2log (problem 3.3.6)

Percentage Accurate: 23.7% → 99.5%

Time: 9.8s

Alternatives: 8

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: 23.7% 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

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

FPCore

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

C

double code(double N) {
	double tmp;
	if ((log((N + 1.0)) - log(N)) <= 0.001) {
		tmp = 1.0 / (N + (fma(N, fma(N, 0.5, -0.08333333333333333), 0.041666666666666664) / (N * N)));
	} 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 = Float64(1.0 / Float64(N + Float64(fma(N, fma(N, 0.5, -0.08333333333333333), 0.041666666666666664) / Float64(N * N))));
	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[(1.0 / N[(N + N[(N[(N * N[(N * 0.5 + -0.08333333333333333), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] / N[(N * N), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-N[Log[N[(N / N[(N + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 16.3%

      \[\log \left(N + 1\right) - \log N \]
   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 99.7%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. clear-num N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 99.7

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

   7. Applied rewrites 99.7%

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

   8. Taylor expanded in N around inf

      \[\leadsto \frac{1}{\color{blue}{N \cdot \left(\left(1 + \left(\frac{1}{2} \cdot \frac{1}{N} + \frac{1}{24} \cdot \frac{1}{{N}^{3}}\right)\right) - \frac{\frac{1}{12}}{{N}^{2}}\right)}} \]
   9. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \frac{1}{N \cdot \color{blue}{\left(1 + \left(\left(\frac{1}{2} \cdot \frac{1}{N} + \frac{1}{24} \cdot \frac{1}{{N}^{3}}\right) - \frac{\frac{1}{12}}{{N}^{2}}\right)\right)}} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{1}{\color{blue}{N \cdot 1 + N \cdot \left(\left(\frac{1}{2} \cdot \frac{1}{N} + \frac{1}{24} \cdot \frac{1}{{N}^{3}}\right) - \frac{\frac{1}{12}}{{N}^{2}}\right)}} \]

      3. *-rgt-identity N/A

         \[\leadsto \frac{1}{\color{blue}{N} + N \cdot \left(\left(\frac{1}{2} \cdot \frac{1}{N} + \frac{1}{24} \cdot \frac{1}{{N}^{3}}\right) - \frac{\frac{1}{12}}{{N}^{2}}\right)} \]

      4. lower-+.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{N + N \cdot \left(\left(\frac{1}{2} \cdot \frac{1}{N} + \frac{1}{24} \cdot \frac{1}{{N}^{3}}\right) - \frac{\frac{1}{12}}{{N}^{2}}\right)}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{1}{N + \color{blue}{N \cdot \left(\left(\frac{1}{2} \cdot \frac{1}{N} + \frac{1}{24} \cdot \frac{1}{{N}^{3}}\right) - \frac{\frac{1}{12}}{{N}^{2}}\right)}} \]

      6. sub-neg N/A

         \[\leadsto \frac{1}{N + N \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \frac{1}{N} + \frac{1}{24} \cdot \frac{1}{{N}^{3}}\right) + \left(\mathsf{neg}\left(\frac{\frac{1}{12}}{{N}^{2}}\right)\right)\right)}} \]

      7. lower-+.f64 N/A

         \[\leadsto \frac{1}{N + N \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \frac{1}{N} + \frac{1}{24} \cdot \frac{1}{{N}^{3}}\right) + \left(\mathsf{neg}\left(\frac{\frac{1}{12}}{{N}^{2}}\right)\right)\right)}} \]

   10. Applied rewrites 99.8%

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

   11. Taylor expanded in N around 0

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

       1. lower-/.f64 N/A

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

       2. +-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. sub-neg N/A

          \[\leadsto \frac{1}{N + \frac{\mathsf{fma}\left(N, \color{blue}{\frac{1}{2} \cdot N + \left(\mathsf{neg}\left(\frac{1}{12}\right)\right)}, \frac{1}{24}\right)}{{N}^{2}}} \]

       5. *-commutative N/A

          \[\leadsto \frac{1}{N + \frac{\mathsf{fma}\left(N, \color{blue}{N \cdot \frac{1}{2}} + \left(\mathsf{neg}\left(\frac{1}{12}\right)\right), \frac{1}{24}\right)}{{N}^{2}}} \]

       6. metadata-eval N/A

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

       7. lower-fma.f64 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 99.8

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

   13. Applied rewrites 99.8%

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

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

   1. Initial program 93.1%

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

      1. lift-+.f64 N/A

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

      2. diff-log N/A

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

      3. clear-num N/A

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

      4. log-rec N/A

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

      5. diff-log N/A

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

      6. lift-log.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. lift-log.f64 N/A

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

      11. diff-log N/A

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

      12. lower-log.f64 N/A

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

      13. lower-/.f64 96.2

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

   4. Applied rewrites 96.2%

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

Alternative 2: 99.4% accurate, 0.6× speedup

Math

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

FPCore

(FPCore (N)
 :precision binary64
 (if (<= (- (log (+ N 1.0)) (log N)) 0.001)
   (/
    1.0
    (+
     N
     (/
      (fma N (fma N 0.5 -0.08333333333333333) 0.041666666666666664)
      (* 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 / (N + (fma(N, fma(N, 0.5, -0.08333333333333333), 0.041666666666666664) / (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 / Float64(N + Float64(fma(N, fma(N, 0.5, -0.08333333333333333), 0.041666666666666664) / Float64(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[(N * N[(N * 0.5 + -0.08333333333333333), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] / N[(N * N), $MachinePrecision]), $MachinePrecision]), $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 16.3%

      \[\log \left(N + 1\right) - \log N \]
   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 99.7%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. clear-num N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 99.7

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

   7. Applied rewrites 99.7%

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

   8. Taylor expanded in N around inf

      \[\leadsto \frac{1}{\color{blue}{N \cdot \left(\left(1 + \left(\frac{1}{2} \cdot \frac{1}{N} + \frac{1}{24} \cdot \frac{1}{{N}^{3}}\right)\right) - \frac{\frac{1}{12}}{{N}^{2}}\right)}} \]
   9. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \frac{1}{N \cdot \color{blue}{\left(1 + \left(\left(\frac{1}{2} \cdot \frac{1}{N} + \frac{1}{24} \cdot \frac{1}{{N}^{3}}\right) - \frac{\frac{1}{12}}{{N}^{2}}\right)\right)}} \]

      2. distribute-lft-in N/A

         \[\leadsto \frac{1}{\color{blue}{N \cdot 1 + N \cdot \left(\left(\frac{1}{2} \cdot \frac{1}{N} + \frac{1}{24} \cdot \frac{1}{{N}^{3}}\right) - \frac{\frac{1}{12}}{{N}^{2}}\right)}} \]

      3. *-rgt-identity N/A

         \[\leadsto \frac{1}{\color{blue}{N} + N \cdot \left(\left(\frac{1}{2} \cdot \frac{1}{N} + \frac{1}{24} \cdot \frac{1}{{N}^{3}}\right) - \frac{\frac{1}{12}}{{N}^{2}}\right)} \]

      4. lower-+.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{N + N \cdot \left(\left(\frac{1}{2} \cdot \frac{1}{N} + \frac{1}{24} \cdot \frac{1}{{N}^{3}}\right) - \frac{\frac{1}{12}}{{N}^{2}}\right)}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{1}{N + \color{blue}{N \cdot \left(\left(\frac{1}{2} \cdot \frac{1}{N} + \frac{1}{24} \cdot \frac{1}{{N}^{3}}\right) - \frac{\frac{1}{12}}{{N}^{2}}\right)}} \]

      6. sub-neg N/A

         \[\leadsto \frac{1}{N + N \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \frac{1}{N} + \frac{1}{24} \cdot \frac{1}{{N}^{3}}\right) + \left(\mathsf{neg}\left(\frac{\frac{1}{12}}{{N}^{2}}\right)\right)\right)}} \]

      7. lower-+.f64 N/A

         \[\leadsto \frac{1}{N + N \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \frac{1}{N} + \frac{1}{24} \cdot \frac{1}{{N}^{3}}\right) + \left(\mathsf{neg}\left(\frac{\frac{1}{12}}{{N}^{2}}\right)\right)\right)}} \]

   10. Applied rewrites 99.8%

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

   11. Taylor expanded in N around 0

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

       1. lower-/.f64 N/A

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

       2. +-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. sub-neg N/A

          \[\leadsto \frac{1}{N + \frac{\mathsf{fma}\left(N, \color{blue}{\frac{1}{2} \cdot N + \left(\mathsf{neg}\left(\frac{1}{12}\right)\right)}, \frac{1}{24}\right)}{{N}^{2}}} \]

       5. *-commutative N/A

          \[\leadsto \frac{1}{N + \frac{\mathsf{fma}\left(N, \color{blue}{N \cdot \frac{1}{2}} + \left(\mathsf{neg}\left(\frac{1}{12}\right)\right), \frac{1}{24}\right)}{{N}^{2}}} \]

       6. metadata-eval N/A

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

       7. lower-fma.f64 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f64 99.8

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

   13. Applied rewrites 99.8%

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

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

   1. Initial program 93.1%

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

      1. lift-+.f64 N/A

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

      2. diff-log N/A

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

      3. lower-log.f64 N/A

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

      4. lower-/.f64 95.2

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

   4. Applied rewrites 95.2%

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

Alternative 3: 96.7% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Julia

function code(N)
	return Float64(1.0 / Float64(N + Float64(fma(N, fma(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 + -0.08333333333333333), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] / N[(N * N), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 23.5%

   \[\log \left(N + 1\right) - \log N \]
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 94.9%

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

   1. lift-/.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift--.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-+.f64 N/A

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

   7. clear-num N/A

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

   8. lower-/.f64 N/A

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

   9. lower-/.f64 94.9

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

7. Applied rewrites 94.9%

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

8. Taylor expanded in N around inf

   \[\leadsto \frac{1}{\color{blue}{N \cdot \left(\left(1 + \left(\frac{1}{2} \cdot \frac{1}{N} + \frac{1}{24} \cdot \frac{1}{{N}^{3}}\right)\right) - \frac{\frac{1}{12}}{{N}^{2}}\right)}} \]
9. Step-by-step derivation

   1. associate--l+ N/A

      \[\leadsto \frac{1}{N \cdot \color{blue}{\left(1 + \left(\left(\frac{1}{2} \cdot \frac{1}{N} + \frac{1}{24} \cdot \frac{1}{{N}^{3}}\right) - \frac{\frac{1}{12}}{{N}^{2}}\right)\right)}} \]

   2. distribute-lft-in N/A

      \[\leadsto \frac{1}{\color{blue}{N \cdot 1 + N \cdot \left(\left(\frac{1}{2} \cdot \frac{1}{N} + \frac{1}{24} \cdot \frac{1}{{N}^{3}}\right) - \frac{\frac{1}{12}}{{N}^{2}}\right)}} \]

   3. *-rgt-identity N/A

      \[\leadsto \frac{1}{\color{blue}{N} + N \cdot \left(\left(\frac{1}{2} \cdot \frac{1}{N} + \frac{1}{24} \cdot \frac{1}{{N}^{3}}\right) - \frac{\frac{1}{12}}{{N}^{2}}\right)} \]

   4. lower-+.f64 N/A

      \[\leadsto \frac{1}{\color{blue}{N + N \cdot \left(\left(\frac{1}{2} \cdot \frac{1}{N} + \frac{1}{24} \cdot \frac{1}{{N}^{3}}\right) - \frac{\frac{1}{12}}{{N}^{2}}\right)}} \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{1}{N + \color{blue}{N \cdot \left(\left(\frac{1}{2} \cdot \frac{1}{N} + \frac{1}{24} \cdot \frac{1}{{N}^{3}}\right) - \frac{\frac{1}{12}}{{N}^{2}}\right)}} \]

   6. sub-neg N/A

      \[\leadsto \frac{1}{N + N \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \frac{1}{N} + \frac{1}{24} \cdot \frac{1}{{N}^{3}}\right) + \left(\mathsf{neg}\left(\frac{\frac{1}{12}}{{N}^{2}}\right)\right)\right)}} \]

   7. lower-+.f64 N/A

      \[\leadsto \frac{1}{N + N \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \frac{1}{N} + \frac{1}{24} \cdot \frac{1}{{N}^{3}}\right) + \left(\mathsf{neg}\left(\frac{\frac{1}{12}}{{N}^{2}}\right)\right)\right)}} \]

10. Applied rewrites 95.4%

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

11. Taylor expanded in N around 0

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

    1. lower-/.f64 N/A

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

    2. +-commutative N/A

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

    3. lower-fma.f64 N/A

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

    4. sub-neg N/A

       \[\leadsto \frac{1}{N + \frac{\mathsf{fma}\left(N, \color{blue}{\frac{1}{2} \cdot N + \left(\mathsf{neg}\left(\frac{1}{12}\right)\right)}, \frac{1}{24}\right)}{{N}^{2}}} \]

    5. *-commutative N/A

       \[\leadsto \frac{1}{N + \frac{\mathsf{fma}\left(N, \color{blue}{N \cdot \frac{1}{2}} + \left(\mathsf{neg}\left(\frac{1}{12}\right)\right), \frac{1}{24}\right)}{{N}^{2}}} \]

    6. metadata-eval N/A

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

    7. lower-fma.f64 N/A

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

    8. unpow2 N/A

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

    9. lower-*.f64 95.4

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

13. Applied rewrites 95.4%

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

Alternative 4: 95.5% accurate, 7.1× speedup

Math

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

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(N + Float64(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 + N[(-0.08333333333333333 / N), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 23.5%

   \[\log \left(N + 1\right) - \log N \]
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 94.9%

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

   1. lift-/.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift--.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-+.f64 N/A

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

   7. clear-num N/A

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

   8. lower-/.f64 N/A

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

   9. lower-/.f64 94.9

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

7. Applied rewrites 94.9%

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

8. Taylor expanded in N around inf

   \[\leadsto \frac{1}{\color{blue}{N \cdot \left(\left(1 + \left(\frac{1}{2} \cdot \frac{1}{N} + \frac{1}{24} \cdot \frac{1}{{N}^{3}}\right)\right) - \frac{\frac{1}{12}}{{N}^{2}}\right)}} \]
9. Step-by-step derivation

   1. associate--l+ N/A

      \[\leadsto \frac{1}{N \cdot \color{blue}{\left(1 + \left(\left(\frac{1}{2} \cdot \frac{1}{N} + \frac{1}{24} \cdot \frac{1}{{N}^{3}}\right) - \frac{\frac{1}{12}}{{N}^{2}}\right)\right)}} \]

   2. distribute-lft-in N/A

      \[\leadsto \frac{1}{\color{blue}{N \cdot 1 + N \cdot \left(\left(\frac{1}{2} \cdot \frac{1}{N} + \frac{1}{24} \cdot \frac{1}{{N}^{3}}\right) - \frac{\frac{1}{12}}{{N}^{2}}\right)}} \]

   3. *-rgt-identity N/A

      \[\leadsto \frac{1}{\color{blue}{N} + N \cdot \left(\left(\frac{1}{2} \cdot \frac{1}{N} + \frac{1}{24} \cdot \frac{1}{{N}^{3}}\right) - \frac{\frac{1}{12}}{{N}^{2}}\right)} \]

   4. lower-+.f64 N/A

      \[\leadsto \frac{1}{\color{blue}{N + N \cdot \left(\left(\frac{1}{2} \cdot \frac{1}{N} + \frac{1}{24} \cdot \frac{1}{{N}^{3}}\right) - \frac{\frac{1}{12}}{{N}^{2}}\right)}} \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{1}{N + \color{blue}{N \cdot \left(\left(\frac{1}{2} \cdot \frac{1}{N} + \frac{1}{24} \cdot \frac{1}{{N}^{3}}\right) - \frac{\frac{1}{12}}{{N}^{2}}\right)}} \]

   6. sub-neg N/A

      \[\leadsto \frac{1}{N + N \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \frac{1}{N} + \frac{1}{24} \cdot \frac{1}{{N}^{3}}\right) + \left(\mathsf{neg}\left(\frac{\frac{1}{12}}{{N}^{2}}\right)\right)\right)}} \]

   7. lower-+.f64 N/A

      \[\leadsto \frac{1}{N + N \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \frac{1}{N} + \frac{1}{24} \cdot \frac{1}{{N}^{3}}\right) + \left(\mathsf{neg}\left(\frac{\frac{1}{12}}{{N}^{2}}\right)\right)\right)}} \]

10. Applied rewrites 95.4%

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

11. Taylor expanded in N around inf

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

    1. sub-neg N/A

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

    2. lower-+.f64 N/A

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

    3. associate-*r/ N/A

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

    4. metadata-eval N/A

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

    5. distribute-neg-frac N/A

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

    6. metadata-eval N/A

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

    7. lower-/.f64 94.2

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

13. Applied rewrites 94.2%

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

Alternative 5: 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 23.5%

   \[\log \left(N + 1\right) - \log N \]
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 94.9%

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

   1. lift-/.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift--.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-+.f64 N/A

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

   7. clear-num N/A

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

   8. lower-/.f64 N/A

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

   9. lower-/.f64 94.9

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

7. Applied rewrites 94.9%

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

8. Taylor expanded in N around inf

   \[\leadsto \frac{1}{\color{blue}{N \cdot \left(\left(1 + \left(\frac{1}{2} \cdot \frac{1}{N} + \frac{1}{24} \cdot \frac{1}{{N}^{3}}\right)\right) - \frac{\frac{1}{12}}{{N}^{2}}\right)}} \]
9. Step-by-step derivation

   1. associate--l+ N/A

      \[\leadsto \frac{1}{N \cdot \color{blue}{\left(1 + \left(\left(\frac{1}{2} \cdot \frac{1}{N} + \frac{1}{24} \cdot \frac{1}{{N}^{3}}\right) - \frac{\frac{1}{12}}{{N}^{2}}\right)\right)}} \]

   2. distribute-lft-in N/A

      \[\leadsto \frac{1}{\color{blue}{N \cdot 1 + N \cdot \left(\left(\frac{1}{2} \cdot \frac{1}{N} + \frac{1}{24} \cdot \frac{1}{{N}^{3}}\right) - \frac{\frac{1}{12}}{{N}^{2}}\right)}} \]

   3. *-rgt-identity N/A

      \[\leadsto \frac{1}{\color{blue}{N} + N \cdot \left(\left(\frac{1}{2} \cdot \frac{1}{N} + \frac{1}{24} \cdot \frac{1}{{N}^{3}}\right) - \frac{\frac{1}{12}}{{N}^{2}}\right)} \]

   4. lower-+.f64 N/A

      \[\leadsto \frac{1}{\color{blue}{N + N \cdot \left(\left(\frac{1}{2} \cdot \frac{1}{N} + \frac{1}{24} \cdot \frac{1}{{N}^{3}}\right) - \frac{\frac{1}{12}}{{N}^{2}}\right)}} \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{1}{N + \color{blue}{N \cdot \left(\left(\frac{1}{2} \cdot \frac{1}{N} + \frac{1}{24} \cdot \frac{1}{{N}^{3}}\right) - \frac{\frac{1}{12}}{{N}^{2}}\right)}} \]

   6. sub-neg N/A

      \[\leadsto \frac{1}{N + N \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \frac{1}{N} + \frac{1}{24} \cdot \frac{1}{{N}^{3}}\right) + \left(\mathsf{neg}\left(\frac{\frac{1}{12}}{{N}^{2}}\right)\right)\right)}} \]

   7. lower-+.f64 N/A

      \[\leadsto \frac{1}{N + N \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \frac{1}{N} + \frac{1}{24} \cdot \frac{1}{{N}^{3}}\right) + \left(\mathsf{neg}\left(\frac{\frac{1}{12}}{{N}^{2}}\right)\right)\right)}} \]

10. Applied rewrites 95.4%

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

11. Taylor expanded in N around inf

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

13. Applied rewrites 92.1%

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

Alternative 6: 84.5% 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 23.5%

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

3. Taylor expanded in N around inf

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

   1. lower-/.f64 84.6

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

5. Applied rewrites 84.6%

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

Alternative 7: 7.3% accurate, 18.8× speedup

Math

\[\begin{array}{l} \\ \left(N \cdot N\right) \cdot 24 \end{array} \]

FPCore

(FPCore (N) :precision binary64 (* (* N N) 24.0))

C

double code(double N) {
	return (N * N) * 24.0;
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    code = (n * n) * 24.0d0
end function

Java

public static double code(double N) {
	return (N * N) * 24.0;
}

Python

def code(N):
	return (N * N) * 24.0

Julia

function code(N)
	return Float64(Float64(N * N) * 24.0)
end

MATLAB

function tmp = code(N)
	tmp = (N * N) * 24.0;
end

Wolfram

code[N_] := N[(N[(N * N), $MachinePrecision] * 24.0), $MachinePrecision]

Derivation

1. Initial program 23.5%

   \[\log \left(N + 1\right) - \log N \]
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 94.9%

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

   1. lift-/.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift--.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-+.f64 N/A

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

   7. clear-num N/A

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

   8. lower-/.f64 N/A

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

   9. lower-/.f64 94.9

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

7. Applied rewrites 94.9%

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

8. Taylor expanded in N around inf

   \[\leadsto \frac{1}{\color{blue}{N \cdot \left(\left(1 + \left(\frac{1}{2} \cdot \frac{1}{N} + \frac{1}{24} \cdot \frac{1}{{N}^{3}}\right)\right) - \frac{\frac{1}{12}}{{N}^{2}}\right)}} \]
9. Step-by-step derivation

   1. associate--l+ N/A

      \[\leadsto \frac{1}{N \cdot \color{blue}{\left(1 + \left(\left(\frac{1}{2} \cdot \frac{1}{N} + \frac{1}{24} \cdot \frac{1}{{N}^{3}}\right) - \frac{\frac{1}{12}}{{N}^{2}}\right)\right)}} \]

   2. distribute-lft-in N/A

      \[\leadsto \frac{1}{\color{blue}{N \cdot 1 + N \cdot \left(\left(\frac{1}{2} \cdot \frac{1}{N} + \frac{1}{24} \cdot \frac{1}{{N}^{3}}\right) - \frac{\frac{1}{12}}{{N}^{2}}\right)}} \]

   3. *-rgt-identity N/A

      \[\leadsto \frac{1}{\color{blue}{N} + N \cdot \left(\left(\frac{1}{2} \cdot \frac{1}{N} + \frac{1}{24} \cdot \frac{1}{{N}^{3}}\right) - \frac{\frac{1}{12}}{{N}^{2}}\right)} \]

   4. lower-+.f64 N/A

      \[\leadsto \frac{1}{\color{blue}{N + N \cdot \left(\left(\frac{1}{2} \cdot \frac{1}{N} + \frac{1}{24} \cdot \frac{1}{{N}^{3}}\right) - \frac{\frac{1}{12}}{{N}^{2}}\right)}} \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{1}{N + \color{blue}{N \cdot \left(\left(\frac{1}{2} \cdot \frac{1}{N} + \frac{1}{24} \cdot \frac{1}{{N}^{3}}\right) - \frac{\frac{1}{12}}{{N}^{2}}\right)}} \]

   6. sub-neg N/A

      \[\leadsto \frac{1}{N + N \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \frac{1}{N} + \frac{1}{24} \cdot \frac{1}{{N}^{3}}\right) + \left(\mathsf{neg}\left(\frac{\frac{1}{12}}{{N}^{2}}\right)\right)\right)}} \]

   7. lower-+.f64 N/A

      \[\leadsto \frac{1}{N + N \cdot \color{blue}{\left(\left(\frac{1}{2} \cdot \frac{1}{N} + \frac{1}{24} \cdot \frac{1}{{N}^{3}}\right) + \left(\mathsf{neg}\left(\frac{\frac{1}{12}}{{N}^{2}}\right)\right)\right)}} \]

10. Applied rewrites 95.4%

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

11. Taylor expanded in N around 0

    \[\leadsto \color{blue}{24 \cdot {N}^{2}} \]
12. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \color{blue}{{N}^{2} \cdot 24} \]

    2. lower-*.f64 N/A

       \[\leadsto \color{blue}{{N}^{2} \cdot 24} \]

    3. unpow2 N/A

       \[\leadsto \color{blue}{\left(N \cdot N\right)} \cdot 24 \]

    4. lower-*.f64 7.4

       \[\leadsto \color{blue}{\left(N \cdot N\right)} \cdot 24 \]

13. Applied rewrites 7.4%

    \[\leadsto \color{blue}{\left(N \cdot N\right) \cdot 24} \]
14. Add Preprocessing

Alternative 8: 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 23.5%

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

4. Applied rewrites 25.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left({\left(\mathsf{log1p}\left(N\right)\right)}^{1.5}, \frac{{\left(\mathsf{log1p}\left(N\right)\right)}^{1.5}}{\mathsf{fma}\left(\log N, \log \left(\mathsf{fma}\left(N, N, N\right)\right), {\left(\mathsf{log1p}\left(N\right)\right)}^{2}\right)}, -\frac{{\log N}^{3}}{\mathsf{fma}\left(\log N, \log \left(\mathsf{fma}\left(N, N, N\right)\right), {\left(\mathsf{log1p}\left(N\right)\right)}^{2}\right)}\right)} \]
5. Step-by-step derivation

6. Applied rewrites 25.2%

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

7. Taylor expanded in N around inf

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

   1. distribute-lft1-in N/A

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

   2. metadata-eval N/A

      \[\leadsto \color{blue}{0} \cdot \frac{{\log \left(\frac{1}{N}\right)}^{3}}{2 \cdot {\log \left(\frac{1}{N}\right)}^{2} + {\log \left(\frac{1}{N}\right)}^{2}} \]

   3. mul0-lft 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: 26.3% 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 2024214 
(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)))