2log (problem 3.3.6)

Percentage Accurate: 23.3% → 99.3%

Time: 11.8s

Alternatives: 14

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: 23.3% 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: 99.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\frac{N}{N + 1}\right)\\ \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 0.0001:\\ \;\;\;\;\frac{1}{\frac{N}{1 + \frac{-0.5 - \frac{\frac{0.25}{N} + -0.3333333333333333}{N}}{N}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{t\_0}^{2}}{0 - t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (let* ((t_0 (log (/ N (+ N 1.0)))))
   (if (<= (- (log (+ N 1.0)) (log N)) 0.0001)
     (/
      1.0
      (/ N (+ 1.0 (/ (- -0.5 (/ (+ (/ 0.25 N) -0.3333333333333333) N)) N))))
     (/ (pow t_0 2.0) (- 0.0 t_0)))))

C

double code(double N) {
	double t_0 = log((N / (N + 1.0)));
	double tmp;
	if ((log((N + 1.0)) - log(N)) <= 0.0001) {
		tmp = 1.0 / (N / (1.0 + ((-0.5 - (((0.25 / N) + -0.3333333333333333) / N)) / N)));
	} else {
		tmp = pow(t_0, 2.0) / (0.0 - t_0);
	}
	return tmp;
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = log((n / (n + 1.0d0)))
    if ((log((n + 1.0d0)) - log(n)) <= 0.0001d0) then
        tmp = 1.0d0 / (n / (1.0d0 + (((-0.5d0) - (((0.25d0 / n) + (-0.3333333333333333d0)) / n)) / n)))
    else
        tmp = (t_0 ** 2.0d0) / (0.0d0 - t_0)
    end if
    code = tmp
end function

Java

public static double code(double N) {
	double t_0 = Math.log((N / (N + 1.0)));
	double tmp;
	if ((Math.log((N + 1.0)) - Math.log(N)) <= 0.0001) {
		tmp = 1.0 / (N / (1.0 + ((-0.5 - (((0.25 / N) + -0.3333333333333333) / N)) / N)));
	} else {
		tmp = Math.pow(t_0, 2.0) / (0.0 - t_0);
	}
	return tmp;
}

Python

def code(N):
	t_0 = math.log((N / (N + 1.0)))
	tmp = 0
	if (math.log((N + 1.0)) - math.log(N)) <= 0.0001:
		tmp = 1.0 / (N / (1.0 + ((-0.5 - (((0.25 / N) + -0.3333333333333333) / N)) / N)))
	else:
		tmp = math.pow(t_0, 2.0) / (0.0 - t_0)
	return tmp

Julia

function code(N)
	t_0 = log(Float64(N / Float64(N + 1.0)))
	tmp = 0.0
	if (Float64(log(Float64(N + 1.0)) - log(N)) <= 0.0001)
		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((t_0 ^ 2.0) / Float64(0.0 - t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(N)
	t_0 = log((N / (N + 1.0)));
	tmp = 0.0;
	if ((log((N + 1.0)) - log(N)) <= 0.0001)
		tmp = 1.0 / (N / (1.0 + ((-0.5 - (((0.25 / N) + -0.3333333333333333) / N)) / N)));
	else
		tmp = (t_0 ^ 2.0) / (0.0 - t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[N_] := Block[{t$95$0 = N[Log[N[(N / N[(N + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[Log[N[(N + 1.0), $MachinePrecision]], $MachinePrecision] - N[Log[N], $MachinePrecision]), $MachinePrecision], 0.0001], 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[(N[Power[t$95$0, 2.0], $MachinePrecision] / N[(0.0 - t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (log.f64 (+.f64 N #s(literal 1 binary64))) (log.f64 N)) < 1.00000000000000005e-4

   1. Initial program 15.2%

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

      1. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\log \left(N + 1\right), \color{blue}{\log N}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\log \left(1 + N\right), \log N\right) \]

      3. accelerator-lowering-log1p.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{log1p.f64}\left(N\right), \log \color{blue}{N}\right) \]

      4. log-lowering-log.f64 15.2%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{log1p.f64}\left(N\right), \mathsf{log.f64}\left(N\right)\right) \]

   3. Simplified 15.2%

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

   5. Taylor expanded in N around inf

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

   7. Simplified 99.9%

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

      1. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{N}{1 + \frac{\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}{N}}}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{N}{1 + \frac{\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}{N}}\right)}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \color{blue}{\left(1 + \frac{\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}{N}\right)}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}{N}\right)}\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}\right), \color{blue}{N}\right)\right)\right)\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{-1}{2}, \left(\frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}\right)\right), N\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\left(\frac{\frac{1}{4}}{N} + \frac{-1}{3}\right), N\right)\right), N\right)\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{\frac{1}{4}}{N}\right), \frac{-1}{3}\right), N\right)\right), N\right)\right)\right)\right) \]

      9. /-lowering-/.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{4}, N\right), \frac{-1}{3}\right), N\right)\right), N\right)\right)\right)\right) \]

   9. Applied egg-rr 100.0%

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

   if 1.00000000000000005e-4 < (-.f64 (log.f64 (+.f64 N #s(literal 1 binary64))) (log.f64 N))

   1. Initial program 92.2%

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

      1. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\log \left(N + 1\right), \color{blue}{\log N}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\log \left(1 + N\right), \log N\right) \]

      3. accelerator-lowering-log1p.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{log1p.f64}\left(N\right), \log \color{blue}{N}\right) \]

      4. log-lowering-log.f64 92.3%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{log1p.f64}\left(N\right), \mathsf{log.f64}\left(N\right)\right) \]

   3. Simplified 92.3%

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

      1. diff-log N/A

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

      2. clear-num N/A

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

      3. neg-log N/A

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

      4. diff-log N/A

         \[\leadsto \mathsf{neg}\left(\left(\log N - \log \left(1 + N\right)\right)\right) \]

      5. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\log N - \log \left(1 + N\right)\right)\right) \]

      6. diff-log N/A

         \[\leadsto \mathsf{neg.f64}\left(\log \left(\frac{N}{1 + N}\right)\right) \]

      7. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\left(\frac{N}{1 + N}\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(N, \left(1 + N\right)\right)\right)\right) \]

      9. +-lowering-+.f64 94.3%

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, N\right)\right)\right)\right) \]

   6. Applied egg-rr 94.3%

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

      1. neg-sub0 N/A

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

      2. flip-- N/A

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

      3. metadata-eval N/A

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

      4. log-prod N/A

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

      5. clear-num N/A

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

      6. div-inv N/A

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

      7. clear-num N/A

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

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(0 \cdot 0 - \log \left(\frac{N}{1 + N}\right) \cdot \log \left(\frac{N}{1 + N}\right)\right), \color{blue}{\log \left(\frac{N}{1 + N}\right)}\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left(0 - \log \left(\frac{N}{1 + N}\right) \cdot \log \left(\frac{N}{1 + N}\right)\right), \log \left(\frac{\color{blue}{N}}{1 + N}\right)\right) \]

      10. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \left(\log \left(\frac{N}{1 + N}\right) \cdot \log \left(\frac{N}{1 + N}\right)\right)\right), \log \color{blue}{\left(\frac{N}{1 + N}\right)}\right) \]

      11. pow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \left({\log \left(\frac{N}{1 + N}\right)}^{2}\right)\right), \log \left(\frac{N}{\color{blue}{1 + N}}\right)\right) \]

      12. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{pow.f64}\left(\log \left(\frac{N}{1 + N}\right), 2\right)\right), \log \left(\frac{N}{\color{blue}{1 + N}}\right)\right) \]

      13. log-lowering-log.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{pow.f64}\left(\mathsf{log.f64}\left(\left(\frac{N}{1 + N}\right)\right), 2\right)\right), \log \left(\frac{N}{\color{blue}{1} + N}\right)\right) \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{pow.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(N, \left(1 + N\right)\right)\right), 2\right)\right), \log \left(\frac{N}{1 + N}\right)\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{pow.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, N\right)\right)\right), 2\right)\right), \log \left(\frac{N}{1 + N}\right)\right) \]

   8. Applied egg-rr 94.5%

      \[\leadsto \color{blue}{\frac{0 - {\log \left(\frac{N}{1 + N}\right)}^{2}}{\log \left(\frac{N}{1 + N}\right)}} \]
   9. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left({\log \left(\frac{N}{1 + N}\right)}^{2}\right)\right), \mathsf{log.f64}\left(\color{blue}{\mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, N\right)\right)}\right)\right) \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\left({\log \left(\frac{N}{1 + N}\right)}^{2}\right)\right), \mathsf{log.f64}\left(\color{blue}{\mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, N\right)\right)}\right)\right) \]

      3. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{pow.f64}\left(\log \left(\frac{N}{1 + N}\right), 2\right)\right), \mathsf{log.f64}\left(\mathsf{/.f64}\left(\color{blue}{N}, \mathsf{+.f64}\left(1, N\right)\right)\right)\right) \]

      4. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{pow.f64}\left(\mathsf{log.f64}\left(\left(\frac{N}{1 + N}\right)\right), 2\right)\right), \mathsf{log.f64}\left(\mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, N\right)\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{pow.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(N, \left(1 + N\right)\right)\right), 2\right)\right), \mathsf{log.f64}\left(\mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, N\right)\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{pow.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(N, \left(N + 1\right)\right)\right), 2\right)\right), \mathsf{log.f64}\left(\mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, N\right)\right)\right)\right) \]

      7. +-lowering-+.f64 94.5%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{pow.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(N, \mathsf{+.f64}\left(N, 1\right)\right)\right), 2\right)\right), \mathsf{log.f64}\left(\mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, N\right)\right)\right)\right) \]

   10. Applied egg-rr 94.5%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 0.0001:\\ \;\;\;\;\frac{1}{\frac{N}{1 + \frac{-0.5 - \frac{\frac{0.25}{N} + -0.3333333333333333}{N}}{N}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\log \left(\frac{N}{N + 1}\right)}^{2}}{0 - \log \left(\frac{N}{N + 1}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 0.0001:\\ \;\;\;\;\frac{1}{\frac{N}{1 + \frac{-0.5 - \frac{\frac{0.25}{N} + -0.3333333333333333}{N}}{N}}}\\ \mathbf{else}:\\ \;\;\;\;0 - \log \left(\frac{N}{N + 1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= (- (log (+ N 1.0)) (log N)) 0.0001)
   (/
    1.0
    (/ N (+ 1.0 (/ (- -0.5 (/ (+ (/ 0.25 N) -0.3333333333333333) N)) N))))
   (- 0.0 (log (/ N (+ N 1.0))))))

C

double code(double N) {
	double tmp;
	if ((log((N + 1.0)) - log(N)) <= 0.0001) {
		tmp = 1.0 / (N / (1.0 + ((-0.5 - (((0.25 / N) + -0.3333333333333333) / N)) / N)));
	} else {
		tmp = 0.0 - 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)) <= 0.0001d0) then
        tmp = 1.0d0 / (n / (1.0d0 + (((-0.5d0) - (((0.25d0 / n) + (-0.3333333333333333d0)) / n)) / n)))
    else
        tmp = 0.0d0 - 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)) <= 0.0001) {
		tmp = 1.0 / (N / (1.0 + ((-0.5 - (((0.25 / N) + -0.3333333333333333) / N)) / N)));
	} else {
		tmp = 0.0 - Math.log((N / (N + 1.0)));
	}
	return tmp;
}

Python

def code(N):
	tmp = 0
	if (math.log((N + 1.0)) - math.log(N)) <= 0.0001:
		tmp = 1.0 / (N / (1.0 + ((-0.5 - (((0.25 / N) + -0.3333333333333333) / N)) / N)))
	else:
		tmp = 0.0 - math.log((N / (N + 1.0)))
	return tmp

Julia

function code(N)
	tmp = 0.0
	if (Float64(log(Float64(N + 1.0)) - log(N)) <= 0.0001)
		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(0.0 - 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)) <= 0.0001)
		tmp = 1.0 / (N / (1.0 + ((-0.5 - (((0.25 / N) + -0.3333333333333333) / N)) / N)));
	else
		tmp = 0.0 - 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], 0.0001], 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[(0.0 - N[Log[N[(N / N[(N + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (log.f64 (+.f64 N #s(literal 1 binary64))) (log.f64 N)) < 1.00000000000000005e-4

   1. Initial program 15.2%

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

      1. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\log \left(N + 1\right), \color{blue}{\log N}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\log \left(1 + N\right), \log N\right) \]

      3. accelerator-lowering-log1p.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{log1p.f64}\left(N\right), \log \color{blue}{N}\right) \]

      4. log-lowering-log.f64 15.2%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{log1p.f64}\left(N\right), \mathsf{log.f64}\left(N\right)\right) \]

   3. Simplified 15.2%

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

   5. Taylor expanded in N around inf

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

   7. Simplified 99.9%

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

      1. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{N}{1 + \frac{\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}{N}}}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{N}{1 + \frac{\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}{N}}\right)}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \color{blue}{\left(1 + \frac{\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}{N}\right)}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}{N}\right)}\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}\right), \color{blue}{N}\right)\right)\right)\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{-1}{2}, \left(\frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}\right)\right), N\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\left(\frac{\frac{1}{4}}{N} + \frac{-1}{3}\right), N\right)\right), N\right)\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{\frac{1}{4}}{N}\right), \frac{-1}{3}\right), N\right)\right), N\right)\right)\right)\right) \]

      9. /-lowering-/.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{4}, N\right), \frac{-1}{3}\right), N\right)\right), N\right)\right)\right)\right) \]

   9. Applied egg-rr 100.0%

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

   if 1.00000000000000005e-4 < (-.f64 (log.f64 (+.f64 N #s(literal 1 binary64))) (log.f64 N))

   1. Initial program 92.2%

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

      1. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\log \left(N + 1\right), \color{blue}{\log N}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\log \left(1 + N\right), \log N\right) \]

      3. accelerator-lowering-log1p.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{log1p.f64}\left(N\right), \log \color{blue}{N}\right) \]

      4. log-lowering-log.f64 92.3%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{log1p.f64}\left(N\right), \mathsf{log.f64}\left(N\right)\right) \]

   3. Simplified 92.3%

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

      1. diff-log N/A

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

      2. clear-num N/A

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

      3. neg-log N/A

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

      4. diff-log N/A

         \[\leadsto \mathsf{neg}\left(\left(\log N - \log \left(1 + N\right)\right)\right) \]

      5. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\log N - \log \left(1 + N\right)\right)\right) \]

      6. diff-log N/A

         \[\leadsto \mathsf{neg.f64}\left(\log \left(\frac{N}{1 + N}\right)\right) \]

      7. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\left(\frac{N}{1 + N}\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(N, \left(1 + N\right)\right)\right)\right) \]

      9. +-lowering-+.f64 94.3%

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, N\right)\right)\right)\right) \]

   6. Applied egg-rr 94.3%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 0.0001:\\ \;\;\;\;\frac{1}{\frac{N}{1 + \frac{-0.5 - \frac{\frac{0.25}{N} + -0.3333333333333333}{N}}{N}}}\\ \mathbf{else}:\\ \;\;\;\;0 - \log \left(\frac{N}{N + 1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;N \leq 1150:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{N}{1 + \frac{-0.5 - \frac{\frac{0.25}{N} + -0.3333333333333333}{N}}{N}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= N 1150.0)
   (log (/ (+ N 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 <= 1150.0) {
		tmp = log(((N + 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 <= 1150.0d0) then
        tmp = log(((n + 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 <= 1150.0) {
		tmp = Math.log(((N + 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 <= 1150.0:
		tmp = math.log(((N + 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 <= 1150.0)
		tmp = log(Float64(Float64(N + 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 <= 1150.0)
		tmp = log(((N + 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, 1150.0], N[Log[N[(N[(N + 1.0), $MachinePrecision] / N), $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 < 1150

   1. Initial program 92.2%

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

      1. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\log \left(N + 1\right), \color{blue}{\log N}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\log \left(1 + N\right), \log N\right) \]

      3. accelerator-lowering-log1p.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{log1p.f64}\left(N\right), \log \color{blue}{N}\right) \]

      4. log-lowering-log.f64 92.3%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{log1p.f64}\left(N\right), \mathsf{log.f64}\left(N\right)\right) \]

   3. Simplified 92.3%

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

      1. diff-log N/A

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

      2. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{log.f64}\left(\left(\frac{1 + N}{N}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{/.f64}\left(\left(1 + N\right), N\right)\right) \]

      4. +-lowering-+.f64 94.1%

         \[\leadsto \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(1, N\right), N\right)\right) \]

   6. Applied egg-rr 94.1%

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

   if 1150 < N

   1. Initial program 15.2%

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

      1. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\log \left(N + 1\right), \color{blue}{\log N}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\log \left(1 + N\right), \log N\right) \]

      3. accelerator-lowering-log1p.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{log1p.f64}\left(N\right), \log \color{blue}{N}\right) \]

      4. log-lowering-log.f64 15.2%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{log1p.f64}\left(N\right), \mathsf{log.f64}\left(N\right)\right) \]

   3. Simplified 15.2%

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

   5. Taylor expanded in N around inf

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

   7. Simplified 99.9%

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

      1. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{N}{1 + \frac{\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}{N}}}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{N}{1 + \frac{\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}{N}}\right)}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \color{blue}{\left(1 + \frac{\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}{N}\right)}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}{N}\right)}\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}\right), \color{blue}{N}\right)\right)\right)\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{-1}{2}, \left(\frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}\right)\right), N\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\left(\frac{\frac{1}{4}}{N} + \frac{-1}{3}\right), N\right)\right), N\right)\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{\frac{1}{4}}{N}\right), \frac{-1}{3}\right), N\right)\right), N\right)\right)\right)\right) \]

      9. /-lowering-/.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{4}, N\right), \frac{-1}{3}\right), N\right)\right), N\right)\right)\right)\right) \]

   9. Applied egg-rr 100.0%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;N \leq 1150:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{N}{1 + \frac{-0.5 - \frac{\frac{0.25}{N} + -0.3333333333333333}{N}}{N}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.8% accurate, 10.8× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{N}{\frac{1}{\frac{0.5 + \frac{-0.08333333333333333 + \frac{0.041666666666666664}{N}}{N}}{N} - -1}}} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 20.0%

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

   1. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\log \left(N + 1\right), \color{blue}{\log N}\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{\_.f64}\left(\log \left(1 + N\right), \log N\right) \]

   3. accelerator-lowering-log1p.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{log1p.f64}\left(N\right), \log \color{blue}{N}\right) \]

   4. log-lowering-log.f64 20.0%

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{log1p.f64}\left(N\right), \mathsf{log.f64}\left(N\right)\right) \]

3. Simplified 20.0%

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

5. Taylor expanded in N around inf

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

7. Simplified 97.1%

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

   1. clear-num N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{N}{1 + \frac{\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}{N}}}} \]

   2. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{N}{1 + \frac{\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}{N}}\right)}\right) \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \color{blue}{\left(1 + \frac{\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}{N}\right)}\right)\right) \]

   4. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}{N}\right)}\right)\right)\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}\right), \color{blue}{N}\right)\right)\right)\right) \]

   6. --lowering--.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{-1}{2}, \left(\frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}\right)\right), N\right)\right)\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\left(\frac{\frac{1}{4}}{N} + \frac{-1}{3}\right), N\right)\right), N\right)\right)\right)\right) \]

   8. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{\frac{1}{4}}{N}\right), \frac{-1}{3}\right), N\right)\right), N\right)\right)\right)\right) \]

   9. /-lowering-/.f64 97.1%

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{4}, N\right), \frac{-1}{3}\right), N\right)\right), N\right)\right)\right)\right) \]

9. Applied egg-rr 97.1%

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

10. Taylor expanded in N around -inf

    \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(-1 \cdot \left(N \cdot \left(-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N} - 1\right)\right)\right)}\right) \]
11. Step-by-step derivation

    1. associate-*r* N/A

       \[\leadsto \mathsf{/.f64}\left(1, \left(\left(-1 \cdot N\right) \cdot \color{blue}{\left(-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N} - 1\right)}\right)\right) \]

    2. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(1, \left(\left(-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N} - 1\right) \cdot \color{blue}{\left(-1 \cdot N\right)}\right)\right) \]

    3. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N} - 1\right), \color{blue}{\left(-1 \cdot N\right)}\right)\right) \]

12. Simplified 97.3%

    \[\leadsto \frac{1}{\color{blue}{\left(-1 - \frac{0.5 - \frac{0.08333333333333333 - \frac{0.041666666666666664}{N}}{N}}{N}\right) \cdot \left(0 - N\right)}} \]
13. Step-by-step derivation

    1. sub0-neg N/A

       \[\leadsto \frac{1}{\left(-1 - \frac{\frac{1}{2} - \frac{\frac{1}{12} - \frac{\frac{1}{24}}{N}}{N}}{N}\right) \cdot \left(\mathsf{neg}\left(N\right)\right)} \]

    2. associate-/r* N/A

       \[\leadsto \frac{\frac{1}{-1 - \frac{\frac{1}{2} - \frac{\frac{1}{12} - \frac{\frac{1}{24}}{N}}{N}}{N}}}{\color{blue}{\mathsf{neg}\left(N\right)}} \]

    3. clear-num N/A

       \[\leadsto \frac{1}{\color{blue}{\frac{\mathsf{neg}\left(N\right)}{\frac{1}{-1 - \frac{\frac{1}{2} - \frac{\frac{1}{12} - \frac{\frac{1}{24}}{N}}{N}}{N}}}}} \]

    4. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{\mathsf{neg}\left(N\right)}{\frac{1}{-1 - \frac{\frac{1}{2} - \frac{\frac{1}{12} - \frac{\frac{1}{24}}{N}}{N}}{N}}}\right)}\right) \]

    5. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(N\right)\right), \color{blue}{\left(\frac{1}{-1 - \frac{\frac{1}{2} - \frac{\frac{1}{12} - \frac{\frac{1}{24}}{N}}{N}}{N}}\right)}\right)\right) \]

    6. sub0-neg N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(0 - N\right), \left(\frac{\color{blue}{1}}{-1 - \frac{\frac{1}{2} - \frac{\frac{1}{12} - \frac{\frac{1}{24}}{N}}{N}}{N}}\right)\right)\right) \]

    7. --lowering--.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, N\right), \left(\frac{\color{blue}{1}}{-1 - \frac{\frac{1}{2} - \frac{\frac{1}{12} - \frac{\frac{1}{24}}{N}}{N}}{N}}\right)\right)\right) \]

    8. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, N\right), \mathsf{/.f64}\left(1, \color{blue}{\left(-1 - \frac{\frac{1}{2} - \frac{\frac{1}{12} - \frac{\frac{1}{24}}{N}}{N}}{N}\right)}\right)\right)\right) \]

    9. --lowering--.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, N\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(-1, \color{blue}{\left(\frac{\frac{1}{2} - \frac{\frac{1}{12} - \frac{\frac{1}{24}}{N}}{N}}{N}\right)}\right)\right)\right)\right) \]

    10. /-lowering-/.f64 N/A

        \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, N\right), \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(-1, \mathsf{/.f64}\left(\left(\frac{1}{2} - \frac{\frac{1}{12} - \frac{\frac{1}{24}}{N}}{N}\right), \color{blue}{N}\right)\right)\right)\right)\right) \]

14. Applied egg-rr 97.3%

    \[\leadsto \color{blue}{\frac{1}{\frac{0 - N}{\frac{1}{-1 - \frac{0.5 + \frac{-0.08333333333333333 + \frac{0.041666666666666664}{N}}{N}}{N}}}}} \]

15. Final simplification 97.3%

    \[\leadsto \frac{1}{\frac{N}{\frac{1}{\frac{0.5 + \frac{-0.08333333333333333 + \frac{0.041666666666666664}{N}}{N}}{N} - -1}}} \]
16. Add Preprocessing

Alternative 5: 96.8% accurate, 12.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 20.0%

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

   1. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\log \left(N + 1\right), \color{blue}{\log N}\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{\_.f64}\left(\log \left(1 + N\right), \log N\right) \]

   3. accelerator-lowering-log1p.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{log1p.f64}\left(N\right), \log \color{blue}{N}\right) \]

   4. log-lowering-log.f64 20.0%

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{log1p.f64}\left(N\right), \mathsf{log.f64}\left(N\right)\right) \]

3. Simplified 20.0%

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

5. Taylor expanded in N around inf

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

7. Simplified 97.1%

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

   1. clear-num N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{N}{1 + \frac{\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}{N}}}} \]

   2. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{N}{1 + \frac{\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}{N}}\right)}\right) \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \color{blue}{\left(1 + \frac{\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}{N}\right)}\right)\right) \]

   4. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}{N}\right)}\right)\right)\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}\right), \color{blue}{N}\right)\right)\right)\right) \]

   6. --lowering--.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{-1}{2}, \left(\frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}\right)\right), N\right)\right)\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\left(\frac{\frac{1}{4}}{N} + \frac{-1}{3}\right), N\right)\right), N\right)\right)\right)\right) \]

   8. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{\frac{1}{4}}{N}\right), \frac{-1}{3}\right), N\right)\right), N\right)\right)\right)\right) \]

   9. /-lowering-/.f64 97.1%

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{4}, N\right), \frac{-1}{3}\right), N\right)\right), N\right)\right)\right)\right) \]

9. Applied egg-rr 97.1%

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

10. Taylor expanded in N around -inf

    \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(-1 \cdot \left(N \cdot \left(-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N} - 1\right)\right)\right)}\right) \]
11. Step-by-step derivation

    1. associate-*r* N/A

       \[\leadsto \mathsf{/.f64}\left(1, \left(\left(-1 \cdot N\right) \cdot \color{blue}{\left(-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N} - 1\right)}\right)\right) \]

    2. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(1, \left(\left(-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N} - 1\right) \cdot \color{blue}{\left(-1 \cdot N\right)}\right)\right) \]

    3. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N} - 1\right), \color{blue}{\left(-1 \cdot N\right)}\right)\right) \]

12. Simplified 97.3%

    \[\leadsto \frac{1}{\color{blue}{\left(-1 - \frac{0.5 - \frac{0.08333333333333333 - \frac{0.041666666666666664}{N}}{N}}{N}\right) \cdot \left(0 - N\right)}} \]
13. Step-by-step derivation

    1. sub0-neg N/A

       \[\leadsto \mathsf{/.f64}\left(1, \left(\left(-1 - \frac{\frac{1}{2} - \frac{\frac{1}{12} - \frac{\frac{1}{24}}{N}}{N}}{N}\right) \cdot \left(\mathsf{neg}\left(N\right)\right)\right)\right) \]

    2. distribute-rgt-neg-out N/A

       \[\leadsto \mathsf{/.f64}\left(1, \left(\mathsf{neg}\left(\left(-1 - \frac{\frac{1}{2} - \frac{\frac{1}{12} - \frac{\frac{1}{24}}{N}}{N}}{N}\right) \cdot N\right)\right)\right) \]

    3. neg-lowering-neg.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{neg.f64}\left(\left(\left(-1 - \frac{\frac{1}{2} - \frac{\frac{1}{12} - \frac{\frac{1}{24}}{N}}{N}}{N}\right) \cdot N\right)\right)\right) \]

    4. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(-1 - \frac{\frac{1}{2} - \frac{\frac{1}{12} - \frac{\frac{1}{24}}{N}}{N}}{N}\right), N\right)\right)\right) \]

14. Applied egg-rr 97.3%

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

15. Final simplification 97.3%

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

Alternative 6: 96.8% accurate, 12.1× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1}{\frac{0.5 + \frac{-0.08333333333333333 + \frac{0.041666666666666664}{N}}{N}}{N} - -1}}{N} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 20.0%

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

   1. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\log \left(N + 1\right), \color{blue}{\log N}\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{\_.f64}\left(\log \left(1 + N\right), \log N\right) \]

   3. accelerator-lowering-log1p.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{log1p.f64}\left(N\right), \log \color{blue}{N}\right) \]

   4. log-lowering-log.f64 20.0%

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{log1p.f64}\left(N\right), \mathsf{log.f64}\left(N\right)\right) \]

3. Simplified 20.0%

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

5. Taylor expanded in N around inf

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

7. Simplified 97.1%

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

   1. clear-num N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{N}{1 + \frac{\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}{N}}}} \]

   2. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{N}{1 + \frac{\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}{N}}\right)}\right) \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \color{blue}{\left(1 + \frac{\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}{N}\right)}\right)\right) \]

   4. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}{N}\right)}\right)\right)\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}\right), \color{blue}{N}\right)\right)\right)\right) \]

   6. --lowering--.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{-1}{2}, \left(\frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}\right)\right), N\right)\right)\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\left(\frac{\frac{1}{4}}{N} + \frac{-1}{3}\right), N\right)\right), N\right)\right)\right)\right) \]

   8. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{\frac{1}{4}}{N}\right), \frac{-1}{3}\right), N\right)\right), N\right)\right)\right)\right) \]

   9. /-lowering-/.f64 97.1%

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{4}, N\right), \frac{-1}{3}\right), N\right)\right), N\right)\right)\right)\right) \]

9. Applied egg-rr 97.1%

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

10. Taylor expanded in N around -inf

    \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(-1 \cdot \left(N \cdot \left(-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N} - 1\right)\right)\right)}\right) \]
11. Step-by-step derivation

    1. associate-*r* N/A

       \[\leadsto \mathsf{/.f64}\left(1, \left(\left(-1 \cdot N\right) \cdot \color{blue}{\left(-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N} - 1\right)}\right)\right) \]

    2. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(1, \left(\left(-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N} - 1\right) \cdot \color{blue}{\left(-1 \cdot N\right)}\right)\right) \]

    3. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N} - 1\right), \color{blue}{\left(-1 \cdot N\right)}\right)\right) \]

12. Simplified 97.3%

    \[\leadsto \frac{1}{\color{blue}{\left(-1 - \frac{0.5 - \frac{0.08333333333333333 - \frac{0.041666666666666664}{N}}{N}}{N}\right) \cdot \left(0 - N\right)}} \]
13. Step-by-step derivation

    1. sub0-neg N/A

       \[\leadsto \frac{1}{\left(-1 - \frac{\frac{1}{2} - \frac{\frac{1}{12} - \frac{\frac{1}{24}}{N}}{N}}{N}\right) \cdot \left(\mathsf{neg}\left(N\right)\right)} \]

    2. associate-/r* N/A

       \[\leadsto \frac{\frac{1}{-1 - \frac{\frac{1}{2} - \frac{\frac{1}{12} - \frac{\frac{1}{24}}{N}}{N}}{N}}}{\color{blue}{\mathsf{neg}\left(N\right)}} \]

    3. distribute-frac-neg2 N/A

       \[\leadsto \mathsf{neg}\left(\frac{\frac{1}{-1 - \frac{\frac{1}{2} - \frac{\frac{1}{12} - \frac{\frac{1}{24}}{N}}{N}}{N}}}{N}\right) \]

    4. neg-lowering-neg.f64 N/A

       \[\leadsto \mathsf{neg.f64}\left(\left(\frac{\frac{1}{-1 - \frac{\frac{1}{2} - \frac{\frac{1}{12} - \frac{\frac{1}{24}}{N}}{N}}{N}}}{N}\right)\right) \]

    5. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{-1 - \frac{\frac{1}{2} - \frac{\frac{1}{12} - \frac{\frac{1}{24}}{N}}{N}}{N}}\right), N\right)\right) \]

14. Applied egg-rr 97.3%

    \[\leadsto \color{blue}{-\frac{\frac{1}{-1 - \frac{0.5 + \frac{-0.08333333333333333 + \frac{0.041666666666666664}{N}}{N}}{N}}}{N}} \]

15. Final simplification 97.3%

    \[\leadsto \frac{\frac{1}{\frac{0.5 + \frac{-0.08333333333333333 + \frac{0.041666666666666664}{N}}{N}}{N} - -1}}{N} \]
16. Add Preprocessing

Alternative 7: 96.7% accurate, 12.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 20.0%

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

   1. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\log \left(N + 1\right), \color{blue}{\log N}\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{\_.f64}\left(\log \left(1 + N\right), \log N\right) \]

   3. accelerator-lowering-log1p.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{log1p.f64}\left(N\right), \log \color{blue}{N}\right) \]

   4. log-lowering-log.f64 20.0%

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{log1p.f64}\left(N\right), \mathsf{log.f64}\left(N\right)\right) \]

3. Simplified 20.0%

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

5. Taylor expanded in N around inf

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

7. Simplified 97.1%

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

   1. clear-num N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{N}{1 + \frac{\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}{N}}}} \]

   2. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{N}{1 + \frac{\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}{N}}\right)}\right) \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \color{blue}{\left(1 + \frac{\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}{N}\right)}\right)\right) \]

   4. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}{N}\right)}\right)\right)\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}\right), \color{blue}{N}\right)\right)\right)\right) \]

   6. --lowering--.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{-1}{2}, \left(\frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}\right)\right), N\right)\right)\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\left(\frac{\frac{1}{4}}{N} + \frac{-1}{3}\right), N\right)\right), N\right)\right)\right)\right) \]

   8. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{\frac{1}{4}}{N}\right), \frac{-1}{3}\right), N\right)\right), N\right)\right)\right)\right) \]

   9. /-lowering-/.f64 97.1%

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{4}, N\right), \frac{-1}{3}\right), N\right)\right), N\right)\right)\right)\right) \]

9. Applied egg-rr 97.1%

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

10. Taylor expanded in N around -inf

    \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(-1 \cdot \left(N \cdot \left(-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N} - 1\right)\right)\right)}\right) \]
11. Step-by-step derivation

    1. associate-*r* N/A

       \[\leadsto \mathsf{/.f64}\left(1, \left(\left(-1 \cdot N\right) \cdot \color{blue}{\left(-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N} - 1\right)}\right)\right) \]

    2. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(1, \left(\left(-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N} - 1\right) \cdot \color{blue}{\left(-1 \cdot N\right)}\right)\right) \]

    3. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\left(-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N} - 1\right), \color{blue}{\left(-1 \cdot N\right)}\right)\right) \]

12. Simplified 97.3%

    \[\leadsto \frac{1}{\color{blue}{\left(-1 - \frac{0.5 - \frac{0.08333333333333333 - \frac{0.041666666666666664}{N}}{N}}{N}\right) \cdot \left(0 - N\right)}} \]

13. Taylor expanded in N around 0

    \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{\frac{1}{24} + N \cdot \left(N \cdot \left(\frac{1}{2} + N\right) - \frac{1}{12}\right)}{{N}^{2}}\right)}\right) \]
14. Step-by-step derivation

    1. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{1}{24} + N \cdot \left(N \cdot \left(\frac{1}{2} + N\right) - \frac{1}{12}\right)\right), \color{blue}{\left({N}^{2}\right)}\right)\right) \]

    2. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{24}, \left(N \cdot \left(N \cdot \left(\frac{1}{2} + N\right) - \frac{1}{12}\right)\right)\right), \left({\color{blue}{N}}^{2}\right)\right)\right) \]

    3. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(N, \left(N \cdot \left(\frac{1}{2} + N\right) - \frac{1}{12}\right)\right)\right), \left({N}^{2}\right)\right)\right) \]

    4. sub-neg N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(N, \left(N \cdot \left(\frac{1}{2} + N\right) + \left(\mathsf{neg}\left(\frac{1}{12}\right)\right)\right)\right)\right), \left({N}^{2}\right)\right)\right) \]

    5. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(N, \left(N \cdot \left(\frac{1}{2} + N\right) + \frac{-1}{12}\right)\right)\right), \left({N}^{2}\right)\right)\right) \]

    6. +-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(N, \left(\frac{-1}{12} + N \cdot \left(\frac{1}{2} + N\right)\right)\right)\right), \left({N}^{2}\right)\right)\right) \]

    7. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(N, \mathsf{+.f64}\left(\frac{-1}{12}, \left(N \cdot \left(\frac{1}{2} + N\right)\right)\right)\right)\right), \left({N}^{2}\right)\right)\right) \]

    8. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(N, \mathsf{+.f64}\left(\frac{-1}{12}, \mathsf{*.f64}\left(N, \left(\frac{1}{2} + N\right)\right)\right)\right)\right), \left({N}^{2}\right)\right)\right) \]

    9. +-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(N, \mathsf{+.f64}\left(\frac{-1}{12}, \mathsf{*.f64}\left(N, \left(N + \frac{1}{2}\right)\right)\right)\right)\right), \left({N}^{2}\right)\right)\right) \]

    10. +-lowering-+.f64 N/A

        \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(N, \mathsf{+.f64}\left(\frac{-1}{12}, \mathsf{*.f64}\left(N, \mathsf{+.f64}\left(N, \frac{1}{2}\right)\right)\right)\right)\right), \left({N}^{2}\right)\right)\right) \]

    11. unpow2 N/A

        \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(N, \mathsf{+.f64}\left(\frac{-1}{12}, \mathsf{*.f64}\left(N, \mathsf{+.f64}\left(N, \frac{1}{2}\right)\right)\right)\right)\right), \left(N \cdot \color{blue}{N}\right)\right)\right) \]

    12. *-lowering-*.f64 97.2%

        \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(N, \mathsf{+.f64}\left(\frac{-1}{12}, \mathsf{*.f64}\left(N, \mathsf{+.f64}\left(N, \frac{1}{2}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(N, \color{blue}{N}\right)\right)\right) \]

15. Simplified 97.2%

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

Alternative 8: 96.5% 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 20.0%

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

   1. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\log \left(N + 1\right), \color{blue}{\log N}\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{\_.f64}\left(\log \left(1 + N\right), \log N\right) \]

   3. accelerator-lowering-log1p.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{log1p.f64}\left(N\right), \log \color{blue}{N}\right) \]

   4. log-lowering-log.f64 20.0%

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{log1p.f64}\left(N\right), \mathsf{log.f64}\left(N\right)\right) \]

3. Simplified 20.0%

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

5. Taylor expanded in N around inf

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

7. Simplified 97.1%

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

   1. clear-num N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{N}{1 + \frac{\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}{N}}}} \]

   2. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{N}{1 + \frac{\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}{N}}\right)}\right) \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \color{blue}{\left(1 + \frac{\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}{N}\right)}\right)\right) \]

   4. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}{N}\right)}\right)\right)\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}\right), \color{blue}{N}\right)\right)\right)\right) \]

   6. --lowering--.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{-1}{2}, \left(\frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}\right)\right), N\right)\right)\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\left(\frac{\frac{1}{4}}{N} + \frac{-1}{3}\right), N\right)\right), N\right)\right)\right)\right) \]

   8. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{\frac{1}{4}}{N}\right), \frac{-1}{3}\right), N\right)\right), N\right)\right)\right)\right) \]

   9. /-lowering-/.f64 97.1%

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{4}, N\right), \frac{-1}{3}\right), N\right)\right), N\right)\right)\right)\right) \]

9. Applied egg-rr 97.1%

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

Alternative 9: 96.5% 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 20.0%

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

   1. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\log \left(N + 1\right), \color{blue}{\log N}\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{\_.f64}\left(\log \left(1 + N\right), \log N\right) \]

   3. accelerator-lowering-log1p.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{log1p.f64}\left(N\right), \log \color{blue}{N}\right) \]

   4. log-lowering-log.f64 20.0%

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{log1p.f64}\left(N\right), \mathsf{log.f64}\left(N\right)\right) \]

3. Simplified 20.0%

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

5. Taylor expanded in N around inf

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

7. Simplified 97.1%

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

Alternative 10: 95.7% 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 20.0%

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

   1. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\log \left(N + 1\right), \color{blue}{\log N}\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{\_.f64}\left(\log \left(1 + N\right), \log N\right) \]

   3. accelerator-lowering-log1p.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{log1p.f64}\left(N\right), \log \color{blue}{N}\right) \]

   4. log-lowering-log.f64 20.0%

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{log1p.f64}\left(N\right), \mathsf{log.f64}\left(N\right)\right) \]

3. Simplified 20.0%

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

5. Taylor expanded in N around inf

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

7. Simplified 97.1%

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

   1. clear-num N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{N}{1 + \frac{\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}{N}}}} \]

   2. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{N}{1 + \frac{\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}{N}}\right)}\right) \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \color{blue}{\left(1 + \frac{\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}{N}\right)}\right)\right) \]

   4. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}{N}\right)}\right)\right)\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}\right), \color{blue}{N}\right)\right)\right)\right) \]

   6. --lowering--.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{-1}{2}, \left(\frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}\right)\right), N\right)\right)\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\left(\frac{\frac{1}{4}}{N} + \frac{-1}{3}\right), N\right)\right), N\right)\right)\right)\right) \]

   8. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{\frac{1}{4}}{N}\right), \frac{-1}{3}\right), N\right)\right), N\right)\right)\right)\right) \]

   9. /-lowering-/.f64 97.1%

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{4}, N\right), \frac{-1}{3}\right), N\right)\right), N\right)\right)\right)\right) \]

9. Applied egg-rr 97.1%

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

10. Taylor expanded in N around inf

    \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(N \cdot \left(\left(1 + \frac{1}{2} \cdot \frac{1}{N}\right) - \frac{\frac{1}{12}}{{N}^{2}}\right)\right)}\right) \]
11. Step-by-step derivation

    1. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(N, \color{blue}{\left(\left(1 + \frac{1}{2} \cdot \frac{1}{N}\right) - \frac{\frac{1}{12}}{{N}^{2}}\right)}\right)\right) \]

    2. associate--l+ N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(N, \left(1 + \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{N} - \frac{\frac{1}{12}}{{N}^{2}}\right)}\right)\right)\right) \]

    3. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(N, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{N} - \frac{\frac{1}{12}}{{N}^{2}}\right)}\right)\right)\right) \]

    4. associate-*r/ N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(N, \mathsf{+.f64}\left(1, \left(\frac{\frac{1}{2} \cdot 1}{N} - \frac{\color{blue}{\frac{1}{12}}}{{N}^{2}}\right)\right)\right)\right) \]

    5. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(N, \mathsf{+.f64}\left(1, \left(\frac{\frac{1}{2}}{N} - \frac{\frac{1}{12}}{{N}^{2}}\right)\right)\right)\right) \]

    6. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(N, \mathsf{+.f64}\left(1, \left(\frac{\frac{1}{2}}{N} - \frac{\frac{1}{12}}{N \cdot \color{blue}{N}}\right)\right)\right)\right) \]

    7. associate-/r* N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(N, \mathsf{+.f64}\left(1, \left(\frac{\frac{1}{2}}{N} - \frac{\frac{\frac{1}{12}}{N}}{\color{blue}{N}}\right)\right)\right)\right) \]

    8. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(N, \mathsf{+.f64}\left(1, \left(\frac{\frac{1}{2}}{N} - \frac{\frac{\frac{1}{12} \cdot 1}{N}}{N}\right)\right)\right)\right) \]

    9. associate-*r/ N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(N, \mathsf{+.f64}\left(1, \left(\frac{\frac{1}{2}}{N} - \frac{\frac{1}{12} \cdot \frac{1}{N}}{N}\right)\right)\right)\right) \]

    10. div-sub N/A

        \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(N, \mathsf{+.f64}\left(1, \left(\frac{\frac{1}{2} - \frac{1}{12} \cdot \frac{1}{N}}{\color{blue}{N}}\right)\right)\right)\right) \]

    11. /-lowering-/.f64 N/A

        \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{1}{2} - \frac{1}{12} \cdot \frac{1}{N}\right), \color{blue}{N}\right)\right)\right)\right) \]

    12. sub-neg N/A

        \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{1}{2} + \left(\mathsf{neg}\left(\frac{1}{12} \cdot \frac{1}{N}\right)\right)\right), N\right)\right)\right)\right) \]

    13. +-lowering-+.f64 N/A

        \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\mathsf{neg}\left(\frac{1}{12} \cdot \frac{1}{N}\right)\right)\right), N\right)\right)\right)\right) \]

    14. associate-*r/ N/A

        \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\mathsf{neg}\left(\frac{\frac{1}{12} \cdot 1}{N}\right)\right)\right), N\right)\right)\right)\right) \]

    15. metadata-eval N/A

        \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\mathsf{neg}\left(\frac{\frac{1}{12}}{N}\right)\right)\right), N\right)\right)\right)\right) \]

    16. distribute-neg-frac N/A

        \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\mathsf{neg}\left(\frac{1}{12}\right)}{N}\right)\right), N\right)\right)\right)\right) \]

    17. metadata-eval N/A

        \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{\frac{-1}{12}}{N}\right)\right), N\right)\right)\right)\right) \]

    18. /-lowering-/.f64 96.6%

        \[\leadsto \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(\frac{-1}{12}, N\right)\right), N\right)\right)\right)\right) \]

12. Simplified 96.6%

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

Alternative 11: 95.3% 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 20.0%

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

   1. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\log \left(N + 1\right), \color{blue}{\log N}\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{\_.f64}\left(\log \left(1 + N\right), \log N\right) \]

   3. accelerator-lowering-log1p.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{log1p.f64}\left(N\right), \log \color{blue}{N}\right) \]

   4. log-lowering-log.f64 20.0%

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{log1p.f64}\left(N\right), \mathsf{log.f64}\left(N\right)\right) \]

3. Simplified 20.0%

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

5. Taylor expanded in N around inf

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\left(1 + \frac{\frac{1}{3}}{{N}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{N}\right), \color{blue}{N}\right) \]

7. Simplified 96.4%

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

Alternative 12: 93.5% accurate, 41.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 20.0%

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

   1. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\log \left(N + 1\right), \color{blue}{\log N}\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{\_.f64}\left(\log \left(1 + N\right), \log N\right) \]

   3. accelerator-lowering-log1p.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{log1p.f64}\left(N\right), \log \color{blue}{N}\right) \]

   4. log-lowering-log.f64 20.0%

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{log1p.f64}\left(N\right), \mathsf{log.f64}\left(N\right)\right) \]

3. Simplified 20.0%

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

5. Taylor expanded in N around inf

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

7. Simplified 97.1%

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

   1. clear-num N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{N}{1 + \frac{\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}{N}}}} \]

   2. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{N}{1 + \frac{\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}{N}}\right)}\right) \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \color{blue}{\left(1 + \frac{\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}{N}\right)}\right)\right) \]

   4. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}{N}\right)}\right)\right)\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}\right), \color{blue}{N}\right)\right)\right)\right) \]

   6. --lowering--.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{-1}{2}, \left(\frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}\right)\right), N\right)\right)\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\left(\frac{\frac{1}{4}}{N} + \frac{-1}{3}\right), N\right)\right), N\right)\right)\right)\right) \]

   8. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{\frac{1}{4}}{N}\right), \frac{-1}{3}\right), N\right)\right), N\right)\right)\right)\right) \]

   9. /-lowering-/.f64 97.1%

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{4}, N\right), \frac{-1}{3}\right), N\right)\right), N\right)\right)\right)\right) \]

9. Applied egg-rr 97.1%

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

10. Taylor expanded in N around inf

    \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(N \cdot \left(1 + \frac{1}{2} \cdot \frac{1}{N}\right)\right)}\right) \]
11. Step-by-step derivation

    1. distribute-rgt-in N/A

       \[\leadsto \mathsf{/.f64}\left(1, \left(1 \cdot N + \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{N}\right) \cdot N}\right)\right) \]

    2. *-lft-identity N/A

       \[\leadsto \mathsf{/.f64}\left(1, \left(N + \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{N}\right)} \cdot N\right)\right) \]

    3. associate-*l* N/A

       \[\leadsto \mathsf{/.f64}\left(1, \left(N + \frac{1}{2} \cdot \color{blue}{\left(\frac{1}{N} \cdot N\right)}\right)\right) \]

    4. lft-mult-inverse N/A

       \[\leadsto \mathsf{/.f64}\left(1, \left(N + \frac{1}{2} \cdot 1\right)\right) \]

    5. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(1, \left(N + \frac{1}{2}\right)\right) \]

    6. +-lowering-+.f64 94.8%

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(N, \color{blue}{\frac{1}{2}}\right)\right) \]

12. Simplified 94.8%

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

Alternative 13: 84.9% 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 20.0%

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

   1. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\log \left(N + 1\right), \color{blue}{\log N}\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{\_.f64}\left(\log \left(1 + N\right), \log N\right) \]

   3. accelerator-lowering-log1p.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{log1p.f64}\left(N\right), \log \color{blue}{N}\right) \]

   4. log-lowering-log.f64 20.0%

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{log1p.f64}\left(N\right), \mathsf{log.f64}\left(N\right)\right) \]

3. Simplified 20.0%

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

5. Taylor expanded in N around inf

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

   1. /-lowering-/.f64 87.5%

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{N}\right) \]

7. Simplified 87.5%

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

Alternative 14: 9.8% accurate, 205.0× speedup

Math

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

FPCore

(FPCore (N) :precision binary64 2.0)

C

double code(double N) {
	return 2.0;
}

Fortran

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

Java

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

Python

def code(N):
	return 2.0

Julia

function code(N)
	return 2.0
end

MATLAB

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

Wolfram

code[N_] := 2.0

Derivation

1. Initial program 20.0%

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

   1. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\log \left(N + 1\right), \color{blue}{\log N}\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{\_.f64}\left(\log \left(1 + N\right), \log N\right) \]

   3. accelerator-lowering-log1p.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{log1p.f64}\left(N\right), \log \color{blue}{N}\right) \]

   4. log-lowering-log.f64 20.0%

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{log1p.f64}\left(N\right), \mathsf{log.f64}\left(N\right)\right) \]

3. Simplified 20.0%

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

5. Taylor expanded in N around inf

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

7. Simplified 97.1%

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

   1. clear-num N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{N}{1 + \frac{\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}{N}}}} \]

   2. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{N}{1 + \frac{\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}{N}}\right)}\right) \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \color{blue}{\left(1 + \frac{\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}{N}\right)}\right)\right) \]

   4. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}{N}\right)}\right)\right)\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}\right), \color{blue}{N}\right)\right)\right)\right) \]

   6. --lowering--.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{-1}{2}, \left(\frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}\right)\right), N\right)\right)\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\left(\frac{\frac{1}{4}}{N} + \frac{-1}{3}\right), N\right)\right), N\right)\right)\right)\right) \]

   8. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{\frac{1}{4}}{N}\right), \frac{-1}{3}\right), N\right)\right), N\right)\right)\right)\right) \]

   9. /-lowering-/.f64 97.1%

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(N, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{4}, N\right), \frac{-1}{3}\right), N\right)\right), N\right)\right)\right)\right) \]

9. Applied egg-rr 97.1%

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

10. Taylor expanded in N around inf

    \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(N \cdot \left(1 + \frac{1}{2} \cdot \frac{1}{N}\right)\right)}\right) \]
11. Step-by-step derivation

    1. distribute-rgt-in N/A

       \[\leadsto \mathsf{/.f64}\left(1, \left(1 \cdot N + \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{N}\right) \cdot N}\right)\right) \]

    2. *-lft-identity N/A

       \[\leadsto \mathsf{/.f64}\left(1, \left(N + \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{N}\right)} \cdot N\right)\right) \]

    3. associate-*l* N/A

       \[\leadsto \mathsf{/.f64}\left(1, \left(N + \frac{1}{2} \cdot \color{blue}{\left(\frac{1}{N} \cdot N\right)}\right)\right) \]

    4. lft-mult-inverse N/A

       \[\leadsto \mathsf{/.f64}\left(1, \left(N + \frac{1}{2} \cdot 1\right)\right) \]

    5. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(1, \left(N + \frac{1}{2}\right)\right) \]

    6. +-lowering-+.f64 94.8%

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(N, \color{blue}{\frac{1}{2}}\right)\right) \]

12. Simplified 94.8%

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

13. Taylor expanded in N around 0

    \[\leadsto \color{blue}{2} \]
14. Step-by-step derivation

15. Simplified 9.7%

    \[\leadsto \color{blue}{2} \]
16. 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]

Developer Target 2: 26.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Developer Target 3: 96.5% accurate, 0.6× speedup

Math

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

FPCore

(FPCore (N)
 :precision binary64
 (+
  (+ (+ (/ 1.0 N) (/ -1.0 (* 2.0 (pow N 2.0)))) (/ 1.0 (* 3.0 (pow N 3.0))))
  (/ -1.0 (* 4.0 (pow N 4.0)))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024192 
(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)))

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

  :alt
  (! :herbie-platform default (+ (/ 1 N) (/ -1 (* 2 (pow N 2))) (/ 1 (* 3 (pow N 3))) (/ -1 (* 4 (pow N 4)))))

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