}^{2}}\right)\right)} \]

    4. unpow2 95.9%

       \[\leadsto \frac{1}{N \cdot \left(1 + \left(\frac{0.5}{N} - \frac{0.08333333333333333}{\color{blue}{N \cdot N}}\right)\right)} \]

    5. associate-/r* 95.9%

       \[\leadsto \frac{1}{N \cdot \left(1 + \left(\frac{0.5}{N} - \color{blue}{\frac{\frac{0.08333333333333333}{N}}{N}}\right)\right)} \]

    6. metadata-eval 95.9%

       \[\leadsto \frac{1}{N \cdot \left(1 + \left(\frac{0.5}{N} - \frac{\frac{\color{blue}{0.08333333333333333 \cdot 1}}{N}}{N}\right)\right)} \]

    7. associate-*r/ 95.9%

       \[\leadsto \frac{1}{N \cdot \left(1 + \left(\frac{0.5}{N} - \frac{\color{blue}{0.08333333333333333 \cdot \frac{1}{N}}}{N}\right)\right)} \]

    8. div-sub 95.9%

       \[\leadsto \frac{1}{N \cdot \left(1 + \color{blue}{\frac{0.5 - 0.08333333333333333 \cdot \frac{1}{N}}{N}}\right)} \]

    9. sub-neg 95.9%

       \[\leadsto \frac{1}{N \cdot \left(1 + \frac{\color{blue}{0.5 + \left(-0.08333333333333333 \cdot \frac{1}{N}\right)}}{N}\right)} \]

    10. associate-*r/ 95.9%

        \[\leadsto \frac{1}{N \cdot \left(1 + \frac{0.5 + \left(-\color{blue}{\frac{0.08333333333333333 \cdot 1}{N}}\right)}{N}\right)} \]

    11. metadata-eval 95.9%

        \[\leadsto \frac{1}{N \cdot \left(1 + \frac{0.5 + \left(-\frac{\color{blue}{0.08333333333333333}}{N}\right)}{N}\right)} \]

    12. distribute-neg-frac 95.9%

        \[\leadsto \frac{1}{N \cdot \left(1 + \frac{0.5 + \color{blue}{\frac{-0.08333333333333333}{N}}}{N}\right)} \]

    13. metadata-eval 95.9%

        \[\leadsto \frac{1}{N \cdot \left(1 + \frac{0.5 + \frac{\color{blue}{-0.08333333333333333}}{N}}{N}\right)} \]

17. Simplified 95.9%

    \[\leadsto \frac{1}{\color{blue}{N \cdot \left(1 + \frac{0.5 + \frac{-0.08333333333333333}{N}}{N}\right)}} \]
18. Add Preprocessing

Alternative 7: 95.0% accurate, 18.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 23.4%

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

   1. +-commutative 23.4%

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

   2. log1p-define 23.4%

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

3. Simplified 23.4%

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

5. Taylor expanded in N around inf 95.4%

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

   1. associate--l+ 95.5%

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

   2. unpow2 95.5%

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

   3. associate-/r* 95.5%

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

   4. metadata-eval 95.5%

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

   5. associate-*r/ 95.5%

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

   6. associate-*r/ 95.5%

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

   7. metadata-eval 95.5%

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

   8. div-sub 95.5%

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

   9. sub-neg 95.5%

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

   10. metadata-eval 95.5%

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

   11. +-commutative 95.5%

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

   12. associate-*r/ 95.5%

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

   13. metadata-eval 95.5%

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

7. Simplified 95.5%

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

Alternative 8: 93.0% accurate, 22.8× speedup

Math

\[\begin{array}{l} \\ \frac{1}{N \cdot \left(1 + \frac{0.5}{N}\right)} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 23.4%

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

   1. +-commutative 23.4%

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

   2. log1p-define 23.4%

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

3. Simplified 23.4%

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

5. Taylor expanded in N around -inf 96.7%

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

   1. mul-1-neg 96.7%

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

   2. distribute-neg-frac2 96.7%

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

7. Simplified 96.7%

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

8. Taylor expanded in N around inf 96.7%

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

   1. associate-*r/ 96.7%

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

   2. metadata-eval 96.7%

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

   3. associate-*r/ 96.7%

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

   4. metadata-eval 96.7%

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

10. Simplified 96.7%

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

    1. clear-num 96.7%

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

    2. inv-pow 96.7%

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

    3. +-commutative 96.7%

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

    4. div-inv 96.7%

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

    5. fma-define 96.7%

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

    6. pow-flip 96.7%

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

    7. metadata-eval 96.7%

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

    8. +-commutative 96.7%

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

    9. div-inv 96.7%

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

    10. fma-define 96.7%

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

    11. pow-flip 96.7%

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

    12. metadata-eval 96.7%

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

12. Applied egg-rr 96.7%

    \[\leadsto \color{blue}{{\left(\frac{N}{\mathsf{fma}\left(0.3333333333333333, {N}^{-2}, 1\right) - \mathsf{fma}\left(0.25, {N}^{-3}, \frac{0.5}{N}\right)}\right)}^{-1}} \]
13. Step-by-step derivation

    1. unpow-1 96.7%

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

14. Simplified 96.7%

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

15. Taylor expanded in N around inf 93.6%

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

    1. associate-*r/ 93.6%

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

    2. metadata-eval 93.6%

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

17. Simplified 93.6%

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

Alternative 9: 92.4% accurate, 29.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 23.4%

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

   1. +-commutative 23.4%

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

   2. log1p-define 23.4%

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

3. Simplified 23.4%

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

5. Taylor expanded in N around inf 93.1%

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

   1. associate-*r/ 93.1%

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

   2. metadata-eval 93.1%

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

7. Simplified 93.1%

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

Alternative 10: 84.6% 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 23.4%

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

   1. +-commutative 23.4%

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

   2. log1p-define 23.4%

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

3. Simplified 23.4%

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

5. Taylor expanded in N around inf 84.9%

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

Alternative 11: 1.6% accurate, 102.5× speedup

Math

\[\begin{array}{l} \\ -N \end{array} \]

FPCore

(FPCore (N) :precision binary64 (- N))

C

double code(double N) {
	return -N;
}

Fortran

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

Java

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

Python

def code(N):
	return -N

Julia

function code(N)
	return Float64(-N)
end

MATLAB

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

Wolfram

code[N_] := (-N)

Derivation

1. Initial program 23.4%

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

   1. +-commutative 23.4%

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

   2. log1p-define 23.4%

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

3. Simplified 23.4%

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

5. Taylor expanded in N around inf 84.9%

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

   1. add-exp-log 81.4%

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

   2. neg-log 81.4%

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

   3. add-sqr-sqrt 0.0%

      \[\leadsto e^{\color{blue}{\sqrt{-\log N} \cdot \sqrt{-\log N}}} \]

   4. sqrt-unprod 8.4%

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

   5. sqr-neg 8.4%

      \[\leadsto e^{\sqrt{\color{blue}{\log N \cdot \log N}}} \]

   6. sqrt-unprod 8.4%

      \[\leadsto e^{\color{blue}{\sqrt{\log N} \cdot \sqrt{\log N}}} \]

   7. add-sqr-sqrt 8.4%

      \[\leadsto e^{\color{blue}{\log N}} \]

   8. add-exp-log 8.4%

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

   9. add-sqr-sqrt 8.4%

      \[\leadsto \color{blue}{\sqrt{N} \cdot \sqrt{N}} \]

   10. sqrt-unprod 8.4%

       \[\leadsto \color{blue}{\sqrt{N \cdot N}} \]

   11. sqr-neg 8.4%

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

   12. sqrt-unprod 0.0%

       \[\leadsto \color{blue}{\sqrt{-N} \cdot \sqrt{-N}} \]

   13. add-sqr-sqrt 1.6%

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

   14. neg-sub0 1.6%

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

7. Applied egg-rr 1.6%

   \[\leadsto \color{blue}{0 - N} \]
8. Step-by-step derivation

   1. neg-sub0 1.6%

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

9. Simplified 1.6%

   \[\leadsto \color{blue}{-N} \]
10. Add Preprocessing

Developer Target 1: 99.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{log1p}\left(\frac{1}{N}\right) \end{array} \]

FPCore

(FPCore (N) :precision binary64 (log1p (/ 1.0 N)))

C

double code(double N) {
	return log1p((1.0 / N));
}

Java

public static double code(double N) {
	return Math.log1p((1.0 / N));
}

Python

def code(N):
	return math.log1p((1.0 / N))

Julia

function code(N)
	return log1p(Float64(1.0 / N))
end

Wolfram

code[N_] := N[Log[1 + N[(1.0 / N), $MachinePrecision]], $MachinePrecision]

Reproduce

herbie shell --seed 2024141 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  :pre (and (> N 1.0) (< N 1e+40))

  :alt
  (! :herbie-platform default (log1p (/ 1 N)))

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