2log (problem 3.3.6)

Percentage Accurate: 24.0% → 99.5%

Time: 8.0s

Alternatives: 9

Speedup: 8.0×

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: 24.0% 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, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

function code(N)
	tmp = 0.0
	if (Float64(log(Float64(N + 1.0)) - log(N)) <= 0.001)
		tmp = fma(Float64(Float64(0.5 - Float64(Float64(0.08333333333333333 - Float64(0.041666666666666664 / N)) / N)) / N), N, N) ^ -1.0;
	else
		tmp = Float64(-log(Float64(N / Float64(N + 1.0))));
	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[Power[N[(N[(N[(0.5 - N[(N[(0.08333333333333333 - N[(0.041666666666666664 / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision] * N + N), $MachinePrecision], -1.0], $MachinePrecision], (-N[Log[N[(N / N[(N + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 18.9%

      \[\log \left(N + 1\right) - \log N \]
   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 99.8%

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

   7. Applied rewrites 99.8%

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

   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 99.7%

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

   12. Applied rewrites 99.9%

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

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

   1. Initial program 91.7%

      \[\log \left(N + 1\right) - \log N \]
   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. lift-+.f64 N/A

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

      6. flip-+ N/A

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

      7. associate-/l/ N/A

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

      8. clear-num N/A

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

      9. log-rec N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-log.f64 N/A

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

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

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

      13. lower-/.f64 N/A

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

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

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower--.f64 N/A

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

      18. metadata-eval N/A

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

      19. sub-neg N/A

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

      20. lower-fma.f64 N/A

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

      21. metadata-eval 93.5

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

   4. Applied rewrites 93.5%

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

      1. lift-log.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-fma.f64 N/A

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

      6. metadata-eval N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. associate-/l* N/A

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

      10. log-prod N/A

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

      11. lift-log.f64 N/A

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

      12. unpow1 N/A

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

      13. sqr-pow N/A

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

      14. lower-fma.f64 N/A

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

      15. metadata-eval N/A

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

      16. unpow1/2 N/A

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

      17. lower-sqrt.f64 N/A

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

      18. metadata-eval N/A

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

      19. unpow1/2 N/A

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

      20. lower-sqrt.f64 N/A

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

      21. lower-log.f64 N/A

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

   6. Applied rewrites 90.2%

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

      1. lift-fma.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. rem-square-sqrt N/A

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

      5. lift-log.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. lift-fma.f64 N/A

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

      9. metadata-eval N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. lift--.f64 N/A

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

      13. flip-+ N/A

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

      14. neg-log N/A

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

      15. +-commutative N/A

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

      16. lift-log1p.f64 N/A

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

      17. sub-neg N/A

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

      18. lift-log.f64 N/A

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

      19. lift-log1p.f64 N/A

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

      20. +-commutative N/A

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

   8. Applied rewrites 94.6%

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

4. Final simplification 99.5%

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

Alternative 2: 99.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;N \leq 750:\\ \;\;\;\;\log \left(\frac{1 + N}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\frac{0.5 - \frac{0.08333333333333333 - \frac{0.041666666666666664}{N}}{N}}{N}, N, N\right)\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= N 750.0)
   (log (/ (+ 1.0 N) N))
   (pow
    (fma
     (/ (- 0.5 (/ (- 0.08333333333333333 (/ 0.041666666666666664 N)) N)) N)
     N
     N)
    -1.0)))

C

double code(double N) {
	double tmp;
	if (N <= 750.0) {
		tmp = log(((1.0 + N) / N));
	} else {
		tmp = pow(fma(((0.5 - ((0.08333333333333333 - (0.041666666666666664 / N)) / N)) / N), N, N), -1.0);
	}
	return tmp;
}

Julia

function code(N)
	tmp = 0.0
	if (N <= 750.0)
		tmp = log(Float64(Float64(1.0 + N) / N));
	else
		tmp = fma(Float64(Float64(0.5 - Float64(Float64(0.08333333333333333 - Float64(0.041666666666666664 / N)) / N)) / N), N, N) ^ -1.0;
	end
	return tmp
end

Wolfram

code[N_] := If[LessEqual[N, 750.0], N[Log[N[(N[(1.0 + N), $MachinePrecision] / N), $MachinePrecision]], $MachinePrecision], N[Power[N[(N[(N[(0.5 - N[(N[(0.08333333333333333 - N[(0.041666666666666664 / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision] * N + N), $MachinePrecision], -1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if N < 750

   1. Initial program 92.7%

      \[\log \left(N + 1\right) - \log N \]
   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 94.2

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 94.2

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

   4. Applied rewrites 94.2%

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

   if 750 < 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%

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

   7. Applied rewrites 99.7%

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

   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 99.6%

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

   12. Applied rewrites 99.8%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;N \leq 750:\\ \;\;\;\;\log \left(\frac{1 + N}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(\frac{0.5 - \frac{0.08333333333333333 - \frac{0.041666666666666664}{N}}{N}}{N}, N, N\right)\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.7% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 23.7%

   \[\log \left(N + 1\right) - \log N \]
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 96.8%

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

7. Applied rewrites 96.8%

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

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 97.0%

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

12. Applied rewrites 97.2%

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

13. Final simplification 97.2%

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

Alternative 4: 96.6% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 23.7%

   \[\log \left(N + 1\right) - \log N \]
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 96.8%

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

7. Applied rewrites 96.8%

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

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 97.0%

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

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 97.0%

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

14. Final simplification 97.0%

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

Alternative 5: 93.0% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 23.7%

   \[\log \left(N + 1\right) - \log N \]
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 96.8%

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

7. Applied rewrites 96.8%

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

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 93.8%

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

11. Final simplification 93.8%

    \[\leadsto {\left(\mathsf{fma}\left(\frac{0.5}{N}, N, N\right)\right)}^{-1} \]
12. Add Preprocessing

Alternative 6: 84.3% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (N) :precision binary64 (pow N -1.0))

C

double code(double N) {
	return pow(N, -1.0);
}

Fortran

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

Java

public static double code(double N) {
	return Math.pow(N, -1.0);
}

Python

def code(N):
	return math.pow(N, -1.0)

Julia

function code(N)
	return N ^ -1.0
end

MATLAB

function tmp = code(N)
	tmp = N ^ -1.0;
end

Wolfram

code[N_] := N[Power[N, -1.0], $MachinePrecision]

Derivation

1. Initial program 23.7%

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

3. Taylor expanded in N around inf

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

   1. lower-/.f64 84.4

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

5. Applied rewrites 84.4%

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

6. Final simplification 84.4%

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

Alternative 7: 95.0% accurate, 5.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 23.7%

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

3. Taylor expanded in N around inf

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

   1. lower-/.f64 N/A

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

5. Applied rewrites 95.8%

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

Alternative 8: 92.3% accurate, 8.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 23.7%

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

3. Taylor expanded in N around inf

   \[\leadsto \color{blue}{\frac{1 - \frac{1}{2} \cdot \frac{1}{N}}{N}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{1 - \frac{1}{2} \cdot \frac{1}{N}}{N}} \]

   2. lower--.f64 N/A

      \[\leadsto \frac{\color{blue}{1 - \frac{1}{2} \cdot \frac{1}{N}}}{N} \]

   3. associate-*r/ N/A

      \[\leadsto \frac{1 - \color{blue}{\frac{\frac{1}{2} \cdot 1}{N}}}{N} \]

   4. metadata-eval N/A

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

   5. lower-/.f64 93.1

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

5. Applied rewrites 93.1%

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

Alternative 9: 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 23.7%

   \[\log \left(N + 1\right) - \log N \]
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. lift-+.f64 N/A

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

   6. flip-+ N/A

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

   7. associate-/l/ N/A

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

   8. clear-num N/A

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

   9. log-rec N/A

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

   10. lower-neg.f64 N/A

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

   11. lower-log.f64 N/A

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

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

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

   13. lower-/.f64 N/A

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

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

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

   15. *-commutative N/A

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

   16. lower-*.f64 N/A

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

   17. lower--.f64 N/A

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

   18. metadata-eval N/A

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

   19. sub-neg N/A

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

   20. lower-fma.f64 N/A

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

   21. metadata-eval 26.7

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

4. Applied rewrites 26.7%

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

   1. lift-log.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lift-fma.f64 N/A

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

   6. metadata-eval N/A

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

   7. sub-neg N/A

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

   8. metadata-eval N/A

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

   9. associate-/l* N/A

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

   10. log-prod N/A

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

   11. lift-log.f64 N/A

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

   12. unpow1 N/A

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

   13. sqr-pow N/A

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

   14. lower-fma.f64 N/A

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

   15. metadata-eval N/A

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

   16. unpow1/2 N/A

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

   17. lower-sqrt.f64 N/A

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

   18. metadata-eval N/A

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

   19. unpow1/2 N/A

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

   20. lower-sqrt.f64 N/A

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

   21. lower-log.f64 N/A

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

6. Applied rewrites 25.8%

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

7. Taylor expanded in N around inf

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

   1. distribute-rgt1-in N/A

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

   2. metadata-eval N/A

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

   3. mul0-lft N/A

      \[\leadsto -1 \cdot \color{blue}{0} \]

   4. metadata-eval 3.3

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

9. Applied rewrites 3.3%

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

10. Final simplification 3.3%

    \[\leadsto 0 \]
11. Add Preprocessing

Developer Target 1: 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 2024298 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  :pre (and (> N 1.0) (< N 1e+40))

  :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)))