2log (problem 3.3.6)

Percentage Accurate: 23.7% → 99.5%

Time: 9.9s

Alternatives: 10

Speedup: 17.3×

Specification

\[N > 1 \land N < 10^{+40}\]

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 23.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 0.0005:\\ \;\;\;\;\frac{1}{\left(N + 0.5\right) + N \cdot \left(\frac{0.041666666666666664}{N \cdot \left(N \cdot N\right)} + \frac{-0.08333333333333333}{N \cdot N}\right)}\\ \mathbf{else}:\\ \;\;\;\;-\log \left(\frac{N}{N + 1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= (- (log (+ N 1.0)) (log N)) 0.0005)
   (/
    1.0
    (+
     (+ N 0.5)
     (*
      N
      (+
       (/ 0.041666666666666664 (* N (* N N)))
       (/ -0.08333333333333333 (* N N))))))
   (- (log (/ N (+ N 1.0))))))

C

double code(double N) {
	double tmp;
	if ((log((N + 1.0)) - log(N)) <= 0.0005) {
		tmp = 1.0 / ((N + 0.5) + (N * ((0.041666666666666664 / (N * (N * N))) + (-0.08333333333333333 / (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)) <= 0.0005d0) then
        tmp = 1.0d0 / ((n + 0.5d0) + (n * ((0.041666666666666664d0 / (n * (n * n))) + ((-0.08333333333333333d0) / (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)) <= 0.0005) {
		tmp = 1.0 / ((N + 0.5) + (N * ((0.041666666666666664 / (N * (N * N))) + (-0.08333333333333333 / (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)) <= 0.0005:
		tmp = 1.0 / ((N + 0.5) + (N * ((0.041666666666666664 / (N * (N * N))) + (-0.08333333333333333 / (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)) <= 0.0005)
		tmp = Float64(1.0 / Float64(Float64(N + 0.5) + Float64(N * Float64(Float64(0.041666666666666664 / Float64(N * Float64(N * N))) + Float64(-0.08333333333333333 / Float64(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)) <= 0.0005)
		tmp = 1.0 / ((N + 0.5) + (N * ((0.041666666666666664 / (N * (N * N))) + (-0.08333333333333333 / (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], 0.0005], N[(1.0 / N[(N[(N + 0.5), $MachinePrecision] + N[(N * N[(N[(0.041666666666666664 / N[(N * N[(N * N), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.08333333333333333 / N[(N * N), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-N[Log[N[(N / N[(N + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (log.f64 (+.f64 N #s(literal 1 binary64))) (log.f64 N)) < 5.0000000000000001e-4

   1. Initial program 21.1%

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

   3. Taylor expanded in N around inf

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

   5. Simplified 99.9%

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

   6. Taylor expanded in N around 0

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

      1. /-lowering-/.f64 N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. cube-mult N/A

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 99.9

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

   8. Simplified 99.9%

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

      1. clear-num N/A

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

      2. /-lowering-/.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. +-lowering-+.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. *-lowering-*.f64 99.9

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

   10. Applied egg-rr 99.9%

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

   11. Taylor expanded in N around inf

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

       1. associate-+r+ N/A

          \[\leadsto \frac{1}{N \cdot \left(\color{blue}{\left(\left(1 + \frac{1}{2} \cdot \frac{1}{N}\right) + \frac{1}{24} \cdot \frac{1}{{N}^{3}}\right)} - \frac{\frac{1}{12}}{{N}^{2}}\right)} \]

       2. associate--l+ N/A

          \[\leadsto \frac{1}{N \cdot \color{blue}{\left(\left(1 + \frac{1}{2} \cdot \frac{1}{N}\right) + \left(\frac{1}{24} \cdot \frac{1}{{N}^{3}} - \frac{\frac{1}{12}}{{N}^{2}}\right)\right)}} \]

       3. distribute-lft-in N/A

          \[\leadsto \frac{1}{\color{blue}{N \cdot \left(1 + \frac{1}{2} \cdot \frac{1}{N}\right) + N \cdot \left(\frac{1}{24} \cdot \frac{1}{{N}^{3}} - \frac{\frac{1}{12}}{{N}^{2}}\right)}} \]

       4. +-lowering-+.f64 N/A

          \[\leadsto \frac{1}{\color{blue}{N \cdot \left(1 + \frac{1}{2} \cdot \frac{1}{N}\right) + N \cdot \left(\frac{1}{24} \cdot \frac{1}{{N}^{3}} - \frac{\frac{1}{12}}{{N}^{2}}\right)}} \]

       5. distribute-rgt-in N/A

          \[\leadsto \frac{1}{\color{blue}{\left(1 \cdot N + \left(\frac{1}{2} \cdot \frac{1}{N}\right) \cdot N\right)} + N \cdot \left(\frac{1}{24} \cdot \frac{1}{{N}^{3}} - \frac{\frac{1}{12}}{{N}^{2}}\right)} \]

       6. *-lft-identity N/A

          \[\leadsto \frac{1}{\left(\color{blue}{N} + \left(\frac{1}{2} \cdot \frac{1}{N}\right) \cdot N\right) + N \cdot \left(\frac{1}{24} \cdot \frac{1}{{N}^{3}} - \frac{\frac{1}{12}}{{N}^{2}}\right)} \]

       7. associate-*l* N/A

          \[\leadsto \frac{1}{\left(N + \color{blue}{\frac{1}{2} \cdot \left(\frac{1}{N} \cdot N\right)}\right) + N \cdot \left(\frac{1}{24} \cdot \frac{1}{{N}^{3}} - \frac{\frac{1}{12}}{{N}^{2}}\right)} \]

       8. lft-mult-inverse N/A

          \[\leadsto \frac{1}{\left(N + \frac{1}{2} \cdot \color{blue}{1}\right) + N \cdot \left(\frac{1}{24} \cdot \frac{1}{{N}^{3}} - \frac{\frac{1}{12}}{{N}^{2}}\right)} \]

       9. metadata-eval N/A

          \[\leadsto \frac{1}{\left(N + \color{blue}{\frac{1}{2}}\right) + N \cdot \left(\frac{1}{24} \cdot \frac{1}{{N}^{3}} - \frac{\frac{1}{12}}{{N}^{2}}\right)} \]

       10. +-lowering-+.f64 N/A

           \[\leadsto \frac{1}{\color{blue}{\left(N + \frac{1}{2}\right)} + N \cdot \left(\frac{1}{24} \cdot \frac{1}{{N}^{3}} - \frac{\frac{1}{12}}{{N}^{2}}\right)} \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \frac{1}{\left(N + \frac{1}{2}\right) + \color{blue}{N \cdot \left(\frac{1}{24} \cdot \frac{1}{{N}^{3}} - \frac{\frac{1}{12}}{{N}^{2}}\right)}} \]

       12. sub-neg N/A

           \[\leadsto \frac{1}{\left(N + \frac{1}{2}\right) + N \cdot \color{blue}{\left(\frac{1}{24} \cdot \frac{1}{{N}^{3}} + \left(\mathsf{neg}\left(\frac{\frac{1}{12}}{{N}^{2}}\right)\right)\right)}} \]

       13. +-lowering-+.f64 N/A

           \[\leadsto \frac{1}{\left(N + \frac{1}{2}\right) + N \cdot \color{blue}{\left(\frac{1}{24} \cdot \frac{1}{{N}^{3}} + \left(\mathsf{neg}\left(\frac{\frac{1}{12}}{{N}^{2}}\right)\right)\right)}} \]

   13. Simplified 99.9%

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

   if 5.0000000000000001e-4 < (-.f64 (log.f64 (+.f64 N #s(literal 1 binary64))) (log.f64 N))

   1. Initial program 92.3%

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

      1. diff-log N/A

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

      2. clear-num N/A

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

      3. neg-log N/A

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

      4. diff-log N/A

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

      5. neg-lowering-neg.f64 N/A

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

      6. diff-log N/A

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

      7. log-lowering-log.f64 N/A

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

      8. /-lowering-/.f64 N/A

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

      9. +-lowering-+.f64 95.6

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

   4. Applied egg-rr 95.6%

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

Alternative 2: 99.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 0.002:\\ \;\;\;\;\frac{1}{\left(N + 0.5\right) + N \cdot \left(\frac{0.041666666666666664}{N \cdot \left(N \cdot N\right)} + \frac{-0.08333333333333333}{N \cdot N}\right)}\\ \mathbf{else}:\\ \;\;\;\;\log \left(1 + \frac{1}{N}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= (- (log (+ N 1.0)) (log N)) 0.002)
   (/
    1.0
    (+
     (+ N 0.5)
     (*
      N
      (+
       (/ 0.041666666666666664 (* N (* N N)))
       (/ -0.08333333333333333 (* N N))))))
   (log (+ 1.0 (/ 1.0 N)))))

C

double code(double N) {
	double tmp;
	if ((log((N + 1.0)) - log(N)) <= 0.002) {
		tmp = 1.0 / ((N + 0.5) + (N * ((0.041666666666666664 / (N * (N * N))) + (-0.08333333333333333 / (N * N)))));
	} else {
		tmp = log((1.0 + (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)) <= 0.002d0) then
        tmp = 1.0d0 / ((n + 0.5d0) + (n * ((0.041666666666666664d0 / (n * (n * n))) + ((-0.08333333333333333d0) / (n * n)))))
    else
        tmp = log((1.0d0 + (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)) <= 0.002) {
		tmp = 1.0 / ((N + 0.5) + (N * ((0.041666666666666664 / (N * (N * N))) + (-0.08333333333333333 / (N * N)))));
	} else {
		tmp = Math.log((1.0 + (1.0 / N)));
	}
	return tmp;
}

Python

def code(N):
	tmp = 0
	if (math.log((N + 1.0)) - math.log(N)) <= 0.002:
		tmp = 1.0 / ((N + 0.5) + (N * ((0.041666666666666664 / (N * (N * N))) + (-0.08333333333333333 / (N * N)))))
	else:
		tmp = math.log((1.0 + (1.0 / N)))
	return tmp

Julia

function code(N)
	tmp = 0.0
	if (Float64(log(Float64(N + 1.0)) - log(N)) <= 0.002)
		tmp = Float64(1.0 / Float64(Float64(N + 0.5) + Float64(N * Float64(Float64(0.041666666666666664 / Float64(N * Float64(N * N))) + Float64(-0.08333333333333333 / Float64(N * N))))));
	else
		tmp = log(Float64(1.0 + Float64(1.0 / N)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(N)
	tmp = 0.0;
	if ((log((N + 1.0)) - log(N)) <= 0.002)
		tmp = 1.0 / ((N + 0.5) + (N * ((0.041666666666666664 / (N * (N * N))) + (-0.08333333333333333 / (N * N)))));
	else
		tmp = log((1.0 + (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], 0.002], N[(1.0 / N[(N[(N + 0.5), $MachinePrecision] + N[(N * N[(N[(0.041666666666666664 / N[(N * N[(N * N), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.08333333333333333 / N[(N * N), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[N[(1.0 + 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)) < 2e-3

   1. Initial program 21.4%

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

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

   6. Taylor expanded in N around 0

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

      1. /-lowering-/.f64 N/A

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. cube-mult N/A

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 99.8

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

   8. Simplified 99.8%

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

      1. clear-num N/A

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

      2. /-lowering-/.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. +-lowering-+.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. *-lowering-*.f64 99.8

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

   10. Applied egg-rr 99.8%

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

   11. Taylor expanded in N around inf

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

       1. associate-+r+ N/A

          \[\leadsto \frac{1}{N \cdot \left(\color{blue}{\left(\left(1 + \frac{1}{2} \cdot \frac{1}{N}\right) + \frac{1}{24} \cdot \frac{1}{{N}^{3}}\right)} - \frac{\frac{1}{12}}{{N}^{2}}\right)} \]

       2. associate--l+ N/A

          \[\leadsto \frac{1}{N \cdot \color{blue}{\left(\left(1 + \frac{1}{2} \cdot \frac{1}{N}\right) + \left(\frac{1}{24} \cdot \frac{1}{{N}^{3}} - \frac{\frac{1}{12}}{{N}^{2}}\right)\right)}} \]

       3. distribute-lft-in N/A

          \[\leadsto \frac{1}{\color{blue}{N \cdot \left(1 + \frac{1}{2} \cdot \frac{1}{N}\right) + N \cdot \left(\frac{1}{24} \cdot \frac{1}{{N}^{3}} - \frac{\frac{1}{12}}{{N}^{2}}\right)}} \]

       4. +-lowering-+.f64 N/A

          \[\leadsto \frac{1}{\color{blue}{N \cdot \left(1 + \frac{1}{2} \cdot \frac{1}{N}\right) + N \cdot \left(\frac{1}{24} \cdot \frac{1}{{N}^{3}} - \frac{\frac{1}{12}}{{N}^{2}}\right)}} \]

       5. distribute-rgt-in N/A

          \[\leadsto \frac{1}{\color{blue}{\left(1 \cdot N + \left(\frac{1}{2} \cdot \frac{1}{N}\right) \cdot N\right)} + N \cdot \left(\frac{1}{24} \cdot \frac{1}{{N}^{3}} - \frac{\frac{1}{12}}{{N}^{2}}\right)} \]

       6. *-lft-identity N/A

          \[\leadsto \frac{1}{\left(\color{blue}{N} + \left(\frac{1}{2} \cdot \frac{1}{N}\right) \cdot N\right) + N \cdot \left(\frac{1}{24} \cdot \frac{1}{{N}^{3}} - \frac{\frac{1}{12}}{{N}^{2}}\right)} \]

       7. associate-*l* N/A

          \[\leadsto \frac{1}{\left(N + \color{blue}{\frac{1}{2} \cdot \left(\frac{1}{N} \cdot N\right)}\right) + N \cdot \left(\frac{1}{24} \cdot \frac{1}{{N}^{3}} - \frac{\frac{1}{12}}{{N}^{2}}\right)} \]

       8. lft-mult-inverse N/A

          \[\leadsto \frac{1}{\left(N + \frac{1}{2} \cdot \color{blue}{1}\right) + N \cdot \left(\frac{1}{24} \cdot \frac{1}{{N}^{3}} - \frac{\frac{1}{12}}{{N}^{2}}\right)} \]

       9. metadata-eval N/A

          \[\leadsto \frac{1}{\left(N + \color{blue}{\frac{1}{2}}\right) + N \cdot \left(\frac{1}{24} \cdot \frac{1}{{N}^{3}} - \frac{\frac{1}{12}}{{N}^{2}}\right)} \]

       10. +-lowering-+.f64 N/A

           \[\leadsto \frac{1}{\color{blue}{\left(N + \frac{1}{2}\right)} + N \cdot \left(\frac{1}{24} \cdot \frac{1}{{N}^{3}} - \frac{\frac{1}{12}}{{N}^{2}}\right)} \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \frac{1}{\left(N + \frac{1}{2}\right) + \color{blue}{N \cdot \left(\frac{1}{24} \cdot \frac{1}{{N}^{3}} - \frac{\frac{1}{12}}{{N}^{2}}\right)}} \]

       12. sub-neg N/A

           \[\leadsto \frac{1}{\left(N + \frac{1}{2}\right) + N \cdot \color{blue}{\left(\frac{1}{24} \cdot \frac{1}{{N}^{3}} + \left(\mathsf{neg}\left(\frac{\frac{1}{12}}{{N}^{2}}\right)\right)\right)}} \]

       13. +-lowering-+.f64 N/A

           \[\leadsto \frac{1}{\left(N + \frac{1}{2}\right) + N \cdot \color{blue}{\left(\frac{1}{24} \cdot \frac{1}{{N}^{3}} + \left(\mathsf{neg}\left(\frac{\frac{1}{12}}{{N}^{2}}\right)\right)\right)}} \]

   13. Simplified 99.9%

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

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

   1. Initial program 92.7%

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

      1. diff-log N/A

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

      2. log-lowering-log.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. +-lowering-+.f64 95.2

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

   4. Applied egg-rr 95.2%

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

   5. Taylor expanded in N around inf

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

      1. +-lowering-+.f64 N/A

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

      2. /-lowering-/.f64 95.3

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

   7. Simplified 95.3%

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

Alternative 3: 96.6% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 27.5%

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

3. Taylor expanded in N around inf

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

5. Simplified 95.6%

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

6. Taylor expanded in N around 0

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

   1. /-lowering-/.f64 N/A

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

   2. sub-neg N/A

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

   3. metadata-eval N/A

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

   4. accelerator-lowering-fma.f64 N/A

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

   5. +-commutative N/A

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

   6. *-commutative N/A

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

   7. accelerator-lowering-fma.f64 N/A

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

   8. cube-mult N/A

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

   9. unpow2 N/A

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

   10. *-lowering-*.f64 N/A

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

   11. unpow2 N/A

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

   12. *-lowering-*.f64 95.6

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

8. Simplified 95.6%

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

   1. clear-num N/A

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

   2. /-lowering-/.f64 N/A

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

   3. /-lowering-/.f64 N/A

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

   4. +-lowering-+.f64 N/A

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

   5. /-lowering-/.f64 N/A

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

   6. accelerator-lowering-fma.f64 N/A

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

   7. accelerator-lowering-fma.f64 N/A

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

   8. *-lowering-*.f64 N/A

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

   9. *-lowering-*.f64 95.6

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

10. Applied egg-rr 95.6%

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

11. Taylor expanded in N around inf

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

    1. associate-+r+ N/A

       \[\leadsto \frac{1}{N \cdot \left(\color{blue}{\left(\left(1 + \frac{1}{2} \cdot \frac{1}{N}\right) + \frac{1}{24} \cdot \frac{1}{{N}^{3}}\right)} - \frac{\frac{1}{12}}{{N}^{2}}\right)} \]

    2. associate--l+ N/A

       \[\leadsto \frac{1}{N \cdot \color{blue}{\left(\left(1 + \frac{1}{2} \cdot \frac{1}{N}\right) + \left(\frac{1}{24} \cdot \frac{1}{{N}^{3}} - \frac{\frac{1}{12}}{{N}^{2}}\right)\right)}} \]

    3. distribute-lft-in N/A

       \[\leadsto \frac{1}{\color{blue}{N \cdot \left(1 + \frac{1}{2} \cdot \frac{1}{N}\right) + N \cdot \left(\frac{1}{24} \cdot \frac{1}{{N}^{3}} - \frac{\frac{1}{12}}{{N}^{2}}\right)}} \]

    4. +-lowering-+.f64 N/A

       \[\leadsto \frac{1}{\color{blue}{N \cdot \left(1 + \frac{1}{2} \cdot \frac{1}{N}\right) + N \cdot \left(\frac{1}{24} \cdot \frac{1}{{N}^{3}} - \frac{\frac{1}{12}}{{N}^{2}}\right)}} \]

    5. distribute-rgt-in N/A

       \[\leadsto \frac{1}{\color{blue}{\left(1 \cdot N + \left(\frac{1}{2} \cdot \frac{1}{N}\right) \cdot N\right)} + N \cdot \left(\frac{1}{24} \cdot \frac{1}{{N}^{3}} - \frac{\frac{1}{12}}{{N}^{2}}\right)} \]

    6. *-lft-identity N/A

       \[\leadsto \frac{1}{\left(\color{blue}{N} + \left(\frac{1}{2} \cdot \frac{1}{N}\right) \cdot N\right) + N \cdot \left(\frac{1}{24} \cdot \frac{1}{{N}^{3}} - \frac{\frac{1}{12}}{{N}^{2}}\right)} \]

    7. associate-*l* N/A

       \[\leadsto \frac{1}{\left(N + \color{blue}{\frac{1}{2} \cdot \left(\frac{1}{N} \cdot N\right)}\right) + N \cdot \left(\frac{1}{24} \cdot \frac{1}{{N}^{3}} - \frac{\frac{1}{12}}{{N}^{2}}\right)} \]

    8. lft-mult-inverse N/A

       \[\leadsto \frac{1}{\left(N + \frac{1}{2} \cdot \color{blue}{1}\right) + N \cdot \left(\frac{1}{24} \cdot \frac{1}{{N}^{3}} - \frac{\frac{1}{12}}{{N}^{2}}\right)} \]

    9. metadata-eval N/A

       \[\leadsto \frac{1}{\left(N + \color{blue}{\frac{1}{2}}\right) + N \cdot \left(\frac{1}{24} \cdot \frac{1}{{N}^{3}} - \frac{\frac{1}{12}}{{N}^{2}}\right)} \]

    10. +-lowering-+.f64 N/A

        \[\leadsto \frac{1}{\color{blue}{\left(N + \frac{1}{2}\right)} + N \cdot \left(\frac{1}{24} \cdot \frac{1}{{N}^{3}} - \frac{\frac{1}{12}}{{N}^{2}}\right)} \]

    11. *-lowering-*.f64 N/A

        \[\leadsto \frac{1}{\left(N + \frac{1}{2}\right) + \color{blue}{N \cdot \left(\frac{1}{24} \cdot \frac{1}{{N}^{3}} - \frac{\frac{1}{12}}{{N}^{2}}\right)}} \]

    12. sub-neg N/A

        \[\leadsto \frac{1}{\left(N + \frac{1}{2}\right) + N \cdot \color{blue}{\left(\frac{1}{24} \cdot \frac{1}{{N}^{3}} + \left(\mathsf{neg}\left(\frac{\frac{1}{12}}{{N}^{2}}\right)\right)\right)}} \]

    13. +-lowering-+.f64 N/A

        \[\leadsto \frac{1}{\left(N + \frac{1}{2}\right) + N \cdot \color{blue}{\left(\frac{1}{24} \cdot \frac{1}{{N}^{3}} + \left(\mathsf{neg}\left(\frac{\frac{1}{12}}{{N}^{2}}\right)\right)\right)}} \]

13. Simplified 96.1%

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

Alternative 4: 96.5% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 27.5%

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

3. Taylor expanded in N around inf

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

5. Simplified 95.6%

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

6. Taylor expanded in N around 0

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

   1. /-lowering-/.f64 N/A

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

   2. sub-neg N/A

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

   3. metadata-eval N/A

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

   4. accelerator-lowering-fma.f64 N/A

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

   5. +-commutative N/A

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

   6. *-commutative N/A

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

   7. accelerator-lowering-fma.f64 N/A

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

   8. cube-mult N/A

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

   9. unpow2 N/A

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

   10. *-lowering-*.f64 N/A

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

   11. unpow2 N/A

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

   12. *-lowering-*.f64 95.6

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

8. Simplified 95.6%

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

   1. clear-num N/A

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

   2. /-lowering-/.f64 N/A

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

   3. /-lowering-/.f64 N/A

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

   4. +-lowering-+.f64 N/A

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

   5. /-lowering-/.f64 N/A

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

   6. accelerator-lowering-fma.f64 N/A

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

   7. accelerator-lowering-fma.f64 N/A

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

   8. *-lowering-*.f64 N/A

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

   9. *-lowering-*.f64 95.6

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

10. Applied egg-rr 95.6%

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

11. Taylor expanded in N around -inf

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

    1. associate-*r* N/A

       \[\leadsto \frac{1}{\color{blue}{\left(-1 \cdot N\right) \cdot \left(-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N} - 1\right)}} \]

    2. *-lowering-*.f64 N/A

       \[\leadsto \frac{1}{\color{blue}{\left(-1 \cdot N\right) \cdot \left(-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N} - 1\right)}} \]

    3. mul-1-neg N/A

       \[\leadsto \frac{1}{\color{blue}{\left(\mathsf{neg}\left(N\right)\right)} \cdot \left(-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N} - 1\right)} \]

    4. neg-lowering-neg.f64 N/A

       \[\leadsto \frac{1}{\color{blue}{\left(\mathsf{neg}\left(N\right)\right)} \cdot \left(-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N} - 1\right)} \]

    5. sub-neg N/A

       \[\leadsto \frac{1}{\left(\mathsf{neg}\left(N\right)\right) \cdot \color{blue}{\left(-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N} + \left(\mathsf{neg}\left(1\right)\right)\right)}} \]

    6. metadata-eval N/A

       \[\leadsto \frac{1}{\left(\mathsf{neg}\left(N\right)\right) \cdot \left(-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N} + \color{blue}{-1}\right)} \]

    7. +-commutative N/A

       \[\leadsto \frac{1}{\left(\mathsf{neg}\left(N\right)\right) \cdot \color{blue}{\left(-1 + -1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N}\right)}} \]

    8. mul-1-neg N/A

       \[\leadsto \frac{1}{\left(\mathsf{neg}\left(N\right)\right) \cdot \left(-1 + \color{blue}{\left(\mathsf{neg}\left(\frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N}\right)\right)}\right)} \]

    9. unsub-neg N/A

       \[\leadsto \frac{1}{\left(\mathsf{neg}\left(N\right)\right) \cdot \color{blue}{\left(-1 - \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N}\right)}} \]

    10. --lowering--.f64 N/A

        \[\leadsto \frac{1}{\left(\mathsf{neg}\left(N\right)\right) \cdot \color{blue}{\left(-1 - \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N}\right)}} \]

    11. /-lowering-/.f64 N/A

        \[\leadsto \frac{1}{\left(\mathsf{neg}\left(N\right)\right) \cdot \left(-1 - \color{blue}{\frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N}}\right)} \]

13. Simplified 96.0%

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

14. Final simplification 96.0%

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

Alternative 5: 96.1% accurate, 4.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 27.5%

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

3. Taylor expanded in N around inf

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

5. Simplified 95.6%

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

6. Taylor expanded in N around 0

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

   1. /-lowering-/.f64 N/A

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

   2. sub-neg N/A

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

   3. metadata-eval N/A

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

   4. accelerator-lowering-fma.f64 N/A

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

   5. +-commutative N/A

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

   6. *-commutative N/A

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

   7. accelerator-lowering-fma.f64 N/A

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

   8. cube-mult N/A

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

   9. unpow2 N/A

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

   10. *-lowering-*.f64 N/A

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

   11. unpow2 N/A

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

   12. *-lowering-*.f64 95.6

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

8. Simplified 95.6%

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

Alternative 6: 95.4% accurate, 5.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 27.5%

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

3. Taylor expanded in N around inf

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

5. Simplified 95.6%

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

6. Taylor expanded in N around 0

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

   1. /-lowering-/.f64 N/A

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

   2. sub-neg N/A

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

   3. metadata-eval N/A

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

   4. accelerator-lowering-fma.f64 N/A

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

   5. +-commutative N/A

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

   6. *-commutative N/A

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

   7. accelerator-lowering-fma.f64 N/A

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

   8. cube-mult N/A

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

   9. unpow2 N/A

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

   10. *-lowering-*.f64 N/A

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

   11. unpow2 N/A

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

   12. *-lowering-*.f64 95.6

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

8. Simplified 95.6%

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

   1. clear-num N/A

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

   2. /-lowering-/.f64 N/A

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

   3. /-lowering-/.f64 N/A

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

   4. +-lowering-+.f64 N/A

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

   5. /-lowering-/.f64 N/A

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

   6. accelerator-lowering-fma.f64 N/A

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

   7. accelerator-lowering-fma.f64 N/A

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

   8. *-lowering-*.f64 N/A

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

   9. *-lowering-*.f64 95.6

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

10. Applied egg-rr 95.6%

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

11. Taylor expanded in N around -inf

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

    1. mul-1-neg N/A

       \[\leadsto \frac{1}{\color{blue}{\mathsf{neg}\left(N \cdot \left(-1 \cdot \frac{\frac{1}{2} - \frac{1}{12} \cdot \frac{1}{N}}{N} - 1\right)\right)}} \]

    2. neg-lowering-neg.f64 N/A

       \[\leadsto \frac{1}{\color{blue}{\mathsf{neg}\left(N \cdot \left(-1 \cdot \frac{\frac{1}{2} - \frac{1}{12} \cdot \frac{1}{N}}{N} - 1\right)\right)}} \]

    3. sub-neg N/A

       \[\leadsto \frac{1}{\mathsf{neg}\left(N \cdot \color{blue}{\left(-1 \cdot \frac{\frac{1}{2} - \frac{1}{12} \cdot \frac{1}{N}}{N} + \left(\mathsf{neg}\left(1\right)\right)\right)}\right)} \]

    4. metadata-eval N/A

       \[\leadsto \frac{1}{\mathsf{neg}\left(N \cdot \left(-1 \cdot \frac{\frac{1}{2} - \frac{1}{12} \cdot \frac{1}{N}}{N} + \color{blue}{-1}\right)\right)} \]

    5. distribute-lft-in N/A

       \[\leadsto \frac{1}{\mathsf{neg}\left(\color{blue}{\left(N \cdot \left(-1 \cdot \frac{\frac{1}{2} - \frac{1}{12} \cdot \frac{1}{N}}{N}\right) + N \cdot -1\right)}\right)} \]

    6. *-commutative N/A

       \[\leadsto \frac{1}{\mathsf{neg}\left(\left(N \cdot \left(-1 \cdot \frac{\frac{1}{2} - \frac{1}{12} \cdot \frac{1}{N}}{N}\right) + \color{blue}{-1 \cdot N}\right)\right)} \]

    7. mul-1-neg N/A

       \[\leadsto \frac{1}{\mathsf{neg}\left(\left(N \cdot \left(-1 \cdot \frac{\frac{1}{2} - \frac{1}{12} \cdot \frac{1}{N}}{N}\right) + \color{blue}{\left(\mathsf{neg}\left(N\right)\right)}\right)\right)} \]

    8. unsub-neg N/A

       \[\leadsto \frac{1}{\mathsf{neg}\left(\color{blue}{\left(N \cdot \left(-1 \cdot \frac{\frac{1}{2} - \frac{1}{12} \cdot \frac{1}{N}}{N}\right) - N\right)}\right)} \]

    9. --lowering--.f64 N/A

       \[\leadsto \frac{1}{\mathsf{neg}\left(\color{blue}{\left(N \cdot \left(-1 \cdot \frac{\frac{1}{2} - \frac{1}{12} \cdot \frac{1}{N}}{N}\right) - N\right)}\right)} \]

13. Simplified 94.9%

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

14. Final simplification 94.9%

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

Alternative 7: 95.4% accurate, 5.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 27.5%

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

3. Taylor expanded in N around inf

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

5. Simplified 95.6%

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

6. Taylor expanded in N around 0

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

   1. /-lowering-/.f64 N/A

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

   2. sub-neg N/A

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

   3. metadata-eval N/A

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

   4. accelerator-lowering-fma.f64 N/A

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

   5. +-commutative N/A

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

   6. *-commutative N/A

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

   7. accelerator-lowering-fma.f64 N/A

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

   8. cube-mult N/A

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

   9. unpow2 N/A

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

   10. *-lowering-*.f64 N/A

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

   11. unpow2 N/A

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

   12. *-lowering-*.f64 95.6

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

8. Simplified 95.6%

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

   1. clear-num N/A

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

   2. /-lowering-/.f64 N/A

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

   3. /-lowering-/.f64 N/A

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

   4. +-lowering-+.f64 N/A

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

   5. /-lowering-/.f64 N/A

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

   6. accelerator-lowering-fma.f64 N/A

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

   7. accelerator-lowering-fma.f64 N/A

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

   8. *-lowering-*.f64 N/A

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

   9. *-lowering-*.f64 95.6

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

10. Applied egg-rr 95.6%

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

11. Taylor expanded in N around inf

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

    1. sub-neg N/A

       \[\leadsto \frac{1}{N \cdot \color{blue}{\left(\left(1 + \frac{1}{2} \cdot \frac{1}{N}\right) + \left(\mathsf{neg}\left(\frac{\frac{1}{12}}{{N}^{2}}\right)\right)\right)}} \]

    2. +-commutative N/A

       \[\leadsto \frac{1}{N \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\frac{\frac{1}{12}}{{N}^{2}}\right)\right) + \left(1 + \frac{1}{2} \cdot \frac{1}{N}\right)\right)}} \]

    3. distribute-lft-in N/A

       \[\leadsto \frac{1}{\color{blue}{N \cdot \left(\mathsf{neg}\left(\frac{\frac{1}{12}}{{N}^{2}}\right)\right) + N \cdot \left(1 + \frac{1}{2} \cdot \frac{1}{N}\right)}} \]

    4. accelerator-lowering-fma.f64 N/A

       \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(N, \mathsf{neg}\left(\frac{\frac{1}{12}}{{N}^{2}}\right), N \cdot \left(1 + \frac{1}{2} \cdot \frac{1}{N}\right)\right)}} \]

    5. distribute-neg-frac N/A

       \[\leadsto \frac{1}{\mathsf{fma}\left(N, \color{blue}{\frac{\mathsf{neg}\left(\frac{1}{12}\right)}{{N}^{2}}}, N \cdot \left(1 + \frac{1}{2} \cdot \frac{1}{N}\right)\right)} \]

    6. /-lowering-/.f64 N/A

       \[\leadsto \frac{1}{\mathsf{fma}\left(N, \color{blue}{\frac{\mathsf{neg}\left(\frac{1}{12}\right)}{{N}^{2}}}, N \cdot \left(1 + \frac{1}{2} \cdot \frac{1}{N}\right)\right)} \]

    7. metadata-eval N/A

       \[\leadsto \frac{1}{\mathsf{fma}\left(N, \frac{\color{blue}{\frac{-1}{12}}}{{N}^{2}}, N \cdot \left(1 + \frac{1}{2} \cdot \frac{1}{N}\right)\right)} \]

    8. unpow2 N/A

       \[\leadsto \frac{1}{\mathsf{fma}\left(N, \frac{\frac{-1}{12}}{\color{blue}{N \cdot N}}, N \cdot \left(1 + \frac{1}{2} \cdot \frac{1}{N}\right)\right)} \]

    9. *-lowering-*.f64 N/A

       \[\leadsto \frac{1}{\mathsf{fma}\left(N, \frac{\frac{-1}{12}}{\color{blue}{N \cdot N}}, N \cdot \left(1 + \frac{1}{2} \cdot \frac{1}{N}\right)\right)} \]

    10. distribute-rgt-in N/A

        \[\leadsto \frac{1}{\mathsf{fma}\left(N, \frac{\frac{-1}{12}}{N \cdot N}, \color{blue}{1 \cdot N + \left(\frac{1}{2} \cdot \frac{1}{N}\right) \cdot N}\right)} \]

    11. *-lft-identity N/A

        \[\leadsto \frac{1}{\mathsf{fma}\left(N, \frac{\frac{-1}{12}}{N \cdot N}, \color{blue}{N} + \left(\frac{1}{2} \cdot \frac{1}{N}\right) \cdot N\right)} \]

    12. associate-*l* N/A

        \[\leadsto \frac{1}{\mathsf{fma}\left(N, \frac{\frac{-1}{12}}{N \cdot N}, N + \color{blue}{\frac{1}{2} \cdot \left(\frac{1}{N} \cdot N\right)}\right)} \]

    13. lft-mult-inverse N/A

        \[\leadsto \frac{1}{\mathsf{fma}\left(N, \frac{\frac{-1}{12}}{N \cdot N}, N + \frac{1}{2} \cdot \color{blue}{1}\right)} \]

    14. metadata-eval N/A

        \[\leadsto \frac{1}{\mathsf{fma}\left(N, \frac{\frac{-1}{12}}{N \cdot N}, N + \color{blue}{\frac{1}{2}}\right)} \]

    15. +-lowering-+.f64 94.9

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

13. Simplified 94.9%

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

Alternative 8: 93.0% accurate, 13.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 27.5%

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

3. Taylor expanded in N around inf

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

5. Simplified 95.6%

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

6. Taylor expanded in N around 0

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

   1. /-lowering-/.f64 N/A

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

   2. sub-neg N/A

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

   3. metadata-eval N/A

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

   4. accelerator-lowering-fma.f64 N/A

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

   5. +-commutative N/A

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

   6. *-commutative N/A

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

   7. accelerator-lowering-fma.f64 N/A

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

   8. cube-mult N/A

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

   9. unpow2 N/A

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

   10. *-lowering-*.f64 N/A

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

   11. unpow2 N/A

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

   12. *-lowering-*.f64 95.6

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

8. Simplified 95.6%

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

   1. clear-num N/A

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

   2. /-lowering-/.f64 N/A

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

   3. /-lowering-/.f64 N/A

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

   4. +-lowering-+.f64 N/A

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

   5. /-lowering-/.f64 N/A

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

   6. accelerator-lowering-fma.f64 N/A

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

   7. accelerator-lowering-fma.f64 N/A

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

   8. *-lowering-*.f64 N/A

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

   9. *-lowering-*.f64 95.6

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

10. Applied egg-rr 95.6%

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

11. Taylor expanded in N around inf

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

    1. distribute-rgt-in N/A

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

    2. *-lft-identity N/A

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

    3. associate-*l* N/A

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

    4. lft-mult-inverse N/A

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

    5. metadata-eval N/A

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

    6. +-lowering-+.f64 91.8

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

13. Simplified 91.8%

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

Alternative 9: 84.6% 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 27.5%

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

3. Taylor expanded in N around inf

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

   1. /-lowering-/.f64 81.6

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

5. Simplified 81.6%

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

Alternative 10: 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 27.5%

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

   1. diff-log N/A

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

   2. log-lowering-log.f64 N/A

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

   3. /-lowering-/.f64 N/A

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

   4. +-lowering-+.f64 30.1

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

4. Applied egg-rr 30.1%

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

   1. log-div N/A

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

   2. +-commutative N/A

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

   3. sub-neg N/A

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

   4. +-commutative N/A

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

   5. rem-square-sqrt N/A

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

   6. distribute-rgt-neg-in N/A

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

   7. accelerator-lowering-fma.f64 N/A

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

   8. sqrt-lowering-sqrt.f64 N/A

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

   9. log-lowering-log.f64 N/A

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

   10. neg-lowering-neg.f64 N/A

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

   11. sqrt-lowering-sqrt.f64 N/A

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

   12. log-lowering-log.f64 N/A

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

   13. accelerator-lowering-log1p.f64 28.7

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

6. Applied egg-rr 28.7%

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

7. Taylor expanded in N around inf

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

   1. distribute-rgt1-in N/A

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

   2. metadata-eval N/A

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

   3. mul0-lft 3.3

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

9. Simplified 3.3%

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

Developer Target 1: 99.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \mathsf{log1p}\left(\frac{1}{N}\right) \end{array} \]

FPCore

(FPCore (N) :precision binary64 (log1p (/ 1.0 N)))

C

double code(double N) {
	return log1p((1.0 / N));
}

Java

public static double code(double N) {
	return Math.log1p((1.0 / N));
}

Python

def code(N):
	return math.log1p((1.0 / N))

Julia

function code(N)
	return log1p(Float64(1.0 / N))
end

Wolfram

code[N_] := N[Log[1 + N[(1.0 / N), $MachinePrecision]], $MachinePrecision]

Developer Target 2: 26.4% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Developer Target 3: 96.1% 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 2024198 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  :pre (and (> N 1.0) (< N 1e+40))

  :alt
  (! :herbie-platform default (log1p (/ 1 N)))

  :alt
  (! :herbie-platform default (log (+ 1 (/ 1 N))))

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

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