2log (problem 3.3.6)

Percentage Accurate: 22.9% → 99.5%

Time: 7.6s

Alternatives: 9

Speedup: 5.2×

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: 22.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 18.4%

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

   3. Taylor expanded in N around inf

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

   5. Applied rewrites 99.8%

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

   7. Applied rewrites 99.8%

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

   8. Taylor expanded in N around -inf

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

   10. Applied rewrites 99.8%

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

   12. Applied rewrites 99.9%

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

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

   1. Initial program 93.7%

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

      1. lift--.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. diff-log N/A

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

      5. clear-num N/A

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

      6. clear-num N/A

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

      7. log-rec N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 96.3

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 96.3

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

   4. Applied rewrites 96.3%

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

      1. lift-/.f64 N/A

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

      2. /-rgt-identity 96.3

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

   6. Applied rewrites 96.3%

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

4. Final simplification 99.6%

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

Alternative 2: 99.4% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

function code(N)
	tmp = 0.0
	if (Float64(log(Float64(N + 1.0)) - log(N)) <= 0.001)
		tmp = fma(Float64(Float64(-0.5 + Float64(Float64(0.08333333333333333 - Float64(0.041666666666666664 / N)) / N)) / N), Float64(-N), N) ^ -1.0;
	else
		tmp = log(Float64(Float64(1.0 + N) / 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[Power[N[(N[(N[(-0.5 + N[(N[(0.08333333333333333 - N[(0.041666666666666664 / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision] * (-N) + N), $MachinePrecision], -1.0], $MachinePrecision], N[Log[N[(N[(1.0 + N), $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 18.4%

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

   3. Taylor expanded in N around inf

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

   5. Applied rewrites 99.8%

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

   7. Applied rewrites 99.8%

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

   8. Taylor expanded in N around -inf

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

   10. Applied rewrites 99.8%

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

   12. Applied rewrites 99.9%

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

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

   1. Initial program 93.7%

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

      1. lift--.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. diff-log N/A

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

      5. lower-log.f64 N/A

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

      6. lower-/.f64 95.5

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 95.5

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

   4. Applied rewrites 95.5%

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

4. Final simplification 99.5%

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

Alternative 3: 96.9% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 24.9%

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

3. Taylor expanded in N around inf

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

5. Applied rewrites 95.3%

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

7. Applied rewrites 95.3%

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

8. Taylor expanded in N around -inf

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

10. Applied rewrites 95.8%

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

11. Taylor expanded in N around 0

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

13. Applied rewrites 95.8%

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

14. Final simplification 95.8%

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

Alternative 4: 97.0% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 24.9%

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

3. Taylor expanded in N around inf

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

5. Applied rewrites 95.3%

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

7. Applied rewrites 95.3%

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

8. Taylor expanded in N around -inf

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

10. Applied rewrites 95.8%

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

12. Applied rewrites 95.9%

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

13. Final simplification 95.9%

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

Alternative 5: 96.8% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 24.9%

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

3. Taylor expanded in N around inf

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

5. Applied rewrites 95.3%

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

7. Applied rewrites 95.3%

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

8. Taylor expanded in N around -inf

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

10. Applied rewrites 95.8%

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

11. Taylor expanded in N around 0

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

13. Applied rewrites 95.7%

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

14. Final simplification 95.7%

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

Alternative 6: 93.4% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 24.9%

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

3. Taylor expanded in N around inf

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

5. Applied rewrites 95.3%

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

7. Applied rewrites 95.3%

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

8. Taylor expanded in N around inf

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

10. Applied rewrites 92.2%

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

11. Final simplification 92.2%

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

Alternative 7: 93.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

public static double code(double N) {
	return Math.pow((N + 0.5), -1.0);
}

Python

def code(N):
	return math.pow((N + 0.5), -1.0)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 24.9%

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

3. Taylor expanded in N around inf

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

5. Applied rewrites 95.3%

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

7. Applied rewrites 95.3%

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

8. Taylor expanded in N around inf

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

10. Applied rewrites 92.2%

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

12. Applied rewrites 92.1%

    \[\leadsto \frac{1}{N + 0.5} \]

13. Final simplification 92.1%

    \[\leadsto {\left(N + 0.5\right)}^{-1} \]
14. Add Preprocessing

Alternative 8: 85.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(N)
	return N ^ -1.0
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 24.9%

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

3. Taylor expanded in N around inf

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

   1. lower-/.f64 83.7

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

5. Applied rewrites 83.7%

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

6. Final simplification 83.7%

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

Alternative 9: 95.3% accurate, 5.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 24.9%

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

3. Taylor expanded in N around inf

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

   1. lower-/.f64 N/A

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

5. Applied rewrites 94.1%

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

Developer Target 1: 96.5% 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 2024307 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  :pre (and (> N 1.0) (< N 1e+40))

  :alt
  (! :herbie-platform default (+ (/ 1 N) (/ -1 (* 2 (pow N 2))) (/ 1 (* 3 (pow N 3))) (/ -1 (* 4 (pow N 4)))))

  (- (log (+ N 1.0)) (log N)))