2log (problem 3.3.6)

Percentage Accurate: 24.5% → 99.4%

Time: 5.5s

Alternatives: 11

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: 24.5% 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, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((log((n - (-1.0d0))) - log(n)) <= 0.001d0) then
        tmp = 1.0d0 / (n - (((((0.08333333333333333d0 - (0.041666666666666664d0 / n)) / n) + (-0.5d0)) / n) * n))
    else
        tmp = log(((n - 1.0d0) / n)) + log((((-1.0d0) - n) / (1.0d0 - n)))
    end if
    code = tmp
end function

Java

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

Python

def code(N):
	tmp = 0
	if (math.log((N - -1.0)) - math.log(N)) <= 0.001:
		tmp = 1.0 / (N - (((((0.08333333333333333 - (0.041666666666666664 / N)) / N) + -0.5) / N) * N))
	else:
		tmp = math.log(((N - 1.0) / N)) + math.log(((-1.0 - N) / (1.0 - N)))
	return tmp

Julia

function code(N)
	tmp = 0.0
	if (Float64(log(Float64(N - -1.0)) - log(N)) <= 0.001)
		tmp = Float64(1.0 / Float64(N - Float64(Float64(Float64(Float64(Float64(0.08333333333333333 - Float64(0.041666666666666664 / N)) / N) + -0.5) / N) * N)));
	else
		tmp = Float64(log(Float64(Float64(N - 1.0) / N)) + log(Float64(Float64(-1.0 - N) / Float64(1.0 - N))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(N)
	tmp = 0.0;
	if ((log((N - -1.0)) - log(N)) <= 0.001)
		tmp = 1.0 / (N - (((((0.08333333333333333 - (0.041666666666666664 / N)) / N) + -0.5) / N) * N));
	else
		tmp = log(((N - 1.0) / N)) + log(((-1.0 - N) / (1.0 - N)));
	end
	tmp_2 = tmp;
end

Wolfram

code[N_] := If[LessEqual[N[(N[Log[N[(N - -1.0), $MachinePrecision]], $MachinePrecision] - N[Log[N], $MachinePrecision]), $MachinePrecision], 0.001], N[(1.0 / N[(N - N[(N[(N[(N[(N[(0.08333333333333333 - N[(0.041666666666666664 / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision] + -0.5), $MachinePrecision] / N), $MachinePrecision] * N), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Log[N[(N[(N - 1.0), $MachinePrecision] / N), $MachinePrecision]], $MachinePrecision] + N[Log[N[(N[(-1.0 - N), $MachinePrecision] / N[(1.0 - N), $MachinePrecision]), $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 18.7%

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

   3. Taylor expanded in N around inf

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

   5. Applied rewrites 99.6%

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

   7. Applied rewrites 99.7%

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

   8. Taylor expanded in N around -inf

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

   10. Applied rewrites 99.6%

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

   12. Applied rewrites 99.8%

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

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

   1. Initial program 92.6%

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

      1. lift--.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. diff-log N/A

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

      5. lift-+.f64 N/A

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

      6. flip-+ N/A

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

      7. associate-/l/ N/A

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

      8. clear-num N/A

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

      9. log-rec N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-log.f64 N/A

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

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

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

      13. lower-/.f64 N/A

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

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

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower--.f64 N/A

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

      18. metadata-eval N/A

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

      19. sub-neg N/A

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

      20. lower-fma.f64 N/A

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

      21. metadata-eval 94.3

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

   4. Applied rewrites 94.3%

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

      1. lift-neg.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. neg-log N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. lift-fma.f64 N/A

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

      7. difference-of-sqr--1 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. log-prod N/A

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

      12. lower-+.f64 N/A

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

      13. lower-log.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. lower-log.f64 N/A

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

      18. lower-/.f64 95.3

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

   6. Applied rewrites 95.3%

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

4. Final simplification 99.5%

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

Alternative 2: 99.5% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((log((n - (-1.0d0))) - log(n)) <= 0.001d0) then
        tmp = 1.0d0 / (n - (((((0.08333333333333333d0 - (0.041666666666666664d0 / n)) / n) + (-0.5d0)) / n) * n))
    else
        tmp = -log((n / (n - (-1.0d0))))
    end if
    code = tmp
end function

Java

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

Python

def code(N):
	tmp = 0
	if (math.log((N - -1.0)) - math.log(N)) <= 0.001:
		tmp = 1.0 / (N - (((((0.08333333333333333 - (0.041666666666666664 / N)) / N) + -0.5) / N) * N))
	else:
		tmp = -math.log((N / (N - -1.0)))
	return tmp

Julia

function code(N)
	tmp = 0.0
	if (Float64(log(Float64(N - -1.0)) - log(N)) <= 0.001)
		tmp = Float64(1.0 / Float64(N - Float64(Float64(Float64(Float64(Float64(0.08333333333333333 - Float64(0.041666666666666664 / N)) / N) + -0.5) / N) * N)));
	else
		tmp = Float64(-log(Float64(N / Float64(N - -1.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(N)
	tmp = 0.0;
	if ((log((N - -1.0)) - log(N)) <= 0.001)
		tmp = 1.0 / (N - (((((0.08333333333333333 - (0.041666666666666664 / N)) / N) + -0.5) / N) * N));
	else
		tmp = -log((N / (N - -1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[N_] := If[LessEqual[N[(N[Log[N[(N - -1.0), $MachinePrecision]], $MachinePrecision] - N[Log[N], $MachinePrecision]), $MachinePrecision], 0.001], N[(1.0 / N[(N - N[(N[(N[(N[(N[(0.08333333333333333 - N[(0.041666666666666664 / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision] + -0.5), $MachinePrecision] / N), $MachinePrecision] * 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)) < 1e-3

   1. Initial program 18.7%

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

   3. Taylor expanded in N around inf

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

   5. Applied rewrites 99.6%

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

   7. Applied rewrites 99.7%

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

   8. Taylor expanded in N around -inf

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

   10. Applied rewrites 99.6%

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

   12. Applied rewrites 99.8%

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

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

   1. Initial program 92.6%

      \[\log \left(N + 1\right) - \log N \]
   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 95.3

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

      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 95.3

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

   4. Applied rewrites 95.3%

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

4. Final simplification 99.5%

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

Alternative 3: 99.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;N \leq 800:\\ \;\;\;\;-\log \left(\frac{\left(N - 1\right) \cdot N}{\mathsf{fma}\left(N, N, -1\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N - \frac{\frac{0.08333333333333333 - \frac{0.041666666666666664}{N}}{N} + -0.5}{N} \cdot N}\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= N 800.0)
   (- (log (/ (* (- N 1.0) N) (fma N N -1.0))))
   (/
    1.0
    (-
     N
     (*
      (/ (+ (/ (- 0.08333333333333333 (/ 0.041666666666666664 N)) N) -0.5) N)
      N)))))

C

double code(double N) {
	double tmp;
	if (N <= 800.0) {
		tmp = -log((((N - 1.0) * N) / fma(N, N, -1.0)));
	} else {
		tmp = 1.0 / (N - (((((0.08333333333333333 - (0.041666666666666664 / N)) / N) + -0.5) / N) * N));
	}
	return tmp;
}

Julia

function code(N)
	tmp = 0.0
	if (N <= 800.0)
		tmp = Float64(-log(Float64(Float64(Float64(N - 1.0) * N) / fma(N, N, -1.0))));
	else
		tmp = Float64(1.0 / Float64(N - Float64(Float64(Float64(Float64(Float64(0.08333333333333333 - Float64(0.041666666666666664 / N)) / N) + -0.5) / N) * N)));
	end
	return tmp
end

Wolfram

code[N_] := If[LessEqual[N, 800.0], (-N[Log[N[(N[(N[(N - 1.0), $MachinePrecision] * N), $MachinePrecision] / N[(N * N + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), N[(1.0 / N[(N - N[(N[(N[(N[(N[(0.08333333333333333 - N[(0.041666666666666664 / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision] + -0.5), $MachinePrecision] / N), $MachinePrecision] * N), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if N < 800

   1. Initial program 92.6%

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

      1. lift--.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. diff-log N/A

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

      5. lift-+.f64 N/A

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

      6. flip-+ N/A

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

      7. associate-/l/ N/A

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

      8. clear-num N/A

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

      9. log-rec N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-log.f64 N/A

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

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

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

      13. lower-/.f64 N/A

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

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

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower--.f64 N/A

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

      18. metadata-eval N/A

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

      19. sub-neg N/A

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

      20. lower-fma.f64 N/A

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

      21. metadata-eval 94.3

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

   4. Applied rewrites 94.3%

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

   if 800 < N

   1. Initial program 18.7%

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

   3. Taylor expanded in N around inf

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

   5. Applied rewrites 99.6%

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

   7. Applied rewrites 99.7%

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

   8. Taylor expanded in N around -inf

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

   10. Applied rewrites 99.6%

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

   12. Applied rewrites 99.8%

       \[\leadsto \frac{1}{\left(-\left(-N\right)\right) + \frac{-0.5 + \frac{0.08333333333333333 - \frac{0.041666666666666664}{N}}{N}}{N} \cdot \color{blue}{\left(-N\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;N \leq 800:\\ \;\;\;\;-\log \left(\frac{\left(N - 1\right) \cdot N}{\mathsf{fma}\left(N, N, -1\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N - \frac{\frac{0.08333333333333333 - \frac{0.041666666666666664}{N}}{N} + -0.5}{N} \cdot N}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;N \leq 850:\\ \;\;\;\;\log \left(\frac{N - -1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N - \frac{\frac{0.08333333333333333 - \frac{0.041666666666666664}{N}}{N} + -0.5}{N} \cdot N}\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= N 850.0)
   (log (/ (- N -1.0) N))
   (/
    1.0
    (-
     N
     (*
      (/ (+ (/ (- 0.08333333333333333 (/ 0.041666666666666664 N)) N) -0.5) N)
      N)))))

C

double code(double N) {
	double tmp;
	if (N <= 850.0) {
		tmp = log(((N - -1.0) / N));
	} else {
		tmp = 1.0 / (N - (((((0.08333333333333333 - (0.041666666666666664 / N)) / N) + -0.5) / N) * N));
	}
	return tmp;
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= 850.0d0) then
        tmp = log(((n - (-1.0d0)) / n))
    else
        tmp = 1.0d0 / (n - (((((0.08333333333333333d0 - (0.041666666666666664d0 / n)) / n) + (-0.5d0)) / n) * n))
    end if
    code = tmp
end function

Java

public static double code(double N) {
	double tmp;
	if (N <= 850.0) {
		tmp = Math.log(((N - -1.0) / N));
	} else {
		tmp = 1.0 / (N - (((((0.08333333333333333 - (0.041666666666666664 / N)) / N) + -0.5) / N) * N));
	}
	return tmp;
}

Python

def code(N):
	tmp = 0
	if N <= 850.0:
		tmp = math.log(((N - -1.0) / N))
	else:
		tmp = 1.0 / (N - (((((0.08333333333333333 - (0.041666666666666664 / N)) / N) + -0.5) / N) * N))
	return tmp

Julia

function code(N)
	tmp = 0.0
	if (N <= 850.0)
		tmp = log(Float64(Float64(N - -1.0) / N));
	else
		tmp = Float64(1.0 / Float64(N - Float64(Float64(Float64(Float64(Float64(0.08333333333333333 - Float64(0.041666666666666664 / N)) / N) + -0.5) / N) * N)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(N)
	tmp = 0.0;
	if (N <= 850.0)
		tmp = log(((N - -1.0) / N));
	else
		tmp = 1.0 / (N - (((((0.08333333333333333 - (0.041666666666666664 / N)) / N) + -0.5) / N) * N));
	end
	tmp_2 = tmp;
end

Wolfram

code[N_] := If[LessEqual[N, 850.0], N[Log[N[(N[(N - -1.0), $MachinePrecision] / N), $MachinePrecision]], $MachinePrecision], N[(1.0 / N[(N - N[(N[(N[(N[(N[(0.08333333333333333 - N[(0.041666666666666664 / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision] + -0.5), $MachinePrecision] / N), $MachinePrecision] * N), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if N < 850

   1. Initial program 92.6%

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

      1. lift--.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. diff-log N/A

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

      5. lower-log.f64 N/A

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

      6. lower-/.f64 94.3

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 94.3

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

   4. Applied rewrites 94.3%

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

   if 850 < N

   1. Initial program 18.7%

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

   3. Taylor expanded in N around inf

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

   5. Applied rewrites 99.6%

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

   7. Applied rewrites 99.7%

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

   8. Taylor expanded in N around -inf

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

   10. Applied rewrites 99.6%

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

   12. Applied rewrites 99.8%

       \[\leadsto \frac{1}{\left(-\left(-N\right)\right) + \frac{-0.5 + \frac{0.08333333333333333 - \frac{0.041666666666666664}{N}}{N}}{N} \cdot \color{blue}{\left(-N\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;N \leq 850:\\ \;\;\;\;\log \left(\frac{N - -1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N - \frac{\frac{0.08333333333333333 - \frac{0.041666666666666664}{N}}{N} + -0.5}{N} \cdot N}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.3% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \frac{1}{N - \frac{\frac{0.08333333333333333 - \frac{0.041666666666666664}{N}}{N} + -0.5}{N} \cdot N} \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    code = 1.0d0 / (n - (((((0.08333333333333333d0 - (0.041666666666666664d0 / n)) / n) + (-0.5d0)) / n) * n))
end function

Java

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

Python

def code(N):
	return 1.0 / (N - (((((0.08333333333333333 - (0.041666666666666664 / N)) / N) + -0.5) / N) * N))

Julia

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

MATLAB

function tmp = code(N)
	tmp = 1.0 / (N - (((((0.08333333333333333 - (0.041666666666666664 / N)) / N) + -0.5) / N) * N));
end

Wolfram

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

Derivation

1. Initial program 23.4%

   \[\log \left(N + 1\right) - \log N \]
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 96.5%

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

7. Applied rewrites 96.6%

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

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 96.9%

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

12. Applied rewrites 97.0%

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

13. Final simplification 97.0%

    \[\leadsto \frac{1}{N - \frac{\frac{0.08333333333333333 - \frac{0.041666666666666664}{N}}{N} + -0.5}{N} \cdot N} \]
14. Add Preprocessing

Alternative 6: 96.2% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    code = 1.0d0 / ((((0.5d0 - ((0.08333333333333333d0 - (0.041666666666666664d0 / n)) / n)) / n) - (-1.0d0)) * n)
end function

Java

public static double code(double N) {
	return 1.0 / ((((0.5 - ((0.08333333333333333 - (0.041666666666666664 / N)) / N)) / N) - -1.0) * N);
}

Python

def code(N):
	return 1.0 / ((((0.5 - ((0.08333333333333333 - (0.041666666666666664 / N)) / N)) / N) - -1.0) * N)

Julia

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

MATLAB

function tmp = code(N)
	tmp = 1.0 / ((((0.5 - ((0.08333333333333333 - (0.041666666666666664 / N)) / N)) / N) - -1.0) * N);
end

Wolfram

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

Derivation

1. Initial program 23.4%

   \[\log \left(N + 1\right) - \log N \]
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 96.5%

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

7. Applied rewrites 96.6%

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

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 96.9%

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

11. Final simplification 96.9%

    \[\leadsto \frac{1}{\left(\frac{0.5 - \frac{0.08333333333333333 - \frac{0.041666666666666664}{N}}{N}}{N} - -1\right) \cdot N} \]
12. Add Preprocessing

Alternative 7: 96.1% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 23.4%

   \[\log \left(N + 1\right) - \log N \]
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 96.5%

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

7. Applied rewrites 96.6%

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

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 96.9%

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

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 96.8%

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

Alternative 8: 94.9% accurate, 4.6× speedup

Math

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

FPCore

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

C

double code(double N) {
	return 1.0 / ((((0.5 - (0.08333333333333333 / N)) / N) - -1.0) * N);
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    code = 1.0d0 / ((((0.5d0 - (0.08333333333333333d0 / n)) / n) - (-1.0d0)) * n)
end function

Java

public static double code(double N) {
	return 1.0 / ((((0.5 - (0.08333333333333333 / N)) / N) - -1.0) * N);
}

Python

def code(N):
	return 1.0 / ((((0.5 - (0.08333333333333333 / N)) / N) - -1.0) * N)

Julia

function code(N)
	return Float64(1.0 / Float64(Float64(Float64(Float64(0.5 - Float64(0.08333333333333333 / N)) / N) - -1.0) * N))
end

MATLAB

function tmp = code(N)
	tmp = 1.0 / ((((0.5 - (0.08333333333333333 / N)) / N) - -1.0) * N);
end

Wolfram

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

Derivation

1. Initial program 23.4%

   \[\log \left(N + 1\right) - \log N \]
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 96.5%

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

7. Applied rewrites 96.6%

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

8. Taylor expanded in N around inf

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

10. Applied rewrites 95.7%

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

11. Final simplification 95.7%

    \[\leadsto \frac{1}{\left(\frac{0.5 - \frac{0.08333333333333333}{N}}{N} - -1\right) \cdot N} \]
12. Add Preprocessing

Alternative 9: 94.5% accurate, 5.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 23.4%

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

3. Taylor expanded in N around inf

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

   1. lower-/.f64 N/A

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

   2. associate--l+ N/A

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

   3. +-commutative N/A

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

   4. unpow2 N/A

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

   5. associate-/r* N/A

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

   6. metadata-eval N/A

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

   7. associate-*r/ N/A

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

   8. associate-*r/ N/A

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

   9. metadata-eval N/A

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

   10. div-sub N/A

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

   11. metadata-eval N/A

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

   12. sub-neg N/A

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

   13. lower--.f64 N/A

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

   14. lower-/.f64 N/A

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

   15. lower--.f64 N/A

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

   16. associate-*r/ N/A

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

   17. metadata-eval N/A

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

   18. lower-/.f64 95.3

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

5. Applied rewrites 95.3%

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

Alternative 10: 92.5% accurate, 13.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 23.4%

   \[\log \left(N + 1\right) - \log N \]
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 96.5%

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

7. Applied rewrites 96.6%

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

8. Taylor expanded in N around inf

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

10. Applied rewrites 93.6%

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

Alternative 11: 83.9% accurate, 17.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(N):
	return 1.0 / N

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 23.4%

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

3. Taylor expanded in N around inf

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

   1. lower-/.f64 84.8

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

5. Applied rewrites 84.8%

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

Developer Target 1: 95.7% 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 2024332 
(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)))