2log (problem 3.3.6)

Percentage Accurate: 23.8% → 99.4%

Time: 9.9s

Alternatives: 13

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: 23.8% 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: 99.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 0.0006:\\ \;\;\;\;\frac{-1}{\frac{N}{-1 - \frac{\mathsf{fma}\left(N, \mathsf{fma}\left(N, -0.5, 0.3333333333333333\right), -0.25\right)}{N \cdot \left(N \cdot N\right)}}}\\ \mathbf{else}:\\ \;\;\;\;-\log \left(\frac{N}{N + 1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= (- (log (+ N 1.0)) (log N)) 0.0006)
   (/
    -1.0
    (/
     N
     (- -1.0 (/ (fma N (fma N -0.5 0.3333333333333333) -0.25) (* N (* N N))))))
   (- (log (/ N (+ N 1.0))))))

C

double code(double N) {
	double tmp;
	if ((log((N + 1.0)) - log(N)) <= 0.0006) {
		tmp = -1.0 / (N / (-1.0 - (fma(N, fma(N, -0.5, 0.3333333333333333), -0.25) / (N * (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)) <= 0.0006)
		tmp = Float64(-1.0 / Float64(N / Float64(-1.0 - Float64(fma(N, fma(N, -0.5, 0.3333333333333333), -0.25) / Float64(N * 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], 0.0006], N[(-1.0 / N[(N / N[(-1.0 - N[(N[(N * N[(N * -0.5 + 0.3333333333333333), $MachinePrecision] + -0.25), $MachinePrecision] / N[(N * N[(N * N), $MachinePrecision]), $MachinePrecision]), $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)) < 5.99999999999999947e-4

   1. Initial program 16.5%

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

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. clear-num N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 99.8

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 99.8

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

   7. Applied egg-rr 99.8%

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

   8. Taylor expanded in N around 0

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

      1. lower-/.f64 N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. cube-mult N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 99.8

          \[\leadsto \frac{1}{\frac{N}{1 + \frac{\mathsf{fma}\left(N, \mathsf{fma}\left(N, -0.5, 0.3333333333333333\right), -0.25\right)}{N \cdot \color{blue}{\left(N \cdot N\right)}}}} \]

   10. Simplified 99.8%

       \[\leadsto \frac{1}{\frac{N}{1 + \color{blue}{\frac{\mathsf{fma}\left(N, \mathsf{fma}\left(N, -0.5, 0.3333333333333333\right), -0.25\right)}{N \cdot \left(N \cdot N\right)}}}} \]

   if 5.99999999999999947e-4 < (-.f64 (log.f64 (+.f64 N #s(literal 1 binary64))) (log.f64 N))

   1. Initial program 91.1%

      \[\log \left(N + 1\right) - \log N \]
   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 94.1

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

   4. Applied egg-rr 94.1%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 0.0006:\\ \;\;\;\;\frac{-1}{\frac{N}{-1 - \frac{\mathsf{fma}\left(N, \mathsf{fma}\left(N, -0.5, 0.3333333333333333\right), -0.25\right)}{N \cdot \left(N \cdot N\right)}}}\\ \mathbf{else}:\\ \;\;\;\;-\log \left(\frac{N}{N + 1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;N \leq 1350:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{N}{-1 - \frac{\mathsf{fma}\left(N, \mathsf{fma}\left(N, -0.5, 0.3333333333333333\right), -0.25\right)}{N \cdot \left(N \cdot N\right)}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= N 1350.0)
   (log (/ (+ N 1.0) N))
   (/
    -1.0
    (/
     N
     (-
      -1.0
      (/ (fma N (fma N -0.5 0.3333333333333333) -0.25) (* N (* N N))))))))

C

double code(double N) {
	double tmp;
	if (N <= 1350.0) {
		tmp = log(((N + 1.0) / N));
	} else {
		tmp = -1.0 / (N / (-1.0 - (fma(N, fma(N, -0.5, 0.3333333333333333), -0.25) / (N * (N * N)))));
	}
	return tmp;
}

Julia

function code(N)
	tmp = 0.0
	if (N <= 1350.0)
		tmp = log(Float64(Float64(N + 1.0) / N));
	else
		tmp = Float64(-1.0 / Float64(N / Float64(-1.0 - Float64(fma(N, fma(N, -0.5, 0.3333333333333333), -0.25) / Float64(N * Float64(N * N))))));
	end
	return tmp
end

Wolfram

code[N_] := If[LessEqual[N, 1350.0], N[Log[N[(N[(N + 1.0), $MachinePrecision] / N), $MachinePrecision]], $MachinePrecision], N[(-1.0 / N[(N / N[(-1.0 - N[(N[(N * N[(N * -0.5 + 0.3333333333333333), $MachinePrecision] + -0.25), $MachinePrecision] / N[(N * N[(N * N), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if N < 1350

   1. Initial program 91.5%

      \[\log \left(N + 1\right) - \log N \]
   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 93.8

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

   4. Applied egg-rr 93.8%

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

   if 1350 < N

   1. Initial program 16.8%

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

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. clear-num N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 99.8

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 99.8

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

   7. Applied egg-rr 99.8%

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

   8. Taylor expanded in N around 0

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

      1. lower-/.f64 N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. cube-mult N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 99.8

          \[\leadsto \frac{1}{\frac{N}{1 + \frac{\mathsf{fma}\left(N, \mathsf{fma}\left(N, -0.5, 0.3333333333333333\right), -0.25\right)}{N \cdot \color{blue}{\left(N \cdot N\right)}}}} \]

   10. Simplified 99.8%

       \[\leadsto \frac{1}{\frac{N}{1 + \color{blue}{\frac{\mathsf{fma}\left(N, \mathsf{fma}\left(N, -0.5, 0.3333333333333333\right), -0.25\right)}{N \cdot \left(N \cdot N\right)}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;N \leq 1350:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{N}{-1 - \frac{\mathsf{fma}\left(N, \mathsf{fma}\left(N, -0.5, 0.3333333333333333\right), -0.25\right)}{N \cdot \left(N \cdot N\right)}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -0.5 + \frac{0.3333333333333333}{N}\\ \frac{1 - \frac{t\_0 \cdot t\_0}{N \cdot N}}{N \cdot \left(1 - \frac{-0.5 + \frac{0.3333333333333333 + \frac{-0.25}{N}}{N}}{N}\right)} \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (let* ((t_0 (+ -0.5 (/ 0.3333333333333333 N))))
   (/
    (- 1.0 (/ (* t_0 t_0) (* N N)))
    (* N (- 1.0 (/ (+ -0.5 (/ (+ 0.3333333333333333 (/ -0.25 N)) N)) N))))))

C

double code(double N) {
	double t_0 = -0.5 + (0.3333333333333333 / N);
	return (1.0 - ((t_0 * t_0) / (N * N))) / (N * (1.0 - ((-0.5 + ((0.3333333333333333 + (-0.25 / N)) / N)) / N)));
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    real(8) :: t_0
    t_0 = (-0.5d0) + (0.3333333333333333d0 / n)
    code = (1.0d0 - ((t_0 * t_0) / (n * n))) / (n * (1.0d0 - (((-0.5d0) + ((0.3333333333333333d0 + ((-0.25d0) / n)) / n)) / n)))
end function

Java

public static double code(double N) {
	double t_0 = -0.5 + (0.3333333333333333 / N);
	return (1.0 - ((t_0 * t_0) / (N * N))) / (N * (1.0 - ((-0.5 + ((0.3333333333333333 + (-0.25 / N)) / N)) / N)));
}

Python

def code(N):
	t_0 = -0.5 + (0.3333333333333333 / N)
	return (1.0 - ((t_0 * t_0) / (N * N))) / (N * (1.0 - ((-0.5 + ((0.3333333333333333 + (-0.25 / N)) / N)) / N)))

Julia

function code(N)
	t_0 = Float64(-0.5 + Float64(0.3333333333333333 / N))
	return Float64(Float64(1.0 - Float64(Float64(t_0 * t_0) / Float64(N * N))) / Float64(N * Float64(1.0 - Float64(Float64(-0.5 + Float64(Float64(0.3333333333333333 + Float64(-0.25 / N)) / N)) / N))))
end

MATLAB

function tmp = code(N)
	t_0 = -0.5 + (0.3333333333333333 / N);
	tmp = (1.0 - ((t_0 * t_0) / (N * N))) / (N * (1.0 - ((-0.5 + ((0.3333333333333333 + (-0.25 / N)) / N)) / N)));
end

Wolfram

code[N_] := Block[{t$95$0 = N[(-0.5 + N[(0.3333333333333333 / N), $MachinePrecision]), $MachinePrecision]}, N[(N[(1.0 - N[(N[(t$95$0 * t$95$0), $MachinePrecision] / N[(N * N), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N * N[(1.0 - N[(N[(-0.5 + N[(N[(0.3333333333333333 + N[(-0.25 / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 22.6%

   \[\log \left(N + 1\right) - \log N \]
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. Simplified 96.2%

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

   1. lift-/.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. flip-+ N/A

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

   7. associate-/l/ N/A

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

   8. lower-/.f64 N/A

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

7. Applied egg-rr 96.2%

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

8. Taylor expanded in N around inf

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

   1. lower-/.f64 96.4

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

10. Simplified 96.4%

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

11. Taylor expanded in N around inf

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

    1. lower-/.f64 96.5

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

13. Simplified 96.5%

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

14. Final simplification 96.5%

    \[\leadsto \frac{1 - \frac{\left(-0.5 + \frac{0.3333333333333333}{N}\right) \cdot \left(-0.5 + \frac{0.3333333333333333}{N}\right)}{N \cdot N}}{N \cdot \left(1 - \frac{-0.5 + \frac{0.3333333333333333 + \frac{-0.25}{N}}{N}}{N}\right)} \]
15. Add Preprocessing

Alternative 4: 96.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\frac{0.3333333333333333 + \frac{-0.2361111111111111}{N}}{N} - 0.25}{N \cdot N} + 1}{N \cdot \left(1 - \frac{-0.5 + \frac{0.3333333333333333 + \frac{-0.25}{N}}{N}}{N}\right)} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (/
  (+
   (/ (- (/ (+ 0.3333333333333333 (/ -0.2361111111111111 N)) N) 0.25) (* N N))
   1.0)
  (* N (- 1.0 (/ (+ -0.5 (/ (+ 0.3333333333333333 (/ -0.25 N)) N)) N)))))

C

double code(double N) {
	return (((((0.3333333333333333 + (-0.2361111111111111 / N)) / N) - 0.25) / (N * N)) + 1.0) / (N * (1.0 - ((-0.5 + ((0.3333333333333333 + (-0.25 / N)) / N)) / N)));
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    code = (((((0.3333333333333333d0 + ((-0.2361111111111111d0) / n)) / n) - 0.25d0) / (n * n)) + 1.0d0) / (n * (1.0d0 - (((-0.5d0) + ((0.3333333333333333d0 + ((-0.25d0) / n)) / n)) / n)))
end function

Java

public static double code(double N) {
	return (((((0.3333333333333333 + (-0.2361111111111111 / N)) / N) - 0.25) / (N * N)) + 1.0) / (N * (1.0 - ((-0.5 + ((0.3333333333333333 + (-0.25 / N)) / N)) / N)));
}

Python

def code(N):
	return (((((0.3333333333333333 + (-0.2361111111111111 / N)) / N) - 0.25) / (N * N)) + 1.0) / (N * (1.0 - ((-0.5 + ((0.3333333333333333 + (-0.25 / N)) / N)) / N)))

Julia

function code(N)
	return Float64(Float64(Float64(Float64(Float64(Float64(0.3333333333333333 + Float64(-0.2361111111111111 / N)) / N) - 0.25) / Float64(N * N)) + 1.0) / Float64(N * Float64(1.0 - Float64(Float64(-0.5 + Float64(Float64(0.3333333333333333 + Float64(-0.25 / N)) / N)) / N))))
end

MATLAB

function tmp = code(N)
	tmp = (((((0.3333333333333333 + (-0.2361111111111111 / N)) / N) - 0.25) / (N * N)) + 1.0) / (N * (1.0 - ((-0.5 + ((0.3333333333333333 + (-0.25 / N)) / N)) / N)));
end

Wolfram

code[N_] := N[(N[(N[(N[(N[(N[(0.3333333333333333 + N[(-0.2361111111111111 / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision] - 0.25), $MachinePrecision] / N[(N * N), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / N[(N * N[(1.0 - N[(N[(-0.5 + N[(N[(0.3333333333333333 + N[(-0.25 / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 22.6%

   \[\log \left(N + 1\right) - \log N \]
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. Simplified 96.2%

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

   1. lift-/.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. flip-+ N/A

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

   7. associate-/l/ N/A

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

   8. lower-/.f64 N/A

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

7. Applied egg-rr 96.2%

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

8. Taylor expanded in N around inf

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

   1. lower-/.f64 96.4

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

10. Simplified 96.4%

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

11. Taylor expanded in N around -inf

    \[\leadsto \frac{1 - \color{blue}{\frac{\frac{1}{4} + -1 \cdot \frac{\frac{1}{3} - \frac{17}{72} \cdot \frac{1}{N}}{N}}{{N}^{2}}}}{N \cdot \left(1 - \frac{\frac{\frac{1}{3} + \frac{\frac{-1}{4}}{N}}{N} + \frac{-1}{2}}{N}\right)} \]
12. Step-by-step derivation

    1. lower-/.f64 N/A

       \[\leadsto \frac{1 - \color{blue}{\frac{\frac{1}{4} + -1 \cdot \frac{\frac{1}{3} - \frac{17}{72} \cdot \frac{1}{N}}{N}}{{N}^{2}}}}{N \cdot \left(1 - \frac{\frac{\frac{1}{3} + \frac{\frac{-1}{4}}{N}}{N} + \frac{-1}{2}}{N}\right)} \]

    2. mul-1-neg N/A

       \[\leadsto \frac{1 - \frac{\frac{1}{4} + \color{blue}{\left(\mathsf{neg}\left(\frac{\frac{1}{3} - \frac{17}{72} \cdot \frac{1}{N}}{N}\right)\right)}}{{N}^{2}}}{N \cdot \left(1 - \frac{\frac{\frac{1}{3} + \frac{\frac{-1}{4}}{N}}{N} + \frac{-1}{2}}{N}\right)} \]

    3. unsub-neg N/A

       \[\leadsto \frac{1 - \frac{\color{blue}{\frac{1}{4} - \frac{\frac{1}{3} - \frac{17}{72} \cdot \frac{1}{N}}{N}}}{{N}^{2}}}{N \cdot \left(1 - \frac{\frac{\frac{1}{3} + \frac{\frac{-1}{4}}{N}}{N} + \frac{-1}{2}}{N}\right)} \]

    4. lower--.f64 N/A

       \[\leadsto \frac{1 - \frac{\color{blue}{\frac{1}{4} - \frac{\frac{1}{3} - \frac{17}{72} \cdot \frac{1}{N}}{N}}}{{N}^{2}}}{N \cdot \left(1 - \frac{\frac{\frac{1}{3} + \frac{\frac{-1}{4}}{N}}{N} + \frac{-1}{2}}{N}\right)} \]

    5. lower-/.f64 N/A

       \[\leadsto \frac{1 - \frac{\frac{1}{4} - \color{blue}{\frac{\frac{1}{3} - \frac{17}{72} \cdot \frac{1}{N}}{N}}}{{N}^{2}}}{N \cdot \left(1 - \frac{\frac{\frac{1}{3} + \frac{\frac{-1}{4}}{N}}{N} + \frac{-1}{2}}{N}\right)} \]

    6. sub-neg N/A

       \[\leadsto \frac{1 - \frac{\frac{1}{4} - \frac{\color{blue}{\frac{1}{3} + \left(\mathsf{neg}\left(\frac{17}{72} \cdot \frac{1}{N}\right)\right)}}{N}}{{N}^{2}}}{N \cdot \left(1 - \frac{\frac{\frac{1}{3} + \frac{\frac{-1}{4}}{N}}{N} + \frac{-1}{2}}{N}\right)} \]

    7. lower-+.f64 N/A

       \[\leadsto \frac{1 - \frac{\frac{1}{4} - \frac{\color{blue}{\frac{1}{3} + \left(\mathsf{neg}\left(\frac{17}{72} \cdot \frac{1}{N}\right)\right)}}{N}}{{N}^{2}}}{N \cdot \left(1 - \frac{\frac{\frac{1}{3} + \frac{\frac{-1}{4}}{N}}{N} + \frac{-1}{2}}{N}\right)} \]

    8. associate-*r/ N/A

       \[\leadsto \frac{1 - \frac{\frac{1}{4} - \frac{\frac{1}{3} + \left(\mathsf{neg}\left(\color{blue}{\frac{\frac{17}{72} \cdot 1}{N}}\right)\right)}{N}}{{N}^{2}}}{N \cdot \left(1 - \frac{\frac{\frac{1}{3} + \frac{\frac{-1}{4}}{N}}{N} + \frac{-1}{2}}{N}\right)} \]

    9. metadata-eval N/A

       \[\leadsto \frac{1 - \frac{\frac{1}{4} - \frac{\frac{1}{3} + \left(\mathsf{neg}\left(\frac{\color{blue}{\frac{17}{72}}}{N}\right)\right)}{N}}{{N}^{2}}}{N \cdot \left(1 - \frac{\frac{\frac{1}{3} + \frac{\frac{-1}{4}}{N}}{N} + \frac{-1}{2}}{N}\right)} \]

    10. distribute-neg-frac N/A

        \[\leadsto \frac{1 - \frac{\frac{1}{4} - \frac{\frac{1}{3} + \color{blue}{\frac{\mathsf{neg}\left(\frac{17}{72}\right)}{N}}}{N}}{{N}^{2}}}{N \cdot \left(1 - \frac{\frac{\frac{1}{3} + \frac{\frac{-1}{4}}{N}}{N} + \frac{-1}{2}}{N}\right)} \]

    11. lower-/.f64 N/A

        \[\leadsto \frac{1 - \frac{\frac{1}{4} - \frac{\frac{1}{3} + \color{blue}{\frac{\mathsf{neg}\left(\frac{17}{72}\right)}{N}}}{N}}{{N}^{2}}}{N \cdot \left(1 - \frac{\frac{\frac{1}{3} + \frac{\frac{-1}{4}}{N}}{N} + \frac{-1}{2}}{N}\right)} \]

    12. metadata-eval N/A

        \[\leadsto \frac{1 - \frac{\frac{1}{4} - \frac{\frac{1}{3} + \frac{\color{blue}{\frac{-17}{72}}}{N}}{N}}{{N}^{2}}}{N \cdot \left(1 - \frac{\frac{\frac{1}{3} + \frac{\frac{-1}{4}}{N}}{N} + \frac{-1}{2}}{N}\right)} \]

    13. unpow2 N/A

        \[\leadsto \frac{1 - \frac{\frac{1}{4} - \frac{\frac{1}{3} + \frac{\frac{-17}{72}}{N}}{N}}{\color{blue}{N \cdot N}}}{N \cdot \left(1 - \frac{\frac{\frac{1}{3} + \frac{\frac{-1}{4}}{N}}{N} + \frac{-1}{2}}{N}\right)} \]

    14. lower-*.f64 96.4

        \[\leadsto \frac{1 - \frac{0.25 - \frac{0.3333333333333333 + \frac{-0.2361111111111111}{N}}{N}}{\color{blue}{N \cdot N}}}{N \cdot \left(1 - \frac{\frac{0.3333333333333333 + \frac{-0.25}{N}}{N} + -0.5}{N}\right)} \]

13. Simplified 96.4%

    \[\leadsto \frac{1 - \color{blue}{\frac{0.25 - \frac{0.3333333333333333 + \frac{-0.2361111111111111}{N}}{N}}{N \cdot N}}}{N \cdot \left(1 - \frac{\frac{0.3333333333333333 + \frac{-0.25}{N}}{N} + -0.5}{N}\right)} \]

14. Final simplification 96.4%

    \[\leadsto \frac{\frac{\frac{0.3333333333333333 + \frac{-0.2361111111111111}{N}}{N} - 0.25}{N \cdot N} + 1}{N \cdot \left(1 - \frac{-0.5 + \frac{0.3333333333333333 + \frac{-0.25}{N}}{N}}{N}\right)} \]
15. Add Preprocessing

Alternative 5: 96.5% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \frac{-1}{\frac{N}{-1 - \frac{\mathsf{fma}\left(N, \mathsf{fma}\left(N, -0.5, 0.3333333333333333\right), -0.25\right)}{N \cdot \left(N \cdot N\right)}}} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (/
  -1.0
  (/
   N
   (- -1.0 (/ (fma N (fma N -0.5 0.3333333333333333) -0.25) (* N (* N N)))))))

C

double code(double N) {
	return -1.0 / (N / (-1.0 - (fma(N, fma(N, -0.5, 0.3333333333333333), -0.25) / (N * (N * N)))));
}

Julia

function code(N)
	return Float64(-1.0 / Float64(N / Float64(-1.0 - Float64(fma(N, fma(N, -0.5, 0.3333333333333333), -0.25) / Float64(N * Float64(N * N))))))
end

Wolfram

code[N_] := N[(-1.0 / N[(N / N[(-1.0 - N[(N[(N * N[(N * -0.5 + 0.3333333333333333), $MachinePrecision] + -0.25), $MachinePrecision] / N[(N * N[(N * N), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 22.6%

   \[\log \left(N + 1\right) - \log N \]
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. Simplified 96.2%

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

   1. lift-/.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-+.f64 N/A

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

   7. clear-num N/A

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

   8. lower-/.f64 N/A

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

   9. lower-/.f64 96.3

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

   10. lift-+.f64 N/A

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

   11. +-commutative N/A

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

   12. lower-+.f64 96.3

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

7. Applied egg-rr 96.3%

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

8. Taylor expanded in N around 0

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

   1. lower-/.f64 N/A

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

   2. sub-neg N/A

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

   3. metadata-eval N/A

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

   4. lower-fma.f64 N/A

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

   5. +-commutative N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. cube-mult N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 96.3

       \[\leadsto \frac{1}{\frac{N}{1 + \frac{\mathsf{fma}\left(N, \mathsf{fma}\left(N, -0.5, 0.3333333333333333\right), -0.25\right)}{N \cdot \color{blue}{\left(N \cdot N\right)}}}} \]

10. Simplified 96.3%

    \[\leadsto \frac{1}{\frac{N}{1 + \color{blue}{\frac{\mathsf{fma}\left(N, \mathsf{fma}\left(N, -0.5, 0.3333333333333333\right), -0.25\right)}{N \cdot \left(N \cdot N\right)}}}} \]

11. Final simplification 96.3%

    \[\leadsto \frac{-1}{\frac{N}{-1 - \frac{\mathsf{fma}\left(N, \mathsf{fma}\left(N, -0.5, 0.3333333333333333\right), -0.25\right)}{N \cdot \left(N \cdot N\right)}}} \]
12. Add Preprocessing

Alternative 6: 96.5% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\mathsf{fma}\left(N, \mathsf{fma}\left(N, -0.5, 0.3333333333333333\right), -0.25\right)}{N \cdot \left(N \cdot N\right)} + 1}{N} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (/ (+ (/ (fma N (fma N -0.5 0.3333333333333333) -0.25) (* N (* N N))) 1.0) N))

C

double code(double N) {
	return ((fma(N, fma(N, -0.5, 0.3333333333333333), -0.25) / (N * (N * N))) + 1.0) / N;
}

Julia

function code(N)
	return Float64(Float64(Float64(fma(N, fma(N, -0.5, 0.3333333333333333), -0.25) / Float64(N * Float64(N * N))) + 1.0) / N)
end

Wolfram

code[N_] := N[(N[(N[(N[(N * N[(N * -0.5 + 0.3333333333333333), $MachinePrecision] + -0.25), $MachinePrecision] / N[(N * N[(N * N), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / N), $MachinePrecision]

Derivation

1. Initial program 22.6%

   \[\log \left(N + 1\right) - \log N \]
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. Simplified 96.2%

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

6. Taylor expanded in N around 0

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

   1. lower-/.f64 N/A

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

   2. sub-neg N/A

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

   3. metadata-eval N/A

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

   4. lower-fma.f64 N/A

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

   5. +-commutative N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. cube-mult N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 96.2

       \[\leadsto \frac{1 + \frac{\mathsf{fma}\left(N, \mathsf{fma}\left(N, -0.5, 0.3333333333333333\right), -0.25\right)}{N \cdot \color{blue}{\left(N \cdot N\right)}}}{N} \]

8. Simplified 96.2%

   \[\leadsto \frac{1 + \color{blue}{\frac{\mathsf{fma}\left(N, \mathsf{fma}\left(N, -0.5, 0.3333333333333333\right), -0.25\right)}{N \cdot \left(N \cdot N\right)}}}{N} \]

9. Final simplification 96.2%

   \[\leadsto \frac{\frac{\mathsf{fma}\left(N, \mathsf{fma}\left(N, -0.5, 0.3333333333333333\right), -0.25\right)}{N \cdot \left(N \cdot N\right)} + 1}{N} \]
10. Add Preprocessing

Alternative 7: 95.1% accurate, 5.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 22.6%

   \[\log \left(N + 1\right) - \log N \]
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. Simplified 95.0%

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

6. Final simplification 95.0%

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

Alternative 8: 92.5% accurate, 5.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 22.6%

   \[\log \left(N + 1\right) - \log N \]
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. Simplified 96.2%

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

6. Taylor expanded in N around -inf

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. sub-neg N/A

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

   4. metadata-eval N/A

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

   5. distribute-lft-in N/A

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

   6. neg-mul-1 N/A

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

   7. metadata-eval N/A

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

   8. lower-+.f64 N/A

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

   9. associate-*r/ N/A

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

   10. metadata-eval N/A

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

   11. distribute-neg-frac N/A

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

   12. metadata-eval N/A

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

   13. lower-/.f64 92.4

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

8. Simplified 92.4%

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

   1. lift-/.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. clear-num N/A

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

   4. lower-/.f64 N/A

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

   5. lower-/.f64 92.4

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

   6. lift-+.f64 N/A

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

   7. +-commutative N/A

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

   8. lower-+.f64 92.4

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

10. Applied egg-rr 92.4%

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

11. Final simplification 92.4%

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

Alternative 9: 92.5% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \frac{1 - \frac{0.25}{N \cdot N}}{N + 0.5} \end{array} \]

FPCore

(FPCore (N) :precision binary64 (/ (- 1.0 (/ 0.25 (* N N))) (+ N 0.5)))

C

double code(double N) {
	return (1.0 - (0.25 / (N * N))) / (N + 0.5);
}

Fortran

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

Java

public static double code(double N) {
	return (1.0 - (0.25 / (N * N))) / (N + 0.5);
}

Python

def code(N):
	return (1.0 - (0.25 / (N * N))) / (N + 0.5)

Julia

function code(N)
	return Float64(Float64(1.0 - Float64(0.25 / Float64(N * N))) / Float64(N + 0.5))
end

MATLAB

function tmp = code(N)
	tmp = (1.0 - (0.25 / (N * N))) / (N + 0.5);
end

Wolfram

code[N_] := N[(N[(1.0 - N[(0.25 / N[(N * N), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N + 0.5), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 22.6%

   \[\log \left(N + 1\right) - \log N \]
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. Simplified 96.2%

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

   1. lift-/.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. flip-+ N/A

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

   7. associate-/l/ N/A

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

   8. lower-/.f64 N/A

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

7. Applied egg-rr 96.2%

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

8. Taylor expanded in N around inf

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

   1. lower-/.f64 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f64 94.9

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

10. Simplified 94.9%

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

11. Taylor expanded in N around inf

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

    1. distribute-rgt-in N/A

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

    2. *-lft-identity N/A

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

    3. associate-*l* N/A

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

    4. lft-mult-inverse N/A

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

    5. metadata-eval N/A

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

    6. lower-+.f64 92.4

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

13. Simplified 92.4%

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

Alternative 10: 92.4% accurate, 8.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 22.6%

   \[\log \left(N + 1\right) - \log N \]
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. sub-neg N/A

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

   3. lower-+.f64 N/A

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

   4. associate-*r/ N/A

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

   5. metadata-eval N/A

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

   6. distribute-neg-frac N/A

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

   7. metadata-eval N/A

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

   8. lower-/.f64 92.4

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

5. Simplified 92.4%

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

6. Final simplification 92.4%

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

Alternative 11: 92.2% accurate, 10.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 22.6%

   \[\log \left(N + 1\right) - \log N \]
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. Simplified 96.2%

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

   1. lift-/.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-+.f64 N/A

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

   7. clear-num N/A

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

   8. lower-/.f64 N/A

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

   9. lower-/.f64 96.3

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

   10. lift-+.f64 N/A

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

   11. +-commutative N/A

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

   12. lower-+.f64 96.3

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

7. Applied egg-rr 96.3%

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

8. Taylor expanded in N around inf

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

   1. div-sub N/A

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

   2. *-inverses N/A

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

   3. associate-/r* N/A

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

   4. unpow2 N/A

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

   5. associate-*r/ N/A

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

   6. metadata-eval N/A

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

   7. associate-/r* N/A

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

   8. unpow2 N/A

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

   9. div-sub N/A

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

   10. lower-/.f64 N/A

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

   11. sub-neg N/A

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

   12. metadata-eval N/A

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

   13. lower-+.f64 N/A

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

   14. unpow2 N/A

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

   15. lower-*.f64 92.1

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

10. Simplified 92.1%

    \[\leadsto \color{blue}{\frac{N + -0.5}{N \cdot N}} \]
11. Add Preprocessing

Alternative 12: 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 22.6%

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

3. Taylor expanded in N around inf

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

   1. lower-/.f64 85.3

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

5. Simplified 85.3%

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

Alternative 13: 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 22.6%

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

   1. lift-+.f64 N/A

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

   2. 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. lift--.f64 22.6

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

   5. lift-log.f64 N/A

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

   6. lift-+.f64 N/A

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

   7. +-commutative N/A

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

   8. lower-log1p.f64 22.6

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

4. Applied egg-rr 22.6%

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

   1. lift-log1p.f64 N/A

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

   2. lift-log.f64 N/A

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

   3. flip-- N/A

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

   4. unpow2 N/A

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

   5. lift-pow.f64 N/A

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

   6. unpow2 N/A

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

   7. lift-pow.f64 N/A

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

   8. lift-log1p.f64 N/A

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

   9. lift-log.f64 N/A

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

   10. sum-log N/A

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

   11. +-commutative N/A

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

   12. distribute-lft1-in N/A

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

   13. lift-fma.f64 N/A

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

   14. lift-log.f64 N/A

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

   15. sub-div N/A

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

   16. lift-pow.f64 N/A

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

   17. unpow2 N/A

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

   18. associate-*r/ N/A

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

   19. lift-/.f64 N/A

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

   20. lift-/.f64 N/A

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

6. Applied egg-rr 22.6%

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

7. Taylor expanded in N around inf

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

   1. distribute-rgt-out N/A

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

   2. metadata-eval N/A

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

   3. mul0-rgt 3.3

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

9. Simplified 3.3%

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

Developer Target 1: 99.9% 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: 26.5% 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: 96.4% 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 2024215 
(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)))