2log (problem 3.3.6)

Percentage Accurate: 23.9% → 99.4%

Time: 9.1s

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: 23.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (N)
 :precision binary64
 (if (<= (- (log (+ N 1.0)) (log N)) 0.0005)
   (/
    -1.0
    (/ N (- -1.0 (/ (- -0.5 (/ (- (/ 0.25 N) 0.3333333333333333) N)) N))))
   (- (log (/ N (+ N 1.0))))))

C

double code(double N) {
	double tmp;
	if ((log((N + 1.0)) - log(N)) <= 0.0005) {
		tmp = -1.0 / (N / (-1.0 - ((-0.5 - (((0.25 / N) - 0.3333333333333333) / N)) / N)));
	} else {
		tmp = -log((N / (N + 1.0)));
	}
	return tmp;
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((log((n + 1.0d0)) - log(n)) <= 0.0005d0) then
        tmp = (-1.0d0) / (n / ((-1.0d0) - (((-0.5d0) - (((0.25d0 / n) - 0.3333333333333333d0) / n)) / n)))
    else
        tmp = -log((n / (n + 1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double N) {
	double tmp;
	if ((Math.log((N + 1.0)) - Math.log(N)) <= 0.0005) {
		tmp = -1.0 / (N / (-1.0 - ((-0.5 - (((0.25 / N) - 0.3333333333333333) / N)) / N)));
	} else {
		tmp = -Math.log((N / (N + 1.0)));
	}
	return tmp;
}

Python

def code(N):
	tmp = 0
	if (math.log((N + 1.0)) - math.log(N)) <= 0.0005:
		tmp = -1.0 / (N / (-1.0 - ((-0.5 - (((0.25 / N) - 0.3333333333333333) / N)) / N)))
	else:
		tmp = -math.log((N / (N + 1.0)))
	return tmp

Julia

function code(N)
	tmp = 0.0
	if (Float64(log(Float64(N + 1.0)) - log(N)) <= 0.0005)
		tmp = Float64(-1.0 / Float64(N / Float64(-1.0 - Float64(Float64(-0.5 - Float64(Float64(Float64(0.25 / N) - 0.3333333333333333) / N)) / N))));
	else
		tmp = Float64(-log(Float64(N / Float64(N + 1.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(N)
	tmp = 0.0;
	if ((log((N + 1.0)) - log(N)) <= 0.0005)
		tmp = -1.0 / (N / (-1.0 - ((-0.5 - (((0.25 / N) - 0.3333333333333333) / N)) / N)));
	else
		tmp = -log((N / (N + 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[N_] := If[LessEqual[N[(N[Log[N[(N + 1.0), $MachinePrecision]], $MachinePrecision] - N[Log[N], $MachinePrecision]), $MachinePrecision], 0.0005], N[(-1.0 / N[(N / N[(-1.0 - N[(N[(-0.5 - N[(N[(N[(0.25 / N), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] / N), $MachinePrecision]), $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)) < 5.0000000000000001e-4

   1. Initial program 17.2%

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

      1. +-commutative 17.2%

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

      2. log1p-define 17.2%

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

   3. Simplified 17.2%

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

   5. Taylor expanded in N around inf 99.8%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in N around -inf 99.8%

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

      1. mul-1-neg 99.8%

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

      2. unsub-neg 99.8%

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

   10. Simplified 99.8%

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

       1. clear-num 99.9%

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

       2. inv-pow 99.9%

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

       3. +-commutative 99.9%

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

   12. Applied egg-rr 99.9%

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

   14. Simplified 99.9%

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

   if 5.0000000000000001e-4 < (-.f64 (log.f64 (+.f64 N #s(literal 1 binary64))) (log.f64 N))

   1. Initial program 90.4%

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

      1. +-commutative 90.4%

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

      2. log1p-define 90.4%

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

   3. Simplified 90.4%

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

      1. add-log-exp 90.4%

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

      2. log1p-expm1-u 90.4%

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

      3. log1p-undefine 90.4%

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

      4. diff-log 90.5%

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

      5. log1p-undefine 90.5%

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

      6. rem-exp-log 90.3%

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

      7. +-commutative 90.3%

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

      8. add-exp-log 90.4%

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

      9. log1p-undefine 90.4%

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

      10. log1p-expm1-u 90.4%

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

      11. add-exp-log 93.2%

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

   6. Applied egg-rr 93.2%

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

      1. clear-num 93.2%

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

      2. log-div 93.9%

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

      3. metadata-eval 93.9%

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

   8. Applied egg-rr 93.9%

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

      1. neg-sub0 93.9%

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

   10. Simplified 93.9%

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

4. Final simplification 99.4%

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

Alternative 2: 99.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;N \leq 1600:\\ \;\;\;\;\log \left(1 + \frac{1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{N}{-1 - \frac{-0.5 - \frac{\frac{0.25}{N} - 0.3333333333333333}{N}}{N}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= N 1600.0)
   (log (+ 1.0 (/ 1.0 N)))
   (/
    -1.0
    (/ N (- -1.0 (/ (- -0.5 (/ (- (/ 0.25 N) 0.3333333333333333) N)) N))))))

C

double code(double N) {
	double tmp;
	if (N <= 1600.0) {
		tmp = log((1.0 + (1.0 / N)));
	} else {
		tmp = -1.0 / (N / (-1.0 - ((-0.5 - (((0.25 / N) - 0.3333333333333333) / N)) / N)));
	}
	return tmp;
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= 1600.0d0) then
        tmp = log((1.0d0 + (1.0d0 / n)))
    else
        tmp = (-1.0d0) / (n / ((-1.0d0) - (((-0.5d0) - (((0.25d0 / n) - 0.3333333333333333d0) / n)) / n)))
    end if
    code = tmp
end function

Java

public static double code(double N) {
	double tmp;
	if (N <= 1600.0) {
		tmp = Math.log((1.0 + (1.0 / N)));
	} else {
		tmp = -1.0 / (N / (-1.0 - ((-0.5 - (((0.25 / N) - 0.3333333333333333) / N)) / N)));
	}
	return tmp;
}

Python

def code(N):
	tmp = 0
	if N <= 1600.0:
		tmp = math.log((1.0 + (1.0 / N)))
	else:
		tmp = -1.0 / (N / (-1.0 - ((-0.5 - (((0.25 / N) - 0.3333333333333333) / N)) / N)))
	return tmp

Julia

function code(N)
	tmp = 0.0
	if (N <= 1600.0)
		tmp = log(Float64(1.0 + Float64(1.0 / N)));
	else
		tmp = Float64(-1.0 / Float64(N / Float64(-1.0 - Float64(Float64(-0.5 - Float64(Float64(Float64(0.25 / N) - 0.3333333333333333) / N)) / N))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(N)
	tmp = 0.0;
	if (N <= 1600.0)
		tmp = log((1.0 + (1.0 / N)));
	else
		tmp = -1.0 / (N / (-1.0 - ((-0.5 - (((0.25 / N) - 0.3333333333333333) / N)) / N)));
	end
	tmp_2 = tmp;
end

Wolfram

code[N_] := If[LessEqual[N, 1600.0], N[Log[N[(1.0 + N[(1.0 / N), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(-1.0 / N[(N / N[(-1.0 - N[(N[(-0.5 - N[(N[(N[(0.25 / N), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if N < 1600

   1. Initial program 90.4%

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

      1. +-commutative 90.4%

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

      2. log1p-define 90.4%

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

   3. Simplified 90.4%

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

      1. add-log-exp 90.4%

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

      2. log1p-expm1-u 90.4%

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

      3. log1p-undefine 90.4%

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

      4. diff-log 90.5%

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

      5. log1p-undefine 90.5%

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

      6. rem-exp-log 90.3%

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

      7. +-commutative 90.3%

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

      8. add-exp-log 90.4%

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

      9. log1p-undefine 90.4%

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

      10. log1p-expm1-u 90.4%

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

      11. add-exp-log 93.2%

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

   6. Applied egg-rr 93.2%

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

   7. Taylor expanded in N around 0 93.2%

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

      1. metadata-eval 93.2%

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

      2. remove-double-neg 93.2%

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

      3. distribute-neg-in 93.2%

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

      4. distribute-neg-frac 93.2%

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

      5. distribute-neg-frac2 93.2%

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

      6. *-rgt-identity 93.2%

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

      7. associate-*r/ 93.0%

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

      8. *-commutative 93.0%

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

      9. +-commutative 93.0%

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

      10. distribute-rgt-in 92.9%

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

      11. *-rgt-identity 92.9%

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

      12. distribute-rgt-neg-in 92.9%

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

      13. metadata-eval 92.9%

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

      14. associate-*r* 92.9%

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

      15. mul-1-neg 92.9%

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

      16. distribute-frac-neg2 92.9%

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

      17. remove-double-neg 92.9%

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

      18. rgt-mult-inverse 93.4%

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

      19. associate-*r/ 93.4%

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

      20. metadata-eval 93.4%

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

      21. distribute-neg-frac2 93.4%

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

      22. distribute-neg-frac 93.4%

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

      23. metadata-eval 93.4%

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

   9. Simplified 93.4%

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

   if 1600 < N

   1. Initial program 17.2%

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

      1. +-commutative 17.2%

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

      2. log1p-define 17.2%

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

   3. Simplified 17.2%

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

   5. Taylor expanded in N around inf 99.8%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in N around -inf 99.8%

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

      1. mul-1-neg 99.8%

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

      2. unsub-neg 99.8%

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

   10. Simplified 99.8%

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

       1. clear-num 99.9%

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

       2. inv-pow 99.9%

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

       3. +-commutative 99.9%

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

   12. Applied egg-rr 99.9%

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

   14. Simplified 99.9%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;N \leq 1600:\\ \;\;\;\;\log \left(1 + \frac{1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{N}{-1 - \frac{-0.5 - \frac{\frac{0.25}{N} - 0.3333333333333333}{N}}{N}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.5% accurate, 10.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 22.9%

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

   1. +-commutative 22.9%

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

   2. log1p-define 22.9%

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

3. Simplified 22.9%

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

5. Taylor expanded in N around inf 96.6%

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

7. Simplified 96.7%

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

8. Taylor expanded in N around -inf 96.7%

   \[\leadsto \frac{\color{blue}{1 + -1 \cdot \frac{0.5 + -1 \cdot \frac{0.3333333333333333 - 0.25 \cdot \frac{1}{N}}{N}}{N}}}{N} \]
9. 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%

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

10. Simplified 96.7%

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

    1. clear-num 96.7%

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

    2. inv-pow 96.7%

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

    3. +-commutative 96.7%

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

12. Applied egg-rr 96.7%

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

14. Simplified 96.7%

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

15. Taylor expanded in N around -inf 97.0%

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

16. Final simplification 97.0%

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

Alternative 4: 96.2% accurate, 12.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 22.9%

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

   1. +-commutative 22.9%

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

   2. log1p-define 22.9%

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

3. Simplified 22.9%

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

5. Taylor expanded in N around inf 96.6%

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

7. Simplified 96.7%

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

8. Taylor expanded in N around -inf 96.7%

   \[\leadsto \frac{\color{blue}{1 + -1 \cdot \frac{0.5 + -1 \cdot \frac{0.3333333333333333 - 0.25 \cdot \frac{1}{N}}{N}}{N}}}{N} \]
9. 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%

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

10. Simplified 96.7%

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

    1. clear-num 96.7%

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

    2. inv-pow 96.7%

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

    3. +-commutative 96.7%

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

12. Applied egg-rr 96.7%

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

14. Simplified 96.7%

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

15. Final simplification 96.7%

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

Alternative 5: 96.2% accurate, 13.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 22.9%

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

   1. +-commutative 22.9%

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

   2. log1p-define 22.9%

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

3. Simplified 22.9%

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

5. Taylor expanded in N around inf 96.6%

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

7. Simplified 96.7%

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

8. Taylor expanded in N around -inf 96.7%

   \[\leadsto \frac{\color{blue}{1 + -1 \cdot \frac{0.5 + -1 \cdot \frac{0.3333333333333333 - 0.25 \cdot \frac{1}{N}}{N}}{N}}}{N} \]
9. 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%

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

10. Simplified 96.7%

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

11. Final simplification 96.7%

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

Alternative 6: 95.3% 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 22.9%

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

   1. +-commutative 22.9%

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

   2. log1p-define 22.9%

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

3. Simplified 22.9%

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

5. Taylor expanded in N around inf 96.6%

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

7. Simplified 96.7%

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

8. Taylor expanded in N around -inf 96.7%

   \[\leadsto \frac{\color{blue}{1 + -1 \cdot \frac{0.5 + -1 \cdot \frac{0.3333333333333333 - 0.25 \cdot \frac{1}{N}}{N}}{N}}}{N} \]
9. 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%

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

10. Simplified 96.7%

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

    1. clear-num 96.7%

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

    2. inv-pow 96.7%

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

    3. +-commutative 96.7%

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

12. Applied egg-rr 96.7%

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

14. Simplified 96.7%

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

15. Taylor expanded in N around inf 95.8%

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

    1. associate--l+ 95.8%

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

    2. associate-*r/ 95.8%

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

    3. metadata-eval 95.8%

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

    4. unpow2 95.8%

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

    5. associate-/r* 95.8%

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

    6. metadata-eval 95.8%

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

    7. associate-*r/ 95.8%

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

    8. div-sub 95.8%

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

    9. sub-neg 95.8%

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

    10. associate-*r/ 95.8%

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

    11. metadata-eval 95.8%

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

    12. distribute-neg-frac 95.8%

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

    13. metadata-eval 95.8%

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

17. Simplified 95.8%

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

18. Final simplification 95.8%

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

Alternative 7: 94.8% 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 22.9%

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

   1. +-commutative 22.9%

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

   2. log1p-define 22.9%

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

3. Simplified 22.9%

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

5. Taylor expanded in N around inf 95.3%

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

   1. associate--l+ 95.3%

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

   2. unpow2 95.3%

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

   3. associate-/r* 95.3%

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

   4. metadata-eval 95.3%

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

   5. associate-*r/ 95.3%

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

   6. associate-*r/ 95.3%

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

   7. metadata-eval 95.3%

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

   8. div-sub 95.3%

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

   9. sub-neg 95.3%

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

   10. metadata-eval 95.3%

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

   11. +-commutative 95.3%

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

   12. associate-*r/ 95.3%

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

   13. metadata-eval 95.3%

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

7. Simplified 95.3%

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

Alternative 8: 92.7% 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 22.9%

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

   1. +-commutative 22.9%

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

   2. log1p-define 22.9%

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

3. Simplified 22.9%

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

5. Taylor expanded in N around inf 96.6%

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

7. Simplified 96.7%

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

8. Taylor expanded in N around -inf 96.7%

   \[\leadsto \frac{\color{blue}{1 + -1 \cdot \frac{0.5 + -1 \cdot \frac{0.3333333333333333 - 0.25 \cdot \frac{1}{N}}{N}}{N}}}{N} \]
9. 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%

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

10. Simplified 96.7%

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

    1. clear-num 96.7%

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

    2. inv-pow 96.7%

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

    3. +-commutative 96.7%

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

12. Applied egg-rr 96.7%

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

14. Simplified 96.7%

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

15. Taylor expanded in N around inf 93.3%

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

    1. associate-*r/ 93.3%

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

    2. metadata-eval 93.3%

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

17. Simplified 93.3%

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

18. Final simplification 93.3%

    \[\leadsto \frac{-1}{N \cdot \left(-1 - \frac{0.5}{N}\right)} \]
19. Add Preprocessing

Alternative 9: 92.2% 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 22.9%

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

   1. +-commutative 22.9%

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

   2. log1p-define 22.9%

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

3. Simplified 22.9%

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

5. Taylor expanded in N around inf 92.7%

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

   1. associate-*r/ 92.7%

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

   2. metadata-eval 92.7%

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

7. Simplified 92.7%

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

Alternative 10: 84.4% 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 22.9%

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

   1. +-commutative 22.9%

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

   2. log1p-define 22.9%

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

3. Simplified 22.9%

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

5. Taylor expanded in N around inf 85.0%

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

Alternative 11: 3.3% accurate, 205.0× speedup

Math

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

FPCore

(FPCore (N) :precision binary64 0.0)

C

double code(double N) {
	return 0.0;
}

Fortran

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

Java

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

Python

def code(N):
	return 0.0

Julia

function code(N)
	return 0.0
end

MATLAB

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

Wolfram

code[N_] := 0.0

Derivation

1. Initial program 22.9%

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

   1. +-commutative 22.9%

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

   2. log1p-define 22.9%

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

3. Simplified 22.9%

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

   1. sub-neg 22.9%

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

   2. +-commutative 22.9%

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

   3. add-sqr-sqrt 23.4%

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

   4. distribute-rgt-neg-in 23.4%

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

   5. fma-define 24.8%

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

6. Applied egg-rr 24.8%

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

7. Taylor expanded in N around inf 3.3%

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

   1. distribute-rgt1-in 3.3%

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

   2. metadata-eval 3.3%

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

   3. mul0-lft 3.3%

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

9. Simplified 3.3%

   \[\leadsto \color{blue}{0} \]
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 2024113 
(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)))