2log (problem 3.3.6)

Percentage Accurate: 23.9% → 99.5%

Time: 9.6s

Alternatives: 11

Speedup: 17.3×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 23.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\frac{N}{N + 1}\right)\\ t_1 := {t\_0}^{2}\\ \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 0.001:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(N, \frac{0.5 - \frac{\mathsf{fma}\left(N, 0.08333333333333333, -0.041666666666666664\right)}{N \cdot N}}{N}, N\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{t\_1 \cdot \frac{-1}{t\_0}}\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (let* ((t_0 (log (/ N (+ N 1.0)))) (t_1 (pow t_0 2.0)))
   (if (<= (- (log (+ N 1.0)) (log N)) 0.001)
     (/
      1.0
      (fma
       N
       (/
        (- 0.5 (/ (fma N 0.08333333333333333 -0.041666666666666664) (* N N)))
        N)
       N))
     (/ t_1 (* t_1 (/ -1.0 t_0))))))

C

double code(double N) {
	double t_0 = log((N / (N + 1.0)));
	double t_1 = pow(t_0, 2.0);
	double tmp;
	if ((log((N + 1.0)) - log(N)) <= 0.001) {
		tmp = 1.0 / fma(N, ((0.5 - (fma(N, 0.08333333333333333, -0.041666666666666664) / (N * N))) / N), N);
	} else {
		tmp = t_1 / (t_1 * (-1.0 / t_0));
	}
	return tmp;
}

Julia

function code(N)
	t_0 = log(Float64(N / Float64(N + 1.0)))
	t_1 = t_0 ^ 2.0
	tmp = 0.0
	if (Float64(log(Float64(N + 1.0)) - log(N)) <= 0.001)
		tmp = Float64(1.0 / fma(N, Float64(Float64(0.5 - Float64(fma(N, 0.08333333333333333, -0.041666666666666664) / Float64(N * N))) / N), N));
	else
		tmp = Float64(t_1 / Float64(t_1 * Float64(-1.0 / t_0)));
	end
	return tmp
end

Wolfram

code[N_] := Block[{t$95$0 = N[Log[N[(N / N[(N + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Power[t$95$0, 2.0], $MachinePrecision]}, If[LessEqual[N[(N[Log[N[(N + 1.0), $MachinePrecision]], $MachinePrecision] - N[Log[N], $MachinePrecision]), $MachinePrecision], 0.001], N[(1.0 / N[(N * N[(N[(0.5 - N[(N[(N * 0.08333333333333333 + -0.041666666666666664), $MachinePrecision] / N[(N * N), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision] + N), $MachinePrecision]), $MachinePrecision], N[(t$95$1 / N[(t$95$1 * N[(-1.0 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (log.f64 (+.f64 N #s(literal 1 binary64))) (log.f64 N)) < 1e-3

   1. Initial program 19.4%

      \[\log \left(N + 1\right) - \log N \]
   2. Add Preprocessing

   3. Taylor expanded in N around inf

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

   5. Applied rewrites 99.6%

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

   7. Applied rewrites 99.6%

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

   8. Taylor expanded in N around -inf

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

   10. Applied rewrites 99.8%

       \[\leadsto \frac{1}{-\mathsf{fma}\left(N, \frac{0.5 - \frac{0.08333333333333333 + \frac{-0.041666666666666664}{N}}{N}}{-N}, -N\right)} \]

   11. Taylor expanded in N around -inf

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

   13. Applied rewrites 99.8%

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

   if 1e-3 < (-.f64 (log.f64 (+.f64 N #s(literal 1 binary64))) (log.f64 N))

   1. Initial program 90.4%

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

      1. lift--.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. diff-log N/A

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

      5. clear-num N/A

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

      6. neg-log N/A

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

      7. diff-log N/A

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

      8. lift-log.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lift-log.f64 N/A

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

      12. lift-log.f64 N/A

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

      13. diff-log N/A

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

      14. lower-log.f64 N/A

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

      15. lower-/.f64 94.6

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

   4. Applied rewrites 94.6%

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

      1. lift-neg.f64 N/A

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

      2. neg-sub0 N/A

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

      3. flip-- N/A

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

      4. metadata-eval N/A

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

      5. lift-log.f64 N/A

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

      6. log-prod N/A

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

      7. lift-/.f64 N/A

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

      8. clear-num N/A

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

      9. div-inv N/A

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

      10. clear-num N/A

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

      11. lift-/.f64 N/A

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

      12. lift-log.f64 N/A

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

      13. frac-2neg N/A

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

      14. lift-log.f64 N/A

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

      15. neg-log N/A

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

      16. lift-/.f64 N/A

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

      17. clear-num N/A

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

      18. diff-log N/A

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

   6. Applied rewrites 92.9%

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

      1. lift-log.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\left(0 - {\log \left(\frac{N}{N + 1}\right)}^{2}\right)\right)}{\color{blue}{\log \left(\frac{N + 1}{N}\right)}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\left(0 - {\log \left(\frac{N}{N + 1}\right)}^{2}\right)\right)}{\log \color{blue}{\left(\frac{N + 1}{N}\right)}} \]

      3. clear-num N/A

         \[\leadsto \frac{\mathsf{neg}\left(\left(0 - {\log \left(\frac{N}{N + 1}\right)}^{2}\right)\right)}{\log \color{blue}{\left(\frac{1}{\frac{N}{N + 1}}\right)}} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\left(0 - {\log \left(\frac{N}{N + 1}\right)}^{2}\right)\right)}{\log \left(\frac{1}{\color{blue}{\frac{N}{N + 1}}}\right)} \]

      5. log-rec N/A

         \[\leadsto \frac{\mathsf{neg}\left(\left(0 - {\log \left(\frac{N}{N + 1}\right)}^{2}\right)\right)}{\color{blue}{\mathsf{neg}\left(\log \left(\frac{N}{N + 1}\right)\right)}} \]

      6. lift-log.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\left(0 - {\log \left(\frac{N}{N + 1}\right)}^{2}\right)\right)}{\mathsf{neg}\left(\color{blue}{\log \left(\frac{N}{N + 1}\right)}\right)} \]

      7. unpow1 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\left(0 - {\log \left(\frac{N}{N + 1}\right)}^{2}\right)\right)}{\mathsf{neg}\left(\color{blue}{{\log \left(\frac{N}{N + 1}\right)}^{1}}\right)} \]

      8. metadata-eval N/A

         \[\leadsto \frac{\mathsf{neg}\left(\left(0 - {\log \left(\frac{N}{N + 1}\right)}^{2}\right)\right)}{\mathsf{neg}\left({\log \left(\frac{N}{N + 1}\right)}^{\color{blue}{\left(2 - 1\right)}}\right)} \]

      9. pow-div N/A

         \[\leadsto \frac{\mathsf{neg}\left(\left(0 - {\log \left(\frac{N}{N + 1}\right)}^{2}\right)\right)}{\mathsf{neg}\left(\color{blue}{\frac{{\log \left(\frac{N}{N + 1}\right)}^{2}}{{\log \left(\frac{N}{N + 1}\right)}^{1}}}\right)} \]

      10. lift-pow.f64 N/A

          \[\leadsto \frac{\mathsf{neg}\left(\left(0 - {\log \left(\frac{N}{N + 1}\right)}^{2}\right)\right)}{\mathsf{neg}\left(\frac{\color{blue}{{\log \left(\frac{N}{N + 1}\right)}^{2}}}{{\log \left(\frac{N}{N + 1}\right)}^{1}}\right)} \]

      11. unpow1 N/A

          \[\leadsto \frac{\mathsf{neg}\left(\left(0 - {\log \left(\frac{N}{N + 1}\right)}^{2}\right)\right)}{\mathsf{neg}\left(\frac{{\log \left(\frac{N}{N + 1}\right)}^{2}}{\color{blue}{\log \left(\frac{N}{N + 1}\right)}}\right)} \]

      12. frac-2neg N/A

          \[\leadsto \frac{\mathsf{neg}\left(\left(0 - {\log \left(\frac{N}{N + 1}\right)}^{2}\right)\right)}{\mathsf{neg}\left(\color{blue}{\frac{\mathsf{neg}\left({\log \left(\frac{N}{N + 1}\right)}^{2}\right)}{\mathsf{neg}\left(\log \left(\frac{N}{N + 1}\right)\right)}}\right)} \]

      13. lift-log.f64 N/A

          \[\leadsto \frac{\mathsf{neg}\left(\left(0 - {\log \left(\frac{N}{N + 1}\right)}^{2}\right)\right)}{\mathsf{neg}\left(\frac{\mathsf{neg}\left({\log \left(\frac{N}{N + 1}\right)}^{2}\right)}{\mathsf{neg}\left(\color{blue}{\log \left(\frac{N}{N + 1}\right)}\right)}\right)} \]

      14. log-rec N/A

          \[\leadsto \frac{\mathsf{neg}\left(\left(0 - {\log \left(\frac{N}{N + 1}\right)}^{2}\right)\right)}{\mathsf{neg}\left(\frac{\mathsf{neg}\left({\log \left(\frac{N}{N + 1}\right)}^{2}\right)}{\color{blue}{\log \left(\frac{1}{\frac{N}{N + 1}}\right)}}\right)} \]

      15. lift-/.f64 N/A

          \[\leadsto \frac{\mathsf{neg}\left(\left(0 - {\log \left(\frac{N}{N + 1}\right)}^{2}\right)\right)}{\mathsf{neg}\left(\frac{\mathsf{neg}\left({\log \left(\frac{N}{N + 1}\right)}^{2}\right)}{\log \left(\frac{1}{\color{blue}{\frac{N}{N + 1}}}\right)}\right)} \]

      16. clear-num N/A

          \[\leadsto \frac{\mathsf{neg}\left(\left(0 - {\log \left(\frac{N}{N + 1}\right)}^{2}\right)\right)}{\mathsf{neg}\left(\frac{\mathsf{neg}\left({\log \left(\frac{N}{N + 1}\right)}^{2}\right)}{\log \color{blue}{\left(\frac{N + 1}{N}\right)}}\right)} \]

      17. diff-log N/A

          \[\leadsto \frac{\mathsf{neg}\left(\left(0 - {\log \left(\frac{N}{N + 1}\right)}^{2}\right)\right)}{\mathsf{neg}\left(\frac{\mathsf{neg}\left({\log \left(\frac{N}{N + 1}\right)}^{2}\right)}{\color{blue}{\log \left(N + 1\right) - \log N}}\right)} \]

      18. lift-+.f64 N/A

          \[\leadsto \frac{\mathsf{neg}\left(\left(0 - {\log \left(\frac{N}{N + 1}\right)}^{2}\right)\right)}{\mathsf{neg}\left(\frac{\mathsf{neg}\left({\log \left(\frac{N}{N + 1}\right)}^{2}\right)}{\log \color{blue}{\left(N + 1\right)} - \log N}\right)} \]

   8. Applied rewrites 94.8%

      \[\leadsto \frac{-\left(0 - {\log \left(\frac{N}{N + 1}\right)}^{2}\right)}{\color{blue}{\left(-{\log \left(\frac{N}{1 + N}\right)}^{2}\right) \cdot \frac{1}{\log \left(\frac{N}{1 + N}\right)}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 0.001:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(N, \frac{0.5 - \frac{\mathsf{fma}\left(N, 0.08333333333333333, -0.041666666666666664\right)}{N \cdot N}}{N}, N\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\log \left(\frac{N}{N + 1}\right)}^{2}}{{\log \left(\frac{N}{N + 1}\right)}^{2} \cdot \frac{-1}{\log \left(\frac{N}{N + 1}\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\frac{N}{N + 1}\right)\\ \mathbf{if}\;N \leq 1150:\\ \;\;\;\;-\frac{{t\_0}^{2}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(N, \frac{0.5 - \frac{\mathsf{fma}\left(N, 0.08333333333333333, -0.041666666666666664\right)}{N \cdot N}}{N}, N\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (let* ((t_0 (log (/ N (+ N 1.0)))))
   (if (<= N 1150.0)
     (- (/ (pow t_0 2.0) t_0))
     (/
      1.0
      (fma
       N
       (/
        (- 0.5 (/ (fma N 0.08333333333333333 -0.041666666666666664) (* N N)))
        N)
       N)))))

C

double code(double N) {
	double t_0 = log((N / (N + 1.0)));
	double tmp;
	if (N <= 1150.0) {
		tmp = -(pow(t_0, 2.0) / t_0);
	} else {
		tmp = 1.0 / fma(N, ((0.5 - (fma(N, 0.08333333333333333, -0.041666666666666664) / (N * N))) / N), N);
	}
	return tmp;
}

Julia

function code(N)
	t_0 = log(Float64(N / Float64(N + 1.0)))
	tmp = 0.0
	if (N <= 1150.0)
		tmp = Float64(-Float64((t_0 ^ 2.0) / t_0));
	else
		tmp = Float64(1.0 / fma(N, Float64(Float64(0.5 - Float64(fma(N, 0.08333333333333333, -0.041666666666666664) / Float64(N * N))) / N), N));
	end
	return tmp
end

Wolfram

code[N_] := Block[{t$95$0 = N[Log[N[(N / N[(N + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N, 1150.0], (-N[(N[Power[t$95$0, 2.0], $MachinePrecision] / t$95$0), $MachinePrecision]), N[(1.0 / N[(N * N[(N[(0.5 - N[(N[(N * 0.08333333333333333 + -0.041666666666666664), $MachinePrecision] / N[(N * N), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision] + N), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if N < 1150

   1. Initial program 90.4%

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

      1. lift--.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. diff-log N/A

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

      5. clear-num N/A

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

      6. neg-log N/A

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

      7. diff-log N/A

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

      8. lift-log.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lift-log.f64 N/A

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

      12. lift-log.f64 N/A

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

      13. diff-log N/A

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

      14. lower-log.f64 N/A

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

      15. lower-/.f64 94.6

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

   4. Applied rewrites 94.6%

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

      1. lift-neg.f64 N/A

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

      2. neg-sub0 N/A

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

      3. flip-- N/A

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

      4. metadata-eval N/A

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

      5. lift-log.f64 N/A

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

      6. log-prod N/A

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

      7. lift-/.f64 N/A

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

      8. clear-num N/A

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

      9. div-inv N/A

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

      10. clear-num N/A

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

      11. lift-/.f64 N/A

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

      12. lift-log.f64 N/A

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

      13. frac-2neg N/A

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

      14. lift-log.f64 N/A

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

      15. neg-log N/A

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

      16. lift-/.f64 N/A

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

      17. clear-num N/A

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

      18. diff-log N/A

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

   6. Applied rewrites 92.9%

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

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\left(0 - {\log \left(\frac{N}{N + 1}\right)}^{2}\right)\right)}{\log \left(\frac{N + 1}{N}\right)}} \]

      2. lift-neg.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(\left(0 - {\log \left(\frac{N}{N + 1}\right)}^{2}\right)\right)}}{\log \left(\frac{N + 1}{N}\right)} \]

      3. lift-log.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\left(0 - {\log \left(\frac{N}{N + 1}\right)}^{2}\right)\right)}{\color{blue}{\log \left(\frac{N + 1}{N}\right)}} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\left(0 - {\log \left(\frac{N}{N + 1}\right)}^{2}\right)\right)}{\log \color{blue}{\left(\frac{N + 1}{N}\right)}} \]

      5. clear-num N/A

         \[\leadsto \frac{\mathsf{neg}\left(\left(0 - {\log \left(\frac{N}{N + 1}\right)}^{2}\right)\right)}{\log \color{blue}{\left(\frac{1}{\frac{N}{N + 1}}\right)}} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\left(0 - {\log \left(\frac{N}{N + 1}\right)}^{2}\right)\right)}{\log \left(\frac{1}{\color{blue}{\frac{N}{N + 1}}}\right)} \]

      7. log-rec N/A

         \[\leadsto \frac{\mathsf{neg}\left(\left(0 - {\log \left(\frac{N}{N + 1}\right)}^{2}\right)\right)}{\color{blue}{\mathsf{neg}\left(\log \left(\frac{N}{N + 1}\right)\right)}} \]

      8. lift-log.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\left(0 - {\log \left(\frac{N}{N + 1}\right)}^{2}\right)\right)}{\mathsf{neg}\left(\color{blue}{\log \left(\frac{N}{N + 1}\right)}\right)} \]

      9. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{0 - {\log \left(\frac{N}{N + 1}\right)}^{2}}{\log \left(\frac{N}{N + 1}\right)}} \]

      10. lower-/.f64 94.6

          \[\leadsto \color{blue}{\frac{0 - {\log \left(\frac{N}{N + 1}\right)}^{2}}{\log \left(\frac{N}{N + 1}\right)}} \]

      11. lift--.f64 N/A

          \[\leadsto \frac{\color{blue}{0 - {\log \left(\frac{N}{N + 1}\right)}^{2}}}{\log \left(\frac{N}{N + 1}\right)} \]

      12. sub0-neg N/A

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

      13. lower-neg.f64 94.6

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 94.6

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 94.6

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

   8. Applied rewrites 94.6%

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

   if 1150 < N

   1. Initial program 19.4%

      \[\log \left(N + 1\right) - \log N \]
   2. Add Preprocessing

   3. Taylor expanded in N around inf

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

   5. Applied rewrites 99.6%

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

   7. Applied rewrites 99.6%

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

   8. Taylor expanded in N around -inf

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

   10. Applied rewrites 99.8%

       \[\leadsto \frac{1}{-\mathsf{fma}\left(N, \frac{0.5 - \frac{0.08333333333333333 + \frac{-0.041666666666666664}{N}}{N}}{-N}, -N\right)} \]

   11. Taylor expanded in N around -inf

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

   13. Applied rewrites 99.8%

       \[\leadsto \frac{1}{\mathsf{fma}\left(N, \color{blue}{\frac{0.5 - \frac{\mathsf{fma}\left(N, 0.08333333333333333, -0.041666666666666664\right)}{N \cdot N}}{N}}, N\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;N \leq 1150:\\ \;\;\;\;-\frac{{\log \left(\frac{N}{N + 1}\right)}^{2}}{\log \left(\frac{N}{N + 1}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(N, \frac{0.5 - \frac{\mathsf{fma}\left(N, 0.08333333333333333, -0.041666666666666664\right)}{N \cdot N}}{N}, N\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 0.001:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(N, \frac{0.5 - \frac{\mathsf{fma}\left(N, 0.08333333333333333, -0.041666666666666664\right)}{N \cdot N}}{N}, N\right)}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= (- (log (+ N 1.0)) (log N)) 0.001)
   (/
    1.0
    (fma
     N
     (/
      (- 0.5 (/ (fma N 0.08333333333333333 -0.041666666666666664) (* N N)))
      N)
     N))
   (log (/ (+ N 1.0) N))))

C

double code(double N) {
	double tmp;
	if ((log((N + 1.0)) - log(N)) <= 0.001) {
		tmp = 1.0 / fma(N, ((0.5 - (fma(N, 0.08333333333333333, -0.041666666666666664) / (N * N))) / N), N);
	} else {
		tmp = log(((N + 1.0) / N));
	}
	return tmp;
}

Julia

function code(N)
	tmp = 0.0
	if (Float64(log(Float64(N + 1.0)) - log(N)) <= 0.001)
		tmp = Float64(1.0 / fma(N, Float64(Float64(0.5 - Float64(fma(N, 0.08333333333333333, -0.041666666666666664) / Float64(N * N))) / N), N));
	else
		tmp = log(Float64(Float64(N + 1.0) / N));
	end
	return tmp
end

Wolfram

code[N_] := If[LessEqual[N[(N[Log[N[(N + 1.0), $MachinePrecision]], $MachinePrecision] - N[Log[N], $MachinePrecision]), $MachinePrecision], 0.001], N[(1.0 / N[(N * N[(N[(0.5 - N[(N[(N * 0.08333333333333333 + -0.041666666666666664), $MachinePrecision] / N[(N * N), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision] + N), $MachinePrecision]), $MachinePrecision], N[Log[N[(N[(N + 1.0), $MachinePrecision] / N), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (log.f64 (+.f64 N #s(literal 1 binary64))) (log.f64 N)) < 1e-3

   1. Initial program 19.4%

      \[\log \left(N + 1\right) - \log N \]
   2. Add Preprocessing

   3. Taylor expanded in N around inf

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

   5. Applied rewrites 99.6%

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

   7. Applied rewrites 99.6%

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

   8. Taylor expanded in N around -inf

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

   10. Applied rewrites 99.8%

       \[\leadsto \frac{1}{-\mathsf{fma}\left(N, \frac{0.5 - \frac{0.08333333333333333 + \frac{-0.041666666666666664}{N}}{N}}{-N}, -N\right)} \]

   11. Taylor expanded in N around -inf

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

   13. Applied rewrites 99.8%

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

   if 1e-3 < (-.f64 (log.f64 (+.f64 N #s(literal 1 binary64))) (log.f64 N))

   1. Initial program 90.4%

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

      1. lift--.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. diff-log N/A

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

      5. lower-log.f64 N/A

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

      6. lower-/.f64 93.4

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

   4. Applied rewrites 93.4%

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

Alternative 4: 99.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;N \leq 1150:\\ \;\;\;\;-\log \left(\frac{N}{N + 1}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(N, \frac{0.5 - \frac{\mathsf{fma}\left(N, 0.08333333333333333, -0.041666666666666664\right)}{N \cdot N}}{N}, N\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= N 1150.0)
   (- (log (/ N (+ N 1.0))))
   (/
    1.0
    (fma
     N
     (/
      (- 0.5 (/ (fma N 0.08333333333333333 -0.041666666666666664) (* N N)))
      N)
     N))))

C

double code(double N) {
	double tmp;
	if (N <= 1150.0) {
		tmp = -log((N / (N + 1.0)));
	} else {
		tmp = 1.0 / fma(N, ((0.5 - (fma(N, 0.08333333333333333, -0.041666666666666664) / (N * N))) / N), N);
	}
	return tmp;
}

Julia

function code(N)
	tmp = 0.0
	if (N <= 1150.0)
		tmp = Float64(-log(Float64(N / Float64(N + 1.0))));
	else
		tmp = Float64(1.0 / fma(N, Float64(Float64(0.5 - Float64(fma(N, 0.08333333333333333, -0.041666666666666664) / Float64(N * N))) / N), N));
	end
	return tmp
end

Wolfram

code[N_] := If[LessEqual[N, 1150.0], (-N[Log[N[(N / N[(N + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), N[(1.0 / N[(N * N[(N[(0.5 - N[(N[(N * 0.08333333333333333 + -0.041666666666666664), $MachinePrecision] / N[(N * N), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision] + N), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if N < 1150

   1. Initial program 90.4%

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

      1. lift--.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. diff-log N/A

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

      5. clear-num N/A

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

      6. neg-log N/A

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

      7. diff-log N/A

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

      8. lift-log.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lift-log.f64 N/A

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

      12. lift-log.f64 N/A

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

      13. diff-log N/A

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

      14. lower-log.f64 N/A

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

      15. lower-/.f64 94.6

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

   4. Applied rewrites 94.6%

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

   if 1150 < N

   1. Initial program 19.4%

      \[\log \left(N + 1\right) - \log N \]
   2. Add Preprocessing

   3. Taylor expanded in N around inf

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

   5. Applied rewrites 99.6%

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

   7. Applied rewrites 99.6%

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

   8. Taylor expanded in N around -inf

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

   10. Applied rewrites 99.8%

       \[\leadsto \frac{1}{-\mathsf{fma}\left(N, \frac{0.5 - \frac{0.08333333333333333 + \frac{-0.041666666666666664}{N}}{N}}{-N}, -N\right)} \]

   11. Taylor expanded in N around -inf

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

   13. Applied rewrites 99.8%

       \[\leadsto \frac{1}{\mathsf{fma}\left(N, \color{blue}{\frac{0.5 - \frac{\mathsf{fma}\left(N, 0.08333333333333333, -0.041666666666666664\right)}{N \cdot N}}{N}}, N\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 96.7% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 25.0%

   \[\log \left(N + 1\right) - \log N \]
2. Add Preprocessing

3. Taylor expanded in N around inf

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

5. Applied rewrites 96.3%

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

7. Applied rewrites 96.3%

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

8. Taylor expanded in N around -inf

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

10. Applied rewrites 96.8%

    \[\leadsto \frac{1}{-\mathsf{fma}\left(N, \frac{0.5 - \frac{0.08333333333333333 + \frac{-0.041666666666666664}{N}}{N}}{-N}, -N\right)} \]

11. Taylor expanded in N around -inf

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

13. Applied rewrites 96.8%

    \[\leadsto \frac{1}{\mathsf{fma}\left(N, \color{blue}{\frac{0.5 - \frac{\mathsf{fma}\left(N, 0.08333333333333333, -0.041666666666666664\right)}{N \cdot N}}{N}}, N\right)} \]
14. Add Preprocessing

Alternative 6: 96.5% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 25.0%

   \[\log \left(N + 1\right) - \log N \]
2. Add Preprocessing

3. Taylor expanded in N around inf

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

5. Applied rewrites 96.3%

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

7. Applied rewrites 96.3%

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

8. Taylor expanded in N around -inf

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

10. Applied rewrites 96.8%

    \[\leadsto \frac{1}{-\mathsf{fma}\left(N, \frac{0.5 - \frac{0.08333333333333333 + \frac{-0.041666666666666664}{N}}{N}}{-N}, -N\right)} \]

11. Taylor expanded in N around 0

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

13. Applied rewrites 96.6%

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

Alternative 7: 95.6% accurate, 7.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 25.0%

   \[\log \left(N + 1\right) - \log N \]
2. Add Preprocessing

3. Taylor expanded in N around inf

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

5. Applied rewrites 96.3%

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

7. Applied rewrites 96.3%

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

8. Taylor expanded in N around -inf

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

10. Applied rewrites 96.8%

    \[\leadsto \frac{1}{-\mathsf{fma}\left(N, \frac{0.5 - \frac{0.08333333333333333 + \frac{-0.041666666666666664}{N}}{N}}{-N}, -N\right)} \]

11. Taylor expanded in N around inf

    \[\leadsto \frac{1}{N \cdot \color{blue}{\left(\left(1 + \frac{1}{2} \cdot \frac{1}{N}\right) - \frac{\frac{1}{12}}{{N}^{2}}\right)}} \]

13. Applied rewrites 94.9%

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

14. Final simplification 94.9%

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

Alternative 8: 93.0% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\mathsf{fma}\left(N, \frac{0.5}{N}, N\right)} \end{array} \]

FPCore

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

C

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

Julia

function code(N)
	return Float64(1.0 / fma(N, Float64(0.5 / N), N))
end

Wolfram

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

Derivation

1. Initial program 25.0%

   \[\log \left(N + 1\right) - \log N \]
2. Add Preprocessing

3. Taylor expanded in N around inf

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

5. Applied rewrites 96.3%

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

7. Applied rewrites 96.3%

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

8. Taylor expanded in N around inf

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

10. Applied rewrites 91.4%

    \[\leadsto \frac{1}{\mathsf{fma}\left(N, \color{blue}{\frac{0.5}{N}}, N\right)} \]
11. Add Preprocessing

Alternative 9: 93.0% accurate, 13.8× 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 25.0%

   \[\log \left(N + 1\right) - \log N \]
2. Add Preprocessing

3. Taylor expanded in N around inf

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

5. Applied rewrites 96.3%

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

7. Applied rewrites 96.3%

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

8. Taylor expanded in N around -inf

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

10. Applied rewrites 96.8%

    \[\leadsto \frac{1}{-\mathsf{fma}\left(N, \frac{0.5 - \frac{0.08333333333333333 + \frac{-0.041666666666666664}{N}}{N}}{-N}, -N\right)} \]

11. Taylor expanded in N around inf

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

13. Applied rewrites 91.4%

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

Alternative 10: 84.4% accurate, 17.3× 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 25.0%

   \[\log \left(N + 1\right) - \log N \]
2. Add Preprocessing

3. Taylor expanded in N around inf

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

   1. lower-/.f64 83.3

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

5. Applied rewrites 83.3%

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

Alternative 11: 3.3% accurate, 207.0× speedup

Math

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

FPCore

(FPCore (N) :precision binary64 0.0)

C

double code(double N) {
	return 0.0;
}

Fortran

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

Java

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

Python

def code(N):
	return 0.0

Julia

function code(N)
	return 0.0
end

MATLAB

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

Wolfram

code[N_] := 0.0

Derivation

1. Initial program 25.0%

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

   1. lift--.f64 N/A

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

   2. flip-- N/A

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

   3. div-sub N/A

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

   4. sub-neg N/A

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

   5. associate-/l* N/A

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

   6. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\log \left(N + 1\right), \frac{\log \left(N + 1\right)}{\log \left(N + 1\right) + \log N}, \mathsf{neg}\left(\frac{\log N \cdot \log N}{\log \left(N + 1\right) + \log N}\right)\right)} \]

4. Applied rewrites 25.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{log1p}\left(N\right), \frac{\mathsf{log1p}\left(N\right)}{\log \left(\mathsf{fma}\left(N, N, N\right)\right)}, -\frac{{\log N}^{2}}{\log \left(\mathsf{fma}\left(N, N, N\right)\right)}\right)} \]
5. Step-by-step derivation

   1. lift-log1p.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\log \left(1 + N\right)}, \frac{\mathsf{log1p}\left(N\right)}{\log \left(\mathsf{fma}\left(N, N, N\right)\right)}, \mathsf{neg}\left(\frac{{\log N}^{2}}{\log \left(\mathsf{fma}\left(N, N, N\right)\right)}\right)\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(\log \color{blue}{\left(N + 1\right)}, \frac{\mathsf{log1p}\left(N\right)}{\log \left(\mathsf{fma}\left(N, N, N\right)\right)}, \mathsf{neg}\left(\frac{{\log N}^{2}}{\log \left(\mathsf{fma}\left(N, N, N\right)\right)}\right)\right) \]

   3. flip-+ N/A

      \[\leadsto \mathsf{fma}\left(\log \color{blue}{\left(\frac{N \cdot N - 1 \cdot 1}{N - 1}\right)}, \frac{\mathsf{log1p}\left(N\right)}{\log \left(\mathsf{fma}\left(N, N, N\right)\right)}, \mathsf{neg}\left(\frac{{\log N}^{2}}{\log \left(\mathsf{fma}\left(N, N, N\right)\right)}\right)\right) \]

   4. log-div N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\log \left(N \cdot N - 1 \cdot 1\right) - \log \left(N - 1\right)}, \frac{\mathsf{log1p}\left(N\right)}{\log \left(\mathsf{fma}\left(N, N, N\right)\right)}, \mathsf{neg}\left(\frac{{\log N}^{2}}{\log \left(\mathsf{fma}\left(N, N, N\right)\right)}\right)\right) \]

   5. lower--.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\log \left(N \cdot N - 1 \cdot 1\right) - \log \left(N - 1\right)}, \frac{\mathsf{log1p}\left(N\right)}{\log \left(\mathsf{fma}\left(N, N, N\right)\right)}, \mathsf{neg}\left(\frac{{\log N}^{2}}{\log \left(\mathsf{fma}\left(N, N, N\right)\right)}\right)\right) \]

   6. lower-log.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\log \left(N \cdot N - 1 \cdot 1\right)} - \log \left(N - 1\right), \frac{\mathsf{log1p}\left(N\right)}{\log \left(\mathsf{fma}\left(N, N, N\right)\right)}, \mathsf{neg}\left(\frac{{\log N}^{2}}{\log \left(\mathsf{fma}\left(N, N, N\right)\right)}\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left(\log \left(N \cdot N - \color{blue}{1}\right) - \log \left(N - 1\right), \frac{\mathsf{log1p}\left(N\right)}{\log \left(\mathsf{fma}\left(N, N, N\right)\right)}, \mathsf{neg}\left(\frac{{\log N}^{2}}{\log \left(\mathsf{fma}\left(N, N, N\right)\right)}\right)\right) \]

   8. sub-neg N/A

      \[\leadsto \mathsf{fma}\left(\log \color{blue}{\left(N \cdot N + \left(\mathsf{neg}\left(1\right)\right)\right)} - \log \left(N - 1\right), \frac{\mathsf{log1p}\left(N\right)}{\log \left(\mathsf{fma}\left(N, N, N\right)\right)}, \mathsf{neg}\left(\frac{{\log N}^{2}}{\log \left(\mathsf{fma}\left(N, N, N\right)\right)}\right)\right) \]

   9. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left(\log \left(N \cdot N + \color{blue}{-1}\right) - \log \left(N - 1\right), \frac{\mathsf{log1p}\left(N\right)}{\log \left(\mathsf{fma}\left(N, N, N\right)\right)}, \mathsf{neg}\left(\frac{{\log N}^{2}}{\log \left(\mathsf{fma}\left(N, N, N\right)\right)}\right)\right) \]

   10. lower-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(\log \color{blue}{\left(\mathsf{fma}\left(N, N, -1\right)\right)} - \log \left(N - 1\right), \frac{\mathsf{log1p}\left(N\right)}{\log \left(\mathsf{fma}\left(N, N, N\right)\right)}, \mathsf{neg}\left(\frac{{\log N}^{2}}{\log \left(\mathsf{fma}\left(N, N, N\right)\right)}\right)\right) \]

   11. lower-log.f64 N/A

       \[\leadsto \mathsf{fma}\left(\log \left(\mathsf{fma}\left(N, N, -1\right)\right) - \color{blue}{\log \left(N - 1\right)}, \frac{\mathsf{log1p}\left(N\right)}{\log \left(\mathsf{fma}\left(N, N, N\right)\right)}, \mathsf{neg}\left(\frac{{\log N}^{2}}{\log \left(\mathsf{fma}\left(N, N, N\right)\right)}\right)\right) \]

   12. sub-neg N/A

       \[\leadsto \mathsf{fma}\left(\log \left(\mathsf{fma}\left(N, N, -1\right)\right) - \log \color{blue}{\left(N + \left(\mathsf{neg}\left(1\right)\right)\right)}, \frac{\mathsf{log1p}\left(N\right)}{\log \left(\mathsf{fma}\left(N, N, N\right)\right)}, \mathsf{neg}\left(\frac{{\log N}^{2}}{\log \left(\mathsf{fma}\left(N, N, N\right)\right)}\right)\right) \]

   13. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(\log \left(\mathsf{fma}\left(N, N, -1\right)\right) - \log \left(N + \color{blue}{-1}\right), \frac{\mathsf{log1p}\left(N\right)}{\log \left(\mathsf{fma}\left(N, N, N\right)\right)}, \mathsf{neg}\left(\frac{{\log N}^{2}}{\log \left(\mathsf{fma}\left(N, N, N\right)\right)}\right)\right) \]

   14. lower-+.f64 25.3

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

6. Applied rewrites 25.3%

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

7. Taylor expanded in N around inf

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

   1. distribute-rgt-out-- N/A

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

   2. metadata-eval N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. metadata-eval N/A

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

   6. +-inverses 3.3

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

9. Applied rewrites 3.3%

   \[\leadsto \color{blue}{0} \]
10. Add Preprocessing

Developer Target 1: 99.8% accurate, 1.8× 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: 26.7% accurate, 1.8× 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: 96.2% 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 2024238 
(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)))