2log (problem 3.3.6)

Percentage Accurate: 23.4% → 99.1%

Time: 10.9s

Alternatives: 6

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: 23.4% 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: 99.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\log \left(1 + N\right) - \log N \leq 0.0001:\\ \;\;\;\;\frac{e^{\frac{-0.5}{N} + \frac{0.20833333333333334}{{N}^{2}}}}{N}\\ \mathbf{else}:\\ \;\;\;\;-\log \left(\frac{N}{1 + N}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= (- (log (+ 1.0 N)) (log N)) 0.0001)
   (/ (exp (+ (/ -0.5 N) (/ 0.20833333333333334 (pow N 2.0)))) N)
   (- (log (/ N (+ 1.0 N))))))

C

double code(double N) {
	double tmp;
	if ((log((1.0 + N)) - log(N)) <= 0.0001) {
		tmp = exp(((-0.5 / N) + (0.20833333333333334 / pow(N, 2.0)))) / N;
	} else {
		tmp = -log((N / (1.0 + N)));
	}
	return tmp;
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((log((1.0d0 + n)) - log(n)) <= 0.0001d0) then
        tmp = exp((((-0.5d0) / n) + (0.20833333333333334d0 / (n ** 2.0d0)))) / n
    else
        tmp = -log((n / (1.0d0 + n)))
    end if
    code = tmp
end function

Java

public static double code(double N) {
	double tmp;
	if ((Math.log((1.0 + N)) - Math.log(N)) <= 0.0001) {
		tmp = Math.exp(((-0.5 / N) + (0.20833333333333334 / Math.pow(N, 2.0)))) / N;
	} else {
		tmp = -Math.log((N / (1.0 + N)));
	}
	return tmp;
}

Python

def code(N):
	tmp = 0
	if (math.log((1.0 + N)) - math.log(N)) <= 0.0001:
		tmp = math.exp(((-0.5 / N) + (0.20833333333333334 / math.pow(N, 2.0)))) / N
	else:
		tmp = -math.log((N / (1.0 + N)))
	return tmp

Julia

function code(N)
	tmp = 0.0
	if (Float64(log(Float64(1.0 + N)) - log(N)) <= 0.0001)
		tmp = Float64(exp(Float64(Float64(-0.5 / N) + Float64(0.20833333333333334 / (N ^ 2.0)))) / N);
	else
		tmp = Float64(-log(Float64(N / Float64(1.0 + N))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(N)
	tmp = 0.0;
	if ((log((1.0 + N)) - log(N)) <= 0.0001)
		tmp = exp(((-0.5 / N) + (0.20833333333333334 / (N ^ 2.0)))) / N;
	else
		tmp = -log((N / (1.0 + N)));
	end
	tmp_2 = tmp;
end

Wolfram

code[N_] := If[LessEqual[N[(N[Log[N[(1.0 + N), $MachinePrecision]], $MachinePrecision] - N[Log[N], $MachinePrecision]), $MachinePrecision], 0.0001], N[(N[Exp[N[(N[(-0.5 / N), $MachinePrecision] + N[(0.20833333333333334 / N[Power[N, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N), $MachinePrecision], (-N[Log[N[(N / N[(1.0 + N), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (log.f64 (+.f64 N 1)) (log.f64 N)) < 1.00000000000000005e-4

   1. Initial program 15.8%

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

      1. +-commutative 15.8%

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

      2. log1p-define 15.8%

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

   3. Simplified 15.8%

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

      1. add-cbrt-cube 15.8%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\left(\mathsf{log1p}\left(N\right) - \log N\right) \cdot \left(\mathsf{log1p}\left(N\right) - \log N\right)\right) \cdot \left(\mathsf{log1p}\left(N\right) - \log N\right)}} \]

      2. pow1/3 15.8%

         \[\leadsto \color{blue}{{\left(\left(\left(\mathsf{log1p}\left(N\right) - \log N\right) \cdot \left(\mathsf{log1p}\left(N\right) - \log N\right)\right) \cdot \left(\mathsf{log1p}\left(N\right) - \log N\right)\right)}^{0.3333333333333333}} \]

      3. pow-to-exp 15.8%

         \[\leadsto \color{blue}{e^{\log \left(\left(\left(\mathsf{log1p}\left(N\right) - \log N\right) \cdot \left(\mathsf{log1p}\left(N\right) - \log N\right)\right) \cdot \left(\mathsf{log1p}\left(N\right) - \log N\right)\right) \cdot 0.3333333333333333}} \]

      4. pow3 15.8%

         \[\leadsto e^{\log \color{blue}{\left({\left(\mathsf{log1p}\left(N\right) - \log N\right)}^{3}\right)} \cdot 0.3333333333333333} \]

      5. log-pow 15.8%

         \[\leadsto e^{\color{blue}{\left(3 \cdot \log \left(\mathsf{log1p}\left(N\right) - \log N\right)\right)} \cdot 0.3333333333333333} \]

   6. Applied egg-rr 15.8%

      \[\leadsto \color{blue}{e^{\left(3 \cdot \log \left(\mathsf{log1p}\left(N\right) - \log N\right)\right) \cdot 0.3333333333333333}} \]

   7. Taylor expanded in N around inf 94.3%

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

      1. log-rec 94.3%

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

      2. +-commutative 94.3%

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

      3. unsub-neg 94.3%

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

      4. associate-*r/ 94.3%

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

      5. metadata-eval 94.3%

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

      6. associate-*r/ 94.3%

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

      7. metadata-eval 94.3%

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

   9. Simplified 94.3%

      \[\leadsto e^{\color{blue}{\left(\frac{0.20833333333333334}{{N}^{2}} - \log N\right) - \frac{0.5}{N}}} \]
   10. Step-by-step derivation

       1. sub-neg 94.3%

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

       2. exp-sum 94.5%

          \[\leadsto \color{blue}{e^{\frac{0.20833333333333334}{{N}^{2}} - \log N} \cdot e^{-\frac{0.5}{N}}} \]

       3. exp-diff 94.5%

          \[\leadsto \color{blue}{\frac{e^{\frac{0.20833333333333334}{{N}^{2}}}}{e^{\log N}}} \cdot e^{-\frac{0.5}{N}} \]

       4. add-exp-log 99.7%

          \[\leadsto \frac{e^{\frac{0.20833333333333334}{{N}^{2}}}}{\color{blue}{N}} \cdot e^{-\frac{0.5}{N}} \]

       5. div-inv 99.7%

          \[\leadsto \frac{e^{\color{blue}{0.20833333333333334 \cdot \frac{1}{{N}^{2}}}}}{N} \cdot e^{-\frac{0.5}{N}} \]

       6. exp-prod 99.7%

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

       7. pow-flip 99.7%

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

       8. metadata-eval 99.7%

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

       9. distribute-neg-frac 99.7%

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

       10. metadata-eval 99.7%

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

   11. Applied egg-rr 99.7%

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

   12. Taylor expanded in N around 0 99.7%

       \[\leadsto \color{blue}{\frac{e^{\frac{-0.5}{N}} \cdot e^{\frac{0.20833333333333334}{{N}^{2}}}}{N}} \]
   13. Step-by-step derivation

       1. prod-exp 99.7%

          \[\leadsto \frac{\color{blue}{e^{\frac{-0.5}{N} + \frac{0.20833333333333334}{{N}^{2}}}}}{N} \]

   14. Simplified 99.7%

       \[\leadsto \color{blue}{\frac{e^{\frac{-0.5}{N} + \frac{0.20833333333333334}{{N}^{2}}}}{N}} \]

   if 1.00000000000000005e-4 < (-.f64 (log.f64 (+.f64 N 1)) (log.f64 N))

   1. Initial program 84.9%

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

      1. +-commutative 84.9%

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

      2. log1p-define 85.0%

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

   3. Simplified 85.0%

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

      1. add-log-exp 84.6%

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

      2. add-cube-cbrt 84.7%

         \[\leadsto \log \color{blue}{\left(\left(\sqrt[3]{e^{\mathsf{log1p}\left(N\right) - \log N}} \cdot \sqrt[3]{e^{\mathsf{log1p}\left(N\right) - \log N}}\right) \cdot \sqrt[3]{e^{\mathsf{log1p}\left(N\right) - \log N}}\right)} \]

      3. log-prod 84.3%

         \[\leadsto \color{blue}{\log \left(\sqrt[3]{e^{\mathsf{log1p}\left(N\right) - \log N}} \cdot \sqrt[3]{e^{\mathsf{log1p}\left(N\right) - \log N}}\right) + \log \left(\sqrt[3]{e^{\mathsf{log1p}\left(N\right) - \log N}}\right)} \]

      4. pow2 84.3%

         \[\leadsto \log \color{blue}{\left({\left(\sqrt[3]{e^{\mathsf{log1p}\left(N\right) - \log N}}\right)}^{2}\right)} + \log \left(\sqrt[3]{e^{\mathsf{log1p}\left(N\right) - \log N}}\right) \]

      5. exp-diff 84.5%

         \[\leadsto \log \left({\left(\sqrt[3]{\color{blue}{\frac{e^{\mathsf{log1p}\left(N\right)}}{e^{\log N}}}}\right)}^{2}\right) + \log \left(\sqrt[3]{e^{\mathsf{log1p}\left(N\right) - \log N}}\right) \]

      6. add-exp-log 84.8%

         \[\leadsto \log \left({\left(\sqrt[3]{\frac{e^{\mathsf{log1p}\left(N\right)}}{\color{blue}{N}}}\right)}^{2}\right) + \log \left(\sqrt[3]{e^{\mathsf{log1p}\left(N\right) - \log N}}\right) \]

      7. log1p-undefine 85.0%

         \[\leadsto \log \left({\left(\sqrt[3]{\frac{e^{\color{blue}{\log \left(1 + N\right)}}}{N}}\right)}^{2}\right) + \log \left(\sqrt[3]{e^{\mathsf{log1p}\left(N\right) - \log N}}\right) \]

      8. rem-exp-log 85.4%

         \[\leadsto \log \left({\left(\sqrt[3]{\frac{\color{blue}{1 + N}}{N}}\right)}^{2}\right) + \log \left(\sqrt[3]{e^{\mathsf{log1p}\left(N\right) - \log N}}\right) \]

      9. +-commutative 85.4%

         \[\leadsto \log \left({\left(\sqrt[3]{\frac{\color{blue}{N + 1}}{N}}\right)}^{2}\right) + \log \left(\sqrt[3]{e^{\mathsf{log1p}\left(N\right) - \log N}}\right) \]

      10. exp-diff 85.8%

          \[\leadsto \log \left({\left(\sqrt[3]{\frac{N + 1}{N}}\right)}^{2}\right) + \log \left(\sqrt[3]{\color{blue}{\frac{e^{\mathsf{log1p}\left(N\right)}}{e^{\log N}}}}\right) \]

      11. add-exp-log 86.3%

          \[\leadsto \log \left({\left(\sqrt[3]{\frac{N + 1}{N}}\right)}^{2}\right) + \log \left(\sqrt[3]{\frac{e^{\mathsf{log1p}\left(N\right)}}{\color{blue}{N}}}\right) \]

   6. Applied egg-rr 86.3%

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

      1. log-pow 86.3%

         \[\leadsto \color{blue}{2 \cdot \log \left(\sqrt[3]{\frac{N + 1}{N}}\right)} + \log \left(\sqrt[3]{\frac{N + 1}{N}}\right) \]

      2. distribute-lft1-in 86.3%

         \[\leadsto \color{blue}{\left(2 + 1\right) \cdot \log \left(\sqrt[3]{\frac{N + 1}{N}}\right)} \]

      3. metadata-eval 86.3%

         \[\leadsto \color{blue}{3} \cdot \log \left(\sqrt[3]{\frac{N + 1}{N}}\right) \]

   8. Simplified 86.3%

      \[\leadsto \color{blue}{3 \cdot \log \left(\sqrt[3]{\frac{N + 1}{N}}\right)} \]
   9. Step-by-step derivation

      1. *-commutative 86.3%

         \[\leadsto \color{blue}{\log \left(\sqrt[3]{\frac{N + 1}{N}}\right) \cdot 3} \]

      2. add-log-exp 86.8%

         \[\leadsto \color{blue}{\log \left(e^{\log \left(\sqrt[3]{\frac{N + 1}{N}}\right) \cdot 3}\right)} \]

      3. exp-to-pow 86.8%

         \[\leadsto \log \color{blue}{\left({\left(\sqrt[3]{\frac{N + 1}{N}}\right)}^{3}\right)} \]

      4. pow3 86.2%

         \[\leadsto \log \color{blue}{\left(\left(\sqrt[3]{\frac{N + 1}{N}} \cdot \sqrt[3]{\frac{N + 1}{N}}\right) \cdot \sqrt[3]{\frac{N + 1}{N}}\right)} \]

      5. add-cube-cbrt 88.8%

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

      6. clear-num 88.6%

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

      7. log-rec 90.3%

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

   10. Applied egg-rr 90.3%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(1 + N\right) - \log N \leq 0.0001:\\ \;\;\;\;\frac{e^{\frac{-0.5}{N} + \frac{0.20833333333333334}{{N}^{2}}}}{N}\\ \mathbf{else}:\\ \;\;\;\;-\log \left(\frac{N}{1 + N}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \frac{1}{N} + \left(\frac{0.3333333333333333}{{N}^{3}} - \left(\frac{0.5}{{N}^{2}} + \frac{0.25}{{N}^{4}}\right)\right) \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (+
  (/ 1.0 N)
  (-
   (/ 0.3333333333333333 (pow N 3.0))
   (+ (/ 0.5 (pow N 2.0)) (/ 0.25 (pow N 4.0))))))

C

double code(double N) {
	return (1.0 / N) + ((0.3333333333333333 / pow(N, 3.0)) - ((0.5 / pow(N, 2.0)) + (0.25 / pow(N, 4.0))));
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    code = (1.0d0 / n) + ((0.3333333333333333d0 / (n ** 3.0d0)) - ((0.5d0 / (n ** 2.0d0)) + (0.25d0 / (n ** 4.0d0))))
end function

Java

public static double code(double N) {
	return (1.0 / N) + ((0.3333333333333333 / Math.pow(N, 3.0)) - ((0.5 / Math.pow(N, 2.0)) + (0.25 / Math.pow(N, 4.0))));
}

Python

def code(N):
	return (1.0 / N) + ((0.3333333333333333 / math.pow(N, 3.0)) - ((0.5 / math.pow(N, 2.0)) + (0.25 / math.pow(N, 4.0))))

Julia

function code(N)
	return Float64(Float64(1.0 / N) + Float64(Float64(0.3333333333333333 / (N ^ 3.0)) - Float64(Float64(0.5 / (N ^ 2.0)) + Float64(0.25 / (N ^ 4.0)))))
end

MATLAB

function tmp = code(N)
	tmp = (1.0 / N) + ((0.3333333333333333 / (N ^ 3.0)) - ((0.5 / (N ^ 2.0)) + (0.25 / (N ^ 4.0))));
end

Wolfram

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

Derivation

1. Initial program 20.6%

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

   1. +-commutative 20.6%

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

   2. log1p-define 20.7%

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

3. Simplified 20.7%

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

5. Taylor expanded in N around inf 98.2%

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

   1. +-commutative 98.2%

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

   2. associate--l+ 98.2%

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

   3. associate-*r/ 98.2%

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

   4. metadata-eval 98.2%

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

   5. +-commutative 98.2%

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

   6. associate-*r/ 98.2%

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

   7. metadata-eval 98.2%

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

   8. associate-*r/ 98.2%

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

   9. metadata-eval 98.2%

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

7. Simplified 98.2%

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

8. Final simplification 98.2%

   \[\leadsto \frac{1}{N} + \left(\frac{0.3333333333333333}{{N}^{3}} - \left(\frac{0.5}{{N}^{2}} + \frac{0.25}{{N}^{4}}\right)\right) \]
9. Add Preprocessing

Alternative 3: 97.9% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 13.5%

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

      1. +-commutative 13.5%

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

      2. log1p-define 13.6%

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

   3. Simplified 13.6%

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

      1. add-cbrt-cube 13.6%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\left(\mathsf{log1p}\left(N\right) - \log N\right) \cdot \left(\mathsf{log1p}\left(N\right) - \log N\right)\right) \cdot \left(\mathsf{log1p}\left(N\right) - \log N\right)}} \]

      2. pow1/3 13.6%

         \[\leadsto \color{blue}{{\left(\left(\left(\mathsf{log1p}\left(N\right) - \log N\right) \cdot \left(\mathsf{log1p}\left(N\right) - \log N\right)\right) \cdot \left(\mathsf{log1p}\left(N\right) - \log N\right)\right)}^{0.3333333333333333}} \]

      3. pow-to-exp 13.6%

         \[\leadsto \color{blue}{e^{\log \left(\left(\left(\mathsf{log1p}\left(N\right) - \log N\right) \cdot \left(\mathsf{log1p}\left(N\right) - \log N\right)\right) \cdot \left(\mathsf{log1p}\left(N\right) - \log N\right)\right) \cdot 0.3333333333333333}} \]

      4. pow3 13.6%

         \[\leadsto e^{\log \color{blue}{\left({\left(\mathsf{log1p}\left(N\right) - \log N\right)}^{3}\right)} \cdot 0.3333333333333333} \]

      5. log-pow 13.6%

         \[\leadsto e^{\color{blue}{\left(3 \cdot \log \left(\mathsf{log1p}\left(N\right) - \log N\right)\right)} \cdot 0.3333333333333333} \]

   6. Applied egg-rr 13.6%

      \[\leadsto \color{blue}{e^{\left(3 \cdot \log \left(\mathsf{log1p}\left(N\right) - \log N\right)\right) \cdot 0.3333333333333333}} \]

   7. Taylor expanded in N around inf 94.3%

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

      1. log-rec 94.3%

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

      2. +-commutative 94.3%

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

      3. unsub-neg 94.3%

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

      4. associate-*r/ 94.3%

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

      5. metadata-eval 94.3%

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

      6. associate-*r/ 94.3%

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

      7. metadata-eval 94.3%

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

   9. Simplified 94.3%

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

   10. Taylor expanded in N around inf 93.9%

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

       1. exp-neg 94.0%

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

       2. log-rec 94.0%

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

       3. mul-1-neg 94.0%

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

       4. remove-double-neg 94.0%

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

       5. exp-neg 93.9%

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

       6. log-rec 93.9%

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

       7. rem-exp-log 99.2%

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

       8. associate-*r/ 99.2%

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

   12. Simplified 99.2%

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

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

   1. Initial program 81.0%

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

      1. +-commutative 81.0%

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

      2. log1p-define 81.0%

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

   3. Simplified 81.0%

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

      1. add-log-exp 80.8%

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

      2. add-cube-cbrt 80.8%

         \[\leadsto \log \color{blue}{\left(\left(\sqrt[3]{e^{\mathsf{log1p}\left(N\right) - \log N}} \cdot \sqrt[3]{e^{\mathsf{log1p}\left(N\right) - \log N}}\right) \cdot \sqrt[3]{e^{\mathsf{log1p}\left(N\right) - \log N}}\right)} \]

      3. log-prod 80.6%

         \[\leadsto \color{blue}{\log \left(\sqrt[3]{e^{\mathsf{log1p}\left(N\right) - \log N}} \cdot \sqrt[3]{e^{\mathsf{log1p}\left(N\right) - \log N}}\right) + \log \left(\sqrt[3]{e^{\mathsf{log1p}\left(N\right) - \log N}}\right)} \]

      4. pow2 80.6%

         \[\leadsto \log \color{blue}{\left({\left(\sqrt[3]{e^{\mathsf{log1p}\left(N\right) - \log N}}\right)}^{2}\right)} + \log \left(\sqrt[3]{e^{\mathsf{log1p}\left(N\right) - \log N}}\right) \]

      5. exp-diff 80.8%

         \[\leadsto \log \left({\left(\sqrt[3]{\color{blue}{\frac{e^{\mathsf{log1p}\left(N\right)}}{e^{\log N}}}}\right)}^{2}\right) + \log \left(\sqrt[3]{e^{\mathsf{log1p}\left(N\right) - \log N}}\right) \]

      6. add-exp-log 81.3%

         \[\leadsto \log \left({\left(\sqrt[3]{\frac{e^{\mathsf{log1p}\left(N\right)}}{\color{blue}{N}}}\right)}^{2}\right) + \log \left(\sqrt[3]{e^{\mathsf{log1p}\left(N\right) - \log N}}\right) \]

      7. log1p-undefine 81.4%

         \[\leadsto \log \left({\left(\sqrt[3]{\frac{e^{\color{blue}{\log \left(1 + N\right)}}}{N}}\right)}^{2}\right) + \log \left(\sqrt[3]{e^{\mathsf{log1p}\left(N\right) - \log N}}\right) \]

      8. rem-exp-log 82.2%

         \[\leadsto \log \left({\left(\sqrt[3]{\frac{\color{blue}{1 + N}}{N}}\right)}^{2}\right) + \log \left(\sqrt[3]{e^{\mathsf{log1p}\left(N\right) - \log N}}\right) \]

      9. +-commutative 82.2%

         \[\leadsto \log \left({\left(\sqrt[3]{\frac{\color{blue}{N + 1}}{N}}\right)}^{2}\right) + \log \left(\sqrt[3]{e^{\mathsf{log1p}\left(N\right) - \log N}}\right) \]

      10. exp-diff 82.5%

          \[\leadsto \log \left({\left(\sqrt[3]{\frac{N + 1}{N}}\right)}^{2}\right) + \log \left(\sqrt[3]{\color{blue}{\frac{e^{\mathsf{log1p}\left(N\right)}}{e^{\log N}}}}\right) \]

      11. add-exp-log 83.1%

          \[\leadsto \log \left({\left(\sqrt[3]{\frac{N + 1}{N}}\right)}^{2}\right) + \log \left(\sqrt[3]{\frac{e^{\mathsf{log1p}\left(N\right)}}{\color{blue}{N}}}\right) \]

   6. Applied egg-rr 82.3%

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

      1. log-pow 82.5%

         \[\leadsto \color{blue}{2 \cdot \log \left(\sqrt[3]{\frac{N + 1}{N}}\right)} + \log \left(\sqrt[3]{\frac{N + 1}{N}}\right) \]

      2. distribute-lft1-in 82.5%

         \[\leadsto \color{blue}{\left(2 + 1\right) \cdot \log \left(\sqrt[3]{\frac{N + 1}{N}}\right)} \]

      3. metadata-eval 82.5%

         \[\leadsto \color{blue}{3} \cdot \log \left(\sqrt[3]{\frac{N + 1}{N}}\right) \]

   8. Simplified 82.5%

      \[\leadsto \color{blue}{3 \cdot \log \left(\sqrt[3]{\frac{N + 1}{N}}\right)} \]
   9. Step-by-step derivation

      1. *-commutative 82.5%

         \[\leadsto \color{blue}{\log \left(\sqrt[3]{\frac{N + 1}{N}}\right) \cdot 3} \]

      2. add-log-exp 82.7%

         \[\leadsto \color{blue}{\log \left(e^{\log \left(\sqrt[3]{\frac{N + 1}{N}}\right) \cdot 3}\right)} \]

      3. exp-to-pow 82.7%

         \[\leadsto \log \color{blue}{\left({\left(\sqrt[3]{\frac{N + 1}{N}}\right)}^{3}\right)} \]

      4. pow3 82.3%

         \[\leadsto \log \color{blue}{\left(\left(\sqrt[3]{\frac{N + 1}{N}} \cdot \sqrt[3]{\frac{N + 1}{N}}\right) \cdot \sqrt[3]{\frac{N + 1}{N}}\right)} \]

      5. add-cube-cbrt 85.5%

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

      6. clear-num 85.4%

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

      7. log-rec 86.9%

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

   10. Applied egg-rr 86.9%

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

4. Final simplification 97.9%

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

Alternative 4: 97.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;N \leq 195000:\\ \;\;\;\;\log \left(\frac{1 + N}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N} - \frac{\frac{0.5}{N}}{N}\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= N 195000.0) (log (/ (+ 1.0 N) N)) (- (/ 1.0 N) (/ (/ 0.5 N) N))))

C

double code(double N) {
	double tmp;
	if (N <= 195000.0) {
		tmp = log(((1.0 + N) / N));
	} else {
		tmp = (1.0 / N) - ((0.5 / N) / N);
	}
	return tmp;
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= 195000.0d0) then
        tmp = log(((1.0d0 + n) / n))
    else
        tmp = (1.0d0 / n) - ((0.5d0 / n) / n)
    end if
    code = tmp
end function

Java

public static double code(double N) {
	double tmp;
	if (N <= 195000.0) {
		tmp = Math.log(((1.0 + N) / N));
	} else {
		tmp = (1.0 / N) - ((0.5 / N) / N);
	}
	return tmp;
}

Python

def code(N):
	tmp = 0
	if N <= 195000.0:
		tmp = math.log(((1.0 + N) / N))
	else:
		tmp = (1.0 / N) - ((0.5 / N) / N)
	return tmp

Julia

function code(N)
	tmp = 0.0
	if (N <= 195000.0)
		tmp = log(Float64(Float64(1.0 + N) / N));
	else
		tmp = Float64(Float64(1.0 / N) - Float64(Float64(0.5 / N) / N));
	end
	return tmp
end

MATLAB

function tmp_2 = code(N)
	tmp = 0.0;
	if (N <= 195000.0)
		tmp = log(((1.0 + N) / N));
	else
		tmp = (1.0 / N) - ((0.5 / N) / N);
	end
	tmp_2 = tmp;
end

Wolfram

code[N_] := If[LessEqual[N, 195000.0], N[Log[N[(N[(1.0 + N), $MachinePrecision] / N), $MachinePrecision]], $MachinePrecision], N[(N[(1.0 / N), $MachinePrecision] - N[(N[(0.5 / N), $MachinePrecision] / N), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if N < 195000

   1. Initial program 81.8%

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

      1. +-commutative 81.8%

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

      2. log1p-define 81.9%

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

   3. Simplified 81.9%

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

      1. add-log-exp 81.9%

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

      2. log1p-expm1-u 81.9%

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

      3. log1p-undefine 81.9%

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

      4. diff-log 81.7%

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

      5. log1p-undefine 81.7%

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

      6. rem-exp-log 82.9%

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

      7. +-commutative 82.9%

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

      8. pow1 82.9%

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

      9. exp-to-pow 82.9%

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

      10. log1p-undefine 82.9%

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

      11. log1p-expm1-u 82.9%

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

      12. pow-to-exp 86.4%

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

      13. pow1 86.4%

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

   6. Applied egg-rr 86.4%

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

   if 195000 < N

   1. Initial program 14.0%

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

      1. +-commutative 14.0%

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

      2. log1p-define 14.1%

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

   3. Simplified 14.1%

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

      1. add-cbrt-cube 14.1%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\left(\mathsf{log1p}\left(N\right) - \log N\right) \cdot \left(\mathsf{log1p}\left(N\right) - \log N\right)\right) \cdot \left(\mathsf{log1p}\left(N\right) - \log N\right)}} \]

      2. pow1/3 14.1%

         \[\leadsto \color{blue}{{\left(\left(\left(\mathsf{log1p}\left(N\right) - \log N\right) \cdot \left(\mathsf{log1p}\left(N\right) - \log N\right)\right) \cdot \left(\mathsf{log1p}\left(N\right) - \log N\right)\right)}^{0.3333333333333333}} \]

      3. pow-to-exp 14.1%

         \[\leadsto \color{blue}{e^{\log \left(\left(\left(\mathsf{log1p}\left(N\right) - \log N\right) \cdot \left(\mathsf{log1p}\left(N\right) - \log N\right)\right) \cdot \left(\mathsf{log1p}\left(N\right) - \log N\right)\right) \cdot 0.3333333333333333}} \]

      4. pow3 14.1%

         \[\leadsto e^{\log \color{blue}{\left({\left(\mathsf{log1p}\left(N\right) - \log N\right)}^{3}\right)} \cdot 0.3333333333333333} \]

      5. log-pow 14.1%

         \[\leadsto e^{\color{blue}{\left(3 \cdot \log \left(\mathsf{log1p}\left(N\right) - \log N\right)\right)} \cdot 0.3333333333333333} \]

   6. Applied egg-rr 14.1%

      \[\leadsto \color{blue}{e^{\left(3 \cdot \log \left(\mathsf{log1p}\left(N\right) - \log N\right)\right) \cdot 0.3333333333333333}} \]

   7. Taylor expanded in N around inf 94.3%

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

      1. log-rec 94.3%

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

      2. +-commutative 94.3%

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

      3. unsub-neg 94.3%

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

      4. associate-*r/ 94.3%

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

      5. metadata-eval 94.3%

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

      6. associate-*r/ 94.3%

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

      7. metadata-eval 94.3%

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

   9. Simplified 94.3%

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

   10. Taylor expanded in N around inf 93.8%

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

       1. exp-neg 93.8%

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

       2. log-rec 93.8%

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

       3. mul-1-neg 93.8%

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

       4. remove-double-neg 93.8%

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

       5. exp-neg 93.8%

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

       6. log-rec 93.8%

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

       7. rem-exp-log 99.0%

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

       8. associate-*r/ 99.0%

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

   12. Simplified 99.0%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;N \leq 195000:\\ \;\;\;\;\log \left(\frac{1 + N}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N} - \frac{\frac{0.5}{N}}{N}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 92.4% accurate, 22.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 20.6%

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

   1. +-commutative 20.6%

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

   2. log1p-define 20.7%

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

3. Simplified 20.7%

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

   1. add-cbrt-cube 20.7%

      \[\leadsto \color{blue}{\sqrt[3]{\left(\left(\mathsf{log1p}\left(N\right) - \log N\right) \cdot \left(\mathsf{log1p}\left(N\right) - \log N\right)\right) \cdot \left(\mathsf{log1p}\left(N\right) - \log N\right)}} \]

   2. pow1/3 20.7%

      \[\leadsto \color{blue}{{\left(\left(\left(\mathsf{log1p}\left(N\right) - \log N\right) \cdot \left(\mathsf{log1p}\left(N\right) - \log N\right)\right) \cdot \left(\mathsf{log1p}\left(N\right) - \log N\right)\right)}^{0.3333333333333333}} \]

   3. pow-to-exp 20.7%

      \[\leadsto \color{blue}{e^{\log \left(\left(\left(\mathsf{log1p}\left(N\right) - \log N\right) \cdot \left(\mathsf{log1p}\left(N\right) - \log N\right)\right) \cdot \left(\mathsf{log1p}\left(N\right) - \log N\right)\right) \cdot 0.3333333333333333}} \]

   4. pow3 20.7%

      \[\leadsto e^{\log \color{blue}{\left({\left(\mathsf{log1p}\left(N\right) - \log N\right)}^{3}\right)} \cdot 0.3333333333333333} \]

   5. log-pow 20.7%

      \[\leadsto e^{\color{blue}{\left(3 \cdot \log \left(\mathsf{log1p}\left(N\right) - \log N\right)\right)} \cdot 0.3333333333333333} \]

6. Applied egg-rr 20.7%

   \[\leadsto \color{blue}{e^{\left(3 \cdot \log \left(\mathsf{log1p}\left(N\right) - \log N\right)\right) \cdot 0.3333333333333333}} \]

7. Taylor expanded in N around inf 92.1%

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

   1. log-rec 92.1%

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

   2. +-commutative 92.1%

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

   3. unsub-neg 92.1%

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

   4. associate-*r/ 92.1%

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

   5. metadata-eval 92.1%

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

   6. associate-*r/ 92.1%

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

   7. metadata-eval 92.1%

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

9. Simplified 92.1%

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

10. Taylor expanded in N around inf 89.8%

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

    1. exp-neg 89.9%

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

    2. log-rec 89.9%

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

    3. mul-1-neg 89.9%

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

    4. remove-double-neg 89.9%

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

    5. exp-neg 89.8%

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

    6. log-rec 89.8%

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

    7. rem-exp-log 94.6%

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

    8. associate-*r/ 94.6%

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

12. Simplified 94.6%

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

13. Final simplification 94.6%

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

Alternative 6: 84.7% 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 20.6%

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

   1. +-commutative 20.6%

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

   2. log1p-define 20.7%

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

3. Simplified 20.7%

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

5. Taylor expanded in N around inf 87.0%

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

6. Final simplification 87.0%

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

Developer target: 99.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Reproduce

herbie shell --seed 2024034 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  :pre (and (> N 1.0) (< N 1e+40))

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

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