2log (problem 3.3.6)

Percentage Accurate: 23.6% → 99.4%

Time: 8.7s

Alternatives: 15

Speedup: 10.4×

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: 23.6% 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

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

FPCore

(FPCore (N)
 :precision binary64
 (if (<= (- (log (+ N 1.0)) (log N)) 0.001)
   (pow
    (*
     (+
      1.0
      (/ (- 0.5 (/ (- 0.08333333333333333 (/ 0.041666666666666664 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(((1.0 + ((0.5 - ((0.08333333333333333 - (0.041666666666666664 / N)) / N)) / N)) * N), -1.0);
	} else {
		tmp = -log((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 + ((0.5d0 - ((0.08333333333333333d0 - (0.041666666666666664d0 / n)) / n)) / n)) * n) ** (-1.0d0)
    else
        tmp = -log((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 = Math.pow(((1.0 + ((0.5 - ((0.08333333333333333 - (0.041666666666666664 / N)) / N)) / N)) * N), -1.0);
	} else {
		tmp = -Math.log((N / (1.0 + N)));
	}
	return tmp;
}

Python

def code(N):
	tmp = 0
	if (math.log((N + 1.0)) - math.log(N)) <= 0.001:
		tmp = math.pow(((1.0 + ((0.5 - ((0.08333333333333333 - (0.041666666666666664 / N)) / N)) / N)) * N), -1.0)
	else:
		tmp = -math.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(Float64(1.0 + Float64(Float64(0.5 - Float64(Float64(0.08333333333333333 - Float64(0.041666666666666664 / N)) / N)) / N)) * N) ^ -1.0;
	else
		tmp = Float64(-log(Float64(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 + ((0.5 - ((0.08333333333333333 - (0.041666666666666664 / N)) / N)) / N)) * N) ^ -1.0;
	else
		tmp = -log((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[Power[N[(N[(1.0 + N[(N[(0.5 - N[(N[(0.08333333333333333 - N[(0.041666666666666664 / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision]), $MachinePrecision] * 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 16.8%

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

   3. Taylor expanded in N around inf

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

   5. 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 99.6%

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

   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 99.7%

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

   12. Applied rewrites 99.7%

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

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

   1. Initial program 93.4%

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

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

      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 96.2

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

   4. Applied rewrites 96.2%

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

4. Final simplification 99.4%

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

Alternative 2: 99.4% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

function code(N)
	tmp = 0.0
	if (Float64(log(Float64(N + 1.0)) - log(N)) <= 0.001)
		tmp = Float64(Float64(1.0 + Float64(Float64(0.5 - Float64(Float64(0.08333333333333333 - Float64(0.041666666666666664 / N)) / N)) / N)) * N) ^ -1.0;
	else
		tmp = Float64(-log(Float64(Float64(Float64(N - 1.0) * 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], 0.001], N[Power[N[(N[(1.0 + N[(N[(0.5 - N[(N[(0.08333333333333333 - N[(0.041666666666666664 / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision]), $MachinePrecision] * N), $MachinePrecision], -1.0], $MachinePrecision], (-N[Log[N[(N[(N[(N - 1.0), $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)) < 1e-3

   1. Initial program 16.8%

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

   3. Taylor expanded in N around inf

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

   5. 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 99.6%

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

   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 99.7%

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

   12. Applied rewrites 99.7%

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

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

   1. Initial program 93.4%

      \[\log \left(N + 1\right) - \log N \]
   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 95.8

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

   4. Applied rewrites 95.8%

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

4. Final simplification 99.4%

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

Alternative 3: 99.4% accurate, 0.6× speedup

Math

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

FPCore

(FPCore (N)
 :precision binary64
 (if (<= (- (log (+ N 1.0)) (log N)) 0.001)
   (pow
    (*
     (+
      1.0
      (/ (- 0.5 (/ (- 0.08333333333333333 (/ 0.041666666666666664 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(((1.0 + ((0.5 - ((0.08333333333333333 - (0.041666666666666664 / N)) / N)) / N)) * N), -1.0);
	} else {
		tmp = log(((1.0 + N) / 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 + ((0.5d0 - ((0.08333333333333333d0 - (0.041666666666666664d0 / n)) / n)) / n)) * n) ** (-1.0d0)
    else
        tmp = log(((1.0d0 + n) / 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 = Math.pow(((1.0 + ((0.5 - ((0.08333333333333333 - (0.041666666666666664 / N)) / N)) / N)) * N), -1.0);
	} else {
		tmp = Math.log(((1.0 + N) / N));
	}
	return tmp;
}

Python

def code(N):
	tmp = 0
	if (math.log((N + 1.0)) - math.log(N)) <= 0.001:
		tmp = math.pow(((1.0 + ((0.5 - ((0.08333333333333333 - (0.041666666666666664 / N)) / N)) / N)) * N), -1.0)
	else:
		tmp = math.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 = Float64(Float64(1.0 + Float64(Float64(0.5 - Float64(Float64(0.08333333333333333 - Float64(0.041666666666666664 / N)) / N)) / N)) * N) ^ -1.0;
	else
		tmp = log(Float64(Float64(1.0 + N) / 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 + ((0.5 - ((0.08333333333333333 - (0.041666666666666664 / N)) / N)) / N)) * N) ^ -1.0;
	else
		tmp = log(((1.0 + N) / 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[Power[N[(N[(1.0 + N[(N[(0.5 - N[(N[(0.08333333333333333 - N[(0.041666666666666664 / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision]), $MachinePrecision] * 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 16.8%

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

   3. Taylor expanded in N around inf

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

   5. 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 99.6%

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

   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 99.7%

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

   12. Applied rewrites 99.7%

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

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

   1. Initial program 93.4%

      \[\log \left(N + 1\right) - \log N \]
   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 95.7

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

      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 95.7

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

   4. Applied rewrites 95.7%

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

4. Final simplification 99.4%

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

Alternative 4: 96.6% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(N):
	return math.pow(((1.0 + ((0.5 - ((0.08333333333333333 - (0.041666666666666664 / N)) / N)) / N)) * N), -1.0)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 22.8%

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

3. Taylor expanded in N around inf

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

5. Applied rewrites 95.5%

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

7. Applied rewrites 95.5%

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

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 95.9%

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

12. Applied rewrites 95.9%

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

13. Final simplification 95.9%

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

Alternative 5: 96.5% accurate, 1.5× speedup

Math

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

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

Wolfram

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

Derivation

1. Initial program 22.8%

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

3. Taylor expanded in N around inf

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

5. Applied rewrites 95.5%

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

7. Applied rewrites 95.5%

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

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 95.9%

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

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 95.8%

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

14. Final simplification 95.8%

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

Alternative 6: 95.5% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

public static double code(double N) {
	return Math.pow((N + (((0.5 - (0.08333333333333333 / N)) / N) * N)), -1.0);
}

Python

def code(N):
	return math.pow((N + (((0.5 - (0.08333333333333333 / N)) / N) * N)), -1.0)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 22.8%

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

3. Taylor expanded in N around inf

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

5. Applied rewrites 95.5%

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

7. Applied rewrites 95.5%

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

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 94.7%

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

11. Final simplification 94.7%

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

Alternative 7: 95.4% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

public static double code(double N) {
	return Math.pow(((((0.5 - (0.08333333333333333 / N)) / N) + 1.0) * N), -1.0);
}

Python

def code(N):
	return math.pow(((((0.5 - (0.08333333333333333 / N)) / N) + 1.0) * N), -1.0)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 22.8%

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

3. Taylor expanded in N around inf

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

5. Applied rewrites 95.5%

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

7. Applied rewrites 95.5%

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

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 95.9%

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

11. Taylor expanded in N around inf

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

13. Applied rewrites 94.6%

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

14. Final simplification 94.6%

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

Alternative 8: 93.1% 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 22.8%

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

3. Taylor expanded in N around inf

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

5. Applied rewrites 95.5%

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

7. Applied rewrites 95.5%

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

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 92.3%

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

11. Final simplification 92.3%

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

Alternative 9: 84.6% 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 22.8%

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

3. Taylor expanded in N around inf

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

   1. lower-/.f64 85.0

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

5. Applied rewrites 85.0%

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

6. Final simplification 85.0%

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

Alternative 10: 95.8% accurate, 4.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 22.8%

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

3. Taylor expanded in N around inf

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

5. Applied rewrites 95.5%

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

7. Applied rewrites 95.3%

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

8. Taylor expanded in N around 0

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

10. Applied rewrites 95.2%

    \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(N - 0.5, N, 0.3333333333333333\right), N, -0.25\right)}{N}}{N}}{\color{blue}{N} \cdot N} \]
11. Step-by-step derivation

12. Applied rewrites 95.1%

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

Alternative 11: 95.0% 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 22.8%

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

3. Taylor expanded in N around inf

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

Alternative 12: 94.7% accurate, 6.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 22.8%

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

3. Taylor expanded in N around inf

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

5. Applied rewrites 95.5%

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

7. Applied rewrites 95.3%

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

8. Taylor expanded in N around inf

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

10. Applied rewrites 93.9%

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

11. Final simplification 93.9%

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

Alternative 13: 92.4% accurate, 8.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 22.8%

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

3. Taylor expanded in N around inf

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

   1. lower-/.f64 N/A

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

   2. lower--.f64 N/A

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

   3. associate-*r/ N/A

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

   4. metadata-eval N/A

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

   5. lower-/.f64 91.6

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

5. Applied rewrites 91.6%

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

Alternative 14: 92.2% accurate, 10.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 22.8%

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

3. Taylor expanded in N around inf

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

5. Applied rewrites 95.5%

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

7. Applied rewrites 95.3%

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

8. Taylor expanded in N around inf

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

10. Applied rewrites 91.3%

    \[\leadsto \frac{-0.5 - N \cdot -1}{N \cdot N} \]
11. Step-by-step derivation

12. Applied rewrites 91.3%

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

Alternative 15: 3.3% accurate, 207.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (N) :precision binary64 0.0)

C

double code(double N) {
	return 0.0;
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    code = 0.0d0
end function

Java

public static double code(double N) {
	return 0.0;
}

Python

def code(N):
	return 0.0

Julia

function code(N)
	return 0.0
end

MATLAB

function tmp = code(N)
	tmp = 0.0;
end

Wolfram

code[N_] := 0.0

Derivation

1. Initial program 22.8%

   \[\log \left(N + 1\right) - \log N \]
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 25.2

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

4. Applied rewrites 25.2%

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

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

   8. sub-neg N/A

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

   9. metadata-eval N/A

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

   10. lift-*.f64 N/A

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

   11. associate-/r* N/A

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

   12. lift--.f64 N/A

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

   13. flip-+ N/A

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

   14. diff-log N/A

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

   15. lift-log.f64 N/A

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

   16. sub-neg N/A

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

   17. +-commutative N/A

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

   18. lift-log1p.f64 N/A

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

   19. unpow1 N/A

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

   20. sqr-pow N/A

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

6. Applied rewrites 24.4%

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

   1. lift-fma.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. lift-sqrt.f64 N/A

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

   4. rem-square-sqrt N/A

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

   5. +-commutative N/A

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

   6. lift-neg.f64 N/A

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

   7. neg-sub0 N/A

      \[\leadsto \color{blue}{\left(0 - \log N\right)} + \mathsf{log1p}\left(N\right) \]

   8. associate-+l- N/A

      \[\leadsto \color{blue}{0 - \left(\log N - \mathsf{log1p}\left(N\right)\right)} \]

   9. flip-- N/A

      \[\leadsto 0 - \color{blue}{\frac{\log N \cdot \log N - \mathsf{log1p}\left(N\right) \cdot \mathsf{log1p}\left(N\right)}{\log N + \mathsf{log1p}\left(N\right)}} \]

   10. unpow2 N/A

       \[\leadsto 0 - \frac{\color{blue}{{\log N}^{2}} - \mathsf{log1p}\left(N\right) \cdot \mathsf{log1p}\left(N\right)}{\log N + \mathsf{log1p}\left(N\right)} \]

   11. lift-pow.f64 N/A

       \[\leadsto 0 - \frac{\color{blue}{{\log N}^{2}} - \mathsf{log1p}\left(N\right) \cdot \mathsf{log1p}\left(N\right)}{\log N + \mathsf{log1p}\left(N\right)} \]

   12. unpow2 N/A

       \[\leadsto 0 - \frac{{\log N}^{2} - \color{blue}{{\left(\mathsf{log1p}\left(N\right)\right)}^{2}}}{\log N + \mathsf{log1p}\left(N\right)} \]

   13. lift-pow.f64 N/A

       \[\leadsto 0 - \frac{{\log N}^{2} - \color{blue}{{\left(\mathsf{log1p}\left(N\right)\right)}^{2}}}{\log N + \mathsf{log1p}\left(N\right)} \]

   14. lift-+.f64 N/A

       \[\leadsto 0 - \frac{{\log N}^{2} - {\left(\mathsf{log1p}\left(N\right)\right)}^{2}}{\color{blue}{\log N + \mathsf{log1p}\left(N\right)}} \]

   15. sub-div N/A

       \[\leadsto 0 - \color{blue}{\left(\frac{{\log N}^{2}}{\log N + \mathsf{log1p}\left(N\right)} - \frac{{\left(\mathsf{log1p}\left(N\right)\right)}^{2}}{\log N + \mathsf{log1p}\left(N\right)}\right)} \]

   16. lift-/.f64 N/A

       \[\leadsto 0 - \left(\color{blue}{\frac{{\log N}^{2}}{\log N + \mathsf{log1p}\left(N\right)}} - \frac{{\left(\mathsf{log1p}\left(N\right)\right)}^{2}}{\log N + \mathsf{log1p}\left(N\right)}\right) \]

   17. lift-/.f64 N/A

       \[\leadsto 0 - \left(\frac{{\log N}^{2}}{\log N + \mathsf{log1p}\left(N\right)} - \color{blue}{\frac{{\left(\mathsf{log1p}\left(N\right)\right)}^{2}}{\log N + \mathsf{log1p}\left(N\right)}}\right) \]

8. Applied rewrites 23.4%

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

9. Taylor expanded in N around inf

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

    1. distribute-rgt-out N/A

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

    2. metadata-eval N/A

       \[\leadsto \frac{1}{4} \cdot \left(\log \left(\frac{1}{N}\right) \cdot \color{blue}{0}\right) \]

    3. mul0-rgt N/A

       \[\leadsto \frac{1}{4} \cdot \color{blue}{0} \]

    4. metadata-eval 3.3

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

11. Applied rewrites 3.3%

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

Developer Target 1: 96.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \left(\left(\frac{1}{N} + \frac{-1}{2 \cdot {N}^{2}}\right) + \frac{1}{3 \cdot {N}^{3}}\right) + \frac{-1}{4 \cdot {N}^{4}} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (+
  (+ (+ (/ 1.0 N) (/ -1.0 (* 2.0 (pow N 2.0)))) (/ 1.0 (* 3.0 (pow N 3.0))))
  (/ -1.0 (* 4.0 (pow N 4.0)))))

C

double code(double N) {
	return (((1.0 / N) + (-1.0 / (2.0 * pow(N, 2.0)))) + (1.0 / (3.0 * pow(N, 3.0)))) + (-1.0 / (4.0 * pow(N, 4.0)));
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    code = (((1.0d0 / n) + ((-1.0d0) / (2.0d0 * (n ** 2.0d0)))) + (1.0d0 / (3.0d0 * (n ** 3.0d0)))) + ((-1.0d0) / (4.0d0 * (n ** 4.0d0)))
end function

Java

public static double code(double N) {
	return (((1.0 / N) + (-1.0 / (2.0 * Math.pow(N, 2.0)))) + (1.0 / (3.0 * Math.pow(N, 3.0)))) + (-1.0 / (4.0 * Math.pow(N, 4.0)));
}

Python

def code(N):
	return (((1.0 / N) + (-1.0 / (2.0 * math.pow(N, 2.0)))) + (1.0 / (3.0 * math.pow(N, 3.0)))) + (-1.0 / (4.0 * math.pow(N, 4.0)))

Julia

function code(N)
	return Float64(Float64(Float64(Float64(1.0 / N) + Float64(-1.0 / Float64(2.0 * (N ^ 2.0)))) + Float64(1.0 / Float64(3.0 * (N ^ 3.0)))) + Float64(-1.0 / Float64(4.0 * (N ^ 4.0))))
end

MATLAB

function tmp = code(N)
	tmp = (((1.0 / N) + (-1.0 / (2.0 * (N ^ 2.0)))) + (1.0 / (3.0 * (N ^ 3.0)))) + (-1.0 / (4.0 * (N ^ 4.0)));
end

Wolfram

code[N_] := N[(N[(N[(N[(1.0 / N), $MachinePrecision] + N[(-1.0 / N[(2.0 * N[Power[N, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(3.0 * N[Power[N, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[(4.0 * N[Power[N, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024322 
(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)))