2log (problem 3.3.6)

Percentage Accurate: 23.5% → 99.4%

Time: 10.4s

Alternatives: 9

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: 23.5% 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

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 0.0008:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(0.041666666666666664, \frac{1}{N \cdot N}, \mathsf{fma}\left(0.5 + \frac{-0.08333333333333333}{N}, 1, N\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;-\log \left(\frac{N \cdot N - N}{\mathsf{fma}\left(N, N, -1\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= (- (log (+ N 1.0)) (log N)) 0.0008)
   (/
    1.0
    (fma
     0.041666666666666664
     (/ 1.0 (* N N))
     (fma (+ 0.5 (/ -0.08333333333333333 N)) 1.0 N)))
   (- (log (/ (- (* N N) N) (fma N N -1.0))))))

C

double code(double N) {
	double tmp;
	if ((log((N + 1.0)) - log(N)) <= 0.0008) {
		tmp = 1.0 / fma(0.041666666666666664, (1.0 / (N * N)), fma((0.5 + (-0.08333333333333333 / N)), 1.0, N));
	} else {
		tmp = -log((((N * N) - N) / fma(N, N, -1.0)));
	}
	return tmp;
}

Julia

function code(N)
	tmp = 0.0
	if (Float64(log(Float64(N + 1.0)) - log(N)) <= 0.0008)
		tmp = Float64(1.0 / fma(0.041666666666666664, Float64(1.0 / Float64(N * N)), fma(Float64(0.5 + Float64(-0.08333333333333333 / N)), 1.0, N)));
	else
		tmp = Float64(-log(Float64(Float64(Float64(N * N) - N) / fma(N, 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], 0.0008], N[(1.0 / N[(0.041666666666666664 * N[(1.0 / N[(N * N), $MachinePrecision]), $MachinePrecision] + N[(N[(0.5 + N[(-0.08333333333333333 / N), $MachinePrecision]), $MachinePrecision] * 1.0 + N), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-N[Log[N[(N[(N[(N * N), $MachinePrecision] - N), $MachinePrecision] / 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)) < 8.00000000000000038e-4

   1. Initial program 21.0%

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

   3. Taylor expanded in N around inf

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

   5. Applied rewrites 99.6%

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

   7. Applied rewrites 99.6%

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

   8. Taylor expanded in N around inf

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

   10. Applied rewrites 99.8%

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

   12. Applied rewrites 99.8%

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

   if 8.00000000000000038e-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

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

      2. lift-log.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. diff-log N/A

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

      5. lift-+.f64 N/A

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

      6. flip-+ N/A

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

      7. associate-/l/ N/A

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

      8. clear-num N/A

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

      9. log-rec N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-log.f64 N/A

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

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

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

      13. lower-/.f64 N/A

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

      14. *-lft-identity N/A

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

      15. lower--.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. metadata-eval N/A

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

      18. sub-neg N/A

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

      19. lower-fma.f64 N/A

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

      20. metadata-eval 93.9

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

   4. Applied rewrites 93.9%

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

Alternative 2: 99.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 0.0008:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(0.041666666666666664, \frac{1}{N \cdot N}, \mathsf{fma}\left(0.5 + \frac{-0.08333333333333333}{N}, 1, N\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;-\log \left(\frac{N}{N + 1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= (- (log (+ N 1.0)) (log N)) 0.0008)
   (/
    1.0
    (fma
     0.041666666666666664
     (/ 1.0 (* N N))
     (fma (+ 0.5 (/ -0.08333333333333333 N)) 1.0 N)))
   (- (log (/ N (+ N 1.0))))))

C

double code(double N) {
	double tmp;
	if ((log((N + 1.0)) - log(N)) <= 0.0008) {
		tmp = 1.0 / fma(0.041666666666666664, (1.0 / (N * N)), fma((0.5 + (-0.08333333333333333 / N)), 1.0, 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)) <= 0.0008)
		tmp = Float64(1.0 / fma(0.041666666666666664, Float64(1.0 / Float64(N * N)), fma(Float64(0.5 + Float64(-0.08333333333333333 / N)), 1.0, 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], 0.0008], N[(1.0 / N[(0.041666666666666664 * N[(1.0 / N[(N * N), $MachinePrecision]), $MachinePrecision] + N[(N[(0.5 + N[(-0.08333333333333333 / N), $MachinePrecision]), $MachinePrecision] * 1.0 + N), $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)) < 8.00000000000000038e-4

   1. Initial program 21.0%

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

   3. Taylor expanded in N around inf

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

   5. Applied rewrites 99.6%

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

   7. Applied rewrites 99.6%

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

   8. Taylor expanded in N around inf

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

   10. Applied rewrites 99.8%

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

   12. Applied rewrites 99.8%

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

   if 8.00000000000000038e-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

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

      2. lift-log.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. diff-log N/A

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

      5. clear-num N/A

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

      6. neg-log N/A

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

      7. diff-log N/A

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

      8. lift-log.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lift-log.f64 N/A

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

      12. lift-log.f64 N/A

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

      13. diff-log N/A

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

      14. lower-log.f64 N/A

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

      15. lower-/.f64 93.8

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

   4. Applied rewrites 93.8%

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

Alternative 3: 99.4% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

function code(N)
	tmp = 0.0
	if (Float64(log(Float64(N + 1.0)) - log(N)) <= 0.001)
		tmp = Float64(1.0 / fma(0.041666666666666664, Float64(1.0 / Float64(N * N)), fma(Float64(0.5 + Float64(-0.08333333333333333 / N)), 1.0, 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[(0.041666666666666664 * N[(1.0 / N[(N * N), $MachinePrecision]), $MachinePrecision] + N[(N[(0.5 + N[(-0.08333333333333333 / N), $MachinePrecision]), $MachinePrecision] * 1.0 + N), $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 21.3%

      \[\log \left(N + 1\right) - \log N \]
   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 99.5%

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

   7. Applied rewrites 99.6%

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

   8. Taylor expanded in N around inf

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

   10. Applied rewrites 99.7%

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

   12. Applied rewrites 99.7%

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

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

   1. Initial program 90.9%

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

      1. lift--.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. diff-log N/A

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

      5. lower-log.f64 N/A

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

      6. lower-/.f64 93.4

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

   4. Applied rewrites 93.4%

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

Alternative 4: 96.9% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 24.6%

   \[\log \left(N + 1\right) - \log N \]
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. Applied rewrites 97.4%

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

7. Applied rewrites 97.5%

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

8. Taylor expanded in N around inf

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

10. Applied rewrites 97.8%

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

12. Applied rewrites 97.8%

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

Alternative 5: 96.9% 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 24.6%

   \[\log \left(N + 1\right) - \log N \]
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. Applied rewrites 97.4%

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

7. Applied rewrites 97.5%

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

8. Taylor expanded in N around inf

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

10. Applied rewrites 97.8%

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

11. Taylor expanded in N around 0

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

13. Applied rewrites 97.8%

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

Alternative 6: 96.1% accurate, 4.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 24.6%

   \[\log \left(N + 1\right) - \log N \]
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. Applied rewrites 97.4%

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

7. Applied rewrites 97.4%

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

8. Taylor expanded in N around 0

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

10. Applied rewrites 97.1%

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

11. Final simplification 97.1%

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

Alternative 7: 95.7% 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 24.6%

   \[\log \left(N + 1\right) - \log N \]
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. Applied rewrites 97.4%

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

7. Applied rewrites 97.5%

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

8. Taylor expanded in N around inf

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

10. Applied rewrites 97.8%

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

11. Taylor expanded in N around inf

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

13. Applied rewrites 96.5%

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

Alternative 8: 93.3% 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 24.6%

   \[\log \left(N + 1\right) - \log N \]
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. Applied rewrites 97.4%

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

7. Applied rewrites 97.5%

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

8. Taylor expanded in N around inf

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

10. Applied rewrites 97.8%

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

11. Taylor expanded in N around inf

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

13. Applied rewrites 93.3%

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

Alternative 9: 84.7% 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 24.6%

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

3. Taylor expanded in N around inf

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

   1. lower-/.f64 84.0

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

5. Applied rewrites 84.0%

   \[\leadsto \color{blue}{\frac{1}{N}} \]
6. 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: 26.2% 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 2024221 
(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)))