2log (problem 3.3.6)

Percentage Accurate: 24.1% → 99.5%

Time: 9.8s

Alternatives: 15

Speedup: 17.3×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 24.1% 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, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 0.001:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(N, \frac{\frac{0.08333333333333333 + \frac{-0.041666666666666664}{N}}{N} - 0.5}{N}, -N\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\log \left(\mathsf{fma}\left(N, N, N\right)\right) \cdot \log \left(\frac{N}{N + 1}\right)\right) \cdot \frac{-1}{\log N + \mathsf{log1p}\left(N\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= (- (log (+ N 1.0)) (log N)) 0.001)
   (/
    -1.0
    (fma
     N
     (/ (- (/ (+ 0.08333333333333333 (/ -0.041666666666666664 N)) N) 0.5) N)
     (- N)))
   (*
    (* (log (fma N N N)) (log (/ N (+ N 1.0))))
    (/ -1.0 (+ (log N) (log1p N))))))

C

double code(double N) {
	double tmp;
	if ((log((N + 1.0)) - log(N)) <= 0.001) {
		tmp = -1.0 / fma(N, ((((0.08333333333333333 + (-0.041666666666666664 / N)) / N) - 0.5) / N), -N);
	} else {
		tmp = (log(fma(N, N, N)) * log((N / (N + 1.0)))) * (-1.0 / (log(N) + log1p(N)));
	}
	return tmp;
}

Julia

function code(N)
	tmp = 0.0
	if (Float64(log(Float64(N + 1.0)) - log(N)) <= 0.001)
		tmp = Float64(-1.0 / fma(N, Float64(Float64(Float64(Float64(0.08333333333333333 + Float64(-0.041666666666666664 / N)) / N) - 0.5) / N), Float64(-N)));
	else
		tmp = Float64(Float64(log(fma(N, N, N)) * log(Float64(N / Float64(N + 1.0)))) * Float64(-1.0 / Float64(log(N) + log1p(N))));
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 19.0%

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

   3. Taylor expanded in N around inf

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

   5. Applied rewrites 99.7%

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

      1. lift-/.f64 N/A

         \[\leadsto \frac{1 + \frac{\frac{-1}{2} - \frac{\color{blue}{\frac{\frac{1}{4}}{N}} + \frac{-1}{3}}{N}}{N}}{N} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{1 + \frac{\frac{-1}{2} - \frac{\color{blue}{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}}{N}}{N}}{N} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{1 + \frac{\frac{-1}{2} - \color{blue}{\frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}}{N}}{N} \]

      4. lift--.f64 N/A

         \[\leadsto \frac{1 + \frac{\color{blue}{\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}}{N}}{N} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{1 + \color{blue}{\frac{\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}{N}}}{N} \]

      6. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{1 + \frac{\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}{N}}}{N} \]

      7. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{N}{1 + \frac{\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}{N}}}} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{N}{1 + \frac{\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}{N}}}} \]

      9. lower-/.f64 99.7

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

   7. Applied rewrites 99.7%

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

   8. Taylor expanded in N around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. distribute-lft-in N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

   10. Applied rewrites 99.8%

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

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

   1. Initial program 91.7%

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

   4. Applied rewrites 95.2%

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

      1. distribute-lft1-in N/A

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

      2. lift-+.f64 N/A

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

      3. log-prod N/A

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

      4. lift-log.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-log1p.f64 95.3

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

   6. Applied rewrites 95.3%

      \[\leadsto \left(\log \left(\mathsf{fma}\left(N, N, N\right)\right) \cdot \log \left(\frac{N}{N + 1}\right)\right) \cdot \frac{1}{-\color{blue}{\left(\mathsf{log1p}\left(N\right) + \log N\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 0.001:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(N, \frac{\frac{0.08333333333333333 + \frac{-0.041666666666666664}{N}}{N} - 0.5}{N}, -N\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\log \left(\mathsf{fma}\left(N, N, N\right)\right) \cdot \log \left(\frac{N}{N + 1}\right)\right) \cdot \frac{-1}{\log N + \mathsf{log1p}\left(N\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 19.0%

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

   3. Taylor expanded in N around inf

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

   5. Applied rewrites 99.7%

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

      1. lift-/.f64 N/A

         \[\leadsto \frac{1 + \frac{\frac{-1}{2} - \frac{\color{blue}{\frac{\frac{1}{4}}{N}} + \frac{-1}{3}}{N}}{N}}{N} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{1 + \frac{\frac{-1}{2} - \frac{\color{blue}{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}}{N}}{N}}{N} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{1 + \frac{\frac{-1}{2} - \color{blue}{\frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}}{N}}{N} \]

      4. lift--.f64 N/A

         \[\leadsto \frac{1 + \frac{\color{blue}{\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}}{N}}{N} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{1 + \color{blue}{\frac{\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}{N}}}{N} \]

      6. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{1 + \frac{\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}{N}}}{N} \]

      7. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{N}{1 + \frac{\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}{N}}}} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{N}{1 + \frac{\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}{N}}}} \]

      9. lower-/.f64 99.7

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

   7. Applied rewrites 99.7%

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

   8. Taylor expanded in N around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. distribute-lft-in N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

   10. Applied rewrites 99.8%

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

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

   1. Initial program 91.7%

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

   4. Applied rewrites 95.2%

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

      1. lift-fma.f64 N/A

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

      2. lift-log.f64 95.2

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

      3. remove-double-neg N/A

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

      4. lift-log.f64 N/A

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

      5. neg-log N/A

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

      6. neg-log N/A

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

      7. lower-log.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 95.2

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

   6. Applied rewrites 95.2%

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

4. Final simplification 99.5%

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

Alternative 3: 99.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\mathsf{fma}\left(N, N, N\right)\right)\\ \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 0.001:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(N, \frac{\frac{0.08333333333333333 + \frac{-0.041666666666666664}{N}}{N} - 0.5}{N}, -N\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(-t\_0\right) \cdot \frac{\log \left(\frac{N}{N + 1}\right)}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (let* ((t_0 (log (fma N N N))))
   (if (<= (- (log (+ N 1.0)) (log N)) 0.001)
     (/
      -1.0
      (fma
       N
       (/ (- (/ (+ 0.08333333333333333 (/ -0.041666666666666664 N)) N) 0.5) N)
       (- N)))
     (* (- t_0) (/ (log (/ N (+ N 1.0))) t_0)))))

C

double code(double N) {
	double t_0 = log(fma(N, N, N));
	double tmp;
	if ((log((N + 1.0)) - log(N)) <= 0.001) {
		tmp = -1.0 / fma(N, ((((0.08333333333333333 + (-0.041666666666666664 / N)) / N) - 0.5) / N), -N);
	} else {
		tmp = -t_0 * (log((N / (N + 1.0))) / t_0);
	}
	return tmp;
}

Julia

function code(N)
	t_0 = log(fma(N, N, N))
	tmp = 0.0
	if (Float64(log(Float64(N + 1.0)) - log(N)) <= 0.001)
		tmp = Float64(-1.0 / fma(N, Float64(Float64(Float64(Float64(0.08333333333333333 + Float64(-0.041666666666666664 / N)) / N) - 0.5) / N), Float64(-N)));
	else
		tmp = Float64(Float64(-t_0) * Float64(log(Float64(N / Float64(N + 1.0))) / t_0));
	end
	return tmp
end

Wolfram

code[N_] := Block[{t$95$0 = N[Log[N[(N * N + N), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[Log[N[(N + 1.0), $MachinePrecision]], $MachinePrecision] - N[Log[N], $MachinePrecision]), $MachinePrecision], 0.001], N[(-1.0 / N[(N * N[(N[(N[(N[(0.08333333333333333 + N[(-0.041666666666666664 / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision] - 0.5), $MachinePrecision] / N), $MachinePrecision] + (-N)), $MachinePrecision]), $MachinePrecision], N[((-t$95$0) * N[(N[Log[N[(N / N[(N + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $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 19.0%

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

   3. Taylor expanded in N around inf

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

   5. Applied rewrites 99.7%

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

      1. lift-/.f64 N/A

         \[\leadsto \frac{1 + \frac{\frac{-1}{2} - \frac{\color{blue}{\frac{\frac{1}{4}}{N}} + \frac{-1}{3}}{N}}{N}}{N} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{1 + \frac{\frac{-1}{2} - \frac{\color{blue}{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}}{N}}{N}}{N} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{1 + \frac{\frac{-1}{2} - \color{blue}{\frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}}{N}}{N} \]

      4. lift--.f64 N/A

         \[\leadsto \frac{1 + \frac{\color{blue}{\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}}{N}}{N} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{1 + \color{blue}{\frac{\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}{N}}}{N} \]

      6. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{1 + \frac{\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}{N}}}{N} \]

      7. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{N}{1 + \frac{\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}{N}}}} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{N}{1 + \frac{\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}{N}}}} \]

      9. lower-/.f64 99.7

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

   7. Applied rewrites 99.7%

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

   8. Taylor expanded in N around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. distribute-lft-in N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

   10. Applied rewrites 99.8%

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

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

   1. Initial program 91.7%

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

   4. Applied rewrites 95.2%

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

      1. lift-fma.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. un-div-inv N/A

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

      11. lift-*.f64 N/A

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

      12. lift-neg.f64 N/A

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

      13. neg-mul-1 N/A

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

      14. times-frac N/A

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

   6. Applied rewrites 95.2%

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

4. Final simplification 99.5%

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

Alternative 4: 99.5% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 19.0%

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

   3. Taylor expanded in N around inf

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

   5. Applied rewrites 99.7%

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

      1. lift-/.f64 N/A

         \[\leadsto \frac{1 + \frac{\frac{-1}{2} - \frac{\color{blue}{\frac{\frac{1}{4}}{N}} + \frac{-1}{3}}{N}}{N}}{N} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{1 + \frac{\frac{-1}{2} - \frac{\color{blue}{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}}{N}}{N}}{N} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{1 + \frac{\frac{-1}{2} - \color{blue}{\frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}}{N}}{N} \]

      4. lift--.f64 N/A

         \[\leadsto \frac{1 + \frac{\color{blue}{\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}}{N}}{N} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{1 + \color{blue}{\frac{\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}{N}}}{N} \]

      6. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{1 + \frac{\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}{N}}}{N} \]

      7. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{N}{1 + \frac{\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}{N}}}} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{N}{1 + \frac{\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}{N}}}} \]

      9. lower-/.f64 99.7

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

   7. Applied rewrites 99.7%

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

   8. Taylor expanded in N around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. distribute-lft-in N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

   10. Applied rewrites 99.8%

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

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

   1. Initial program 91.7%

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

      1. lift-+.f64 N/A

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

      2. diff-log N/A

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

      3. clear-num N/A

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

      4. log-rec N/A

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

      5. diff-log N/A

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

      6. lift-log.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. lift-log.f64 N/A

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

      11. diff-log N/A

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

      12. lower-log.f64 N/A

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

      13. lower-/.f64 95.2

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

   4. Applied rewrites 95.2%

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

4. Final simplification 99.5%

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

Alternative 5: 99.4% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 19.0%

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

   3. Taylor expanded in N around inf

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

   5. Applied rewrites 99.7%

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

      1. lift-/.f64 N/A

         \[\leadsto \frac{1 + \frac{\frac{-1}{2} - \frac{\color{blue}{\frac{\frac{1}{4}}{N}} + \frac{-1}{3}}{N}}{N}}{N} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{1 + \frac{\frac{-1}{2} - \frac{\color{blue}{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}}{N}}{N}}{N} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{1 + \frac{\frac{-1}{2} - \color{blue}{\frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}}{N}}{N} \]

      4. lift--.f64 N/A

         \[\leadsto \frac{1 + \frac{\color{blue}{\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}}{N}}{N} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{1 + \color{blue}{\frac{\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}{N}}}{N} \]

      6. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{1 + \frac{\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}{N}}}{N} \]

      7. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{N}{1 + \frac{\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}{N}}}} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{N}{1 + \frac{\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}{N}}}} \]

      9. lower-/.f64 99.7

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

   7. Applied rewrites 99.7%

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

   8. Taylor expanded in N around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. distribute-lft-in N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

   10. Applied rewrites 99.8%

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

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

   1. Initial program 91.7%

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

      1. lift-+.f64 N/A

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

      2. diff-log N/A

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

      3. lower-log.f64 N/A

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

      4. lower-/.f64 93.9

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

   4. Applied rewrites 93.9%

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

4. Final simplification 99.4%

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

Alternative 6: 96.7% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 24.2%

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

3. Taylor expanded in N around inf

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

5. Applied rewrites 96.5%

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

   1. lift-/.f64 N/A

      \[\leadsto \frac{1 + \frac{\frac{-1}{2} - \frac{\color{blue}{\frac{\frac{1}{4}}{N}} + \frac{-1}{3}}{N}}{N}}{N} \]

   2. lift-+.f64 N/A

      \[\leadsto \frac{1 + \frac{\frac{-1}{2} - \frac{\color{blue}{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}}{N}}{N}}{N} \]

   3. lift-/.f64 N/A

      \[\leadsto \frac{1 + \frac{\frac{-1}{2} - \color{blue}{\frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}}{N}}{N} \]

   4. lift--.f64 N/A

      \[\leadsto \frac{1 + \frac{\color{blue}{\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}}{N}}{N} \]

   5. lift-/.f64 N/A

      \[\leadsto \frac{1 + \color{blue}{\frac{\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}{N}}}{N} \]

   6. lift-+.f64 N/A

      \[\leadsto \frac{\color{blue}{1 + \frac{\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}{N}}}{N} \]

   7. clear-num N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{N}{1 + \frac{\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}{N}}}} \]

   8. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{N}{1 + \frac{\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}{N}}}} \]

   9. lower-/.f64 96.5

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

7. Applied rewrites 96.5%

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

8. Taylor expanded in N around -inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. sub-neg N/A

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

   4. metadata-eval N/A

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

   5. distribute-lft-in N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

10. Applied rewrites 97.0%

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

11. Final simplification 97.0%

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

Alternative 7: 96.7% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 24.2%

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

3. Taylor expanded in N around inf

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

5. Applied rewrites 96.5%

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

   1. lift-/.f64 N/A

      \[\leadsto \frac{1 + \frac{\frac{-1}{2} - \frac{\color{blue}{\frac{\frac{1}{4}}{N}} + \frac{-1}{3}}{N}}{N}}{N} \]

   2. lift-+.f64 N/A

      \[\leadsto \frac{1 + \frac{\frac{-1}{2} - \frac{\color{blue}{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}}{N}}{N}}{N} \]

   3. lift-/.f64 N/A

      \[\leadsto \frac{1 + \frac{\frac{-1}{2} - \color{blue}{\frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}}{N}}{N} \]

   4. lift--.f64 N/A

      \[\leadsto \frac{1 + \frac{\color{blue}{\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}}{N}}{N} \]

   5. lift-/.f64 N/A

      \[\leadsto \frac{1 + \color{blue}{\frac{\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}{N}}}{N} \]

   6. lift-+.f64 N/A

      \[\leadsto \frac{\color{blue}{1 + \frac{\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}{N}}}{N} \]

   7. clear-num N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{N}{1 + \frac{\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}{N}}}} \]

   8. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{N}{1 + \frac{\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}{N}}}} \]

   9. lower-/.f64 96.5

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

7. Applied rewrites 96.5%

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

8. Taylor expanded in N around -inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. sub-neg N/A

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

   4. metadata-eval N/A

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

   5. distribute-lft-in N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

10. Applied rewrites 97.0%

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

11. Taylor expanded in N around 0

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

    1. lower-/.f64 N/A

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

    2. sub-neg N/A

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

    3. metadata-eval N/A

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

    4. lower-fma.f64 N/A

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

    5. +-commutative N/A

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

    6. *-commutative N/A

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

    7. lower-fma.f64 N/A

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

    8. cube-mult N/A

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

    9. unpow2 N/A

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

    10. lower-*.f64 N/A

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

    11. unpow2 N/A

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

    12. lower-*.f64 97.0

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

13. Applied rewrites 97.0%

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

14. Final simplification 97.0%

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

Alternative 8: 96.5% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 24.2%

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

3. Taylor expanded in N around inf

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

5. Applied rewrites 96.5%

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

   1. lift-/.f64 N/A

      \[\leadsto \frac{1 + \frac{\frac{-1}{2} - \frac{\color{blue}{\frac{\frac{1}{4}}{N}} + \frac{-1}{3}}{N}}{N}}{N} \]

   2. lift-+.f64 N/A

      \[\leadsto \frac{1 + \frac{\frac{-1}{2} - \frac{\color{blue}{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}}{N}}{N}}{N} \]

   3. lift-/.f64 N/A

      \[\leadsto \frac{1 + \frac{\frac{-1}{2} - \color{blue}{\frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}}{N}}{N} \]

   4. lift--.f64 N/A

      \[\leadsto \frac{1 + \frac{\color{blue}{\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}}{N}}{N} \]

   5. lift-/.f64 N/A

      \[\leadsto \frac{1 + \color{blue}{\frac{\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}{N}}}{N} \]

   6. lift-+.f64 N/A

      \[\leadsto \frac{\color{blue}{1 + \frac{\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}{N}}}{N} \]

   7. clear-num N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{N}{1 + \frac{\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}{N}}}} \]

   8. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{N}{1 + \frac{\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}{N}}}} \]

   9. lower-/.f64 96.5

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

7. Applied rewrites 96.5%

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

8. Taylor expanded in N around -inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. sub-neg N/A

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

   4. metadata-eval N/A

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

   5. distribute-lft-in N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

10. Applied rewrites 97.0%

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

11. Taylor expanded in N around 0

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

    1. lower-/.f64 N/A

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

    2. +-commutative N/A

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

    3. lower-fma.f64 N/A

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

    4. sub-neg N/A

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

    5. metadata-eval N/A

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

    6. lower-fma.f64 N/A

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

    7. +-commutative N/A

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

    8. lower-+.f64 N/A

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

    9. unpow2 N/A

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

    10. lower-*.f64 96.7

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

13. Applied rewrites 96.7%

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

Alternative 9: 95.8% accurate, 4.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 24.2%

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

3. Taylor expanded in N around inf

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

5. Applied rewrites 96.5%

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

6. Taylor expanded in N around 0

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

   1. lower-/.f64 N/A

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

   2. sub-neg N/A

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

   3. metadata-eval N/A

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

   4. lower-fma.f64 N/A

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

   5. +-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. sub-neg N/A

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

   8. metadata-eval N/A

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

   9. lower-+.f64 N/A

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

   10. lower-pow.f64 96.0

       \[\leadsto \frac{\mathsf{fma}\left(N, \mathsf{fma}\left(N, N + -0.5, 0.3333333333333333\right), -0.25\right)}{\color{blue}{{N}^{4}}} \]

8. Applied rewrites 96.0%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(N, \mathsf{fma}\left(N, N + -0.5, 0.3333333333333333\right), -0.25\right)}{{N}^{4}}} \]
9. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. lift-fma.f64 N/A

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

   4. lift-pow.f64 N/A

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

   5. lift-/.f64 96.0

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(N, \mathsf{fma}\left(N, N + -0.5, 0.3333333333333333\right), -0.25\right)}{{N}^{4}}} \]

   6. lift-pow.f64 N/A

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

   7. metadata-eval N/A

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

   8. pow-plus N/A

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

   9. cube-unmult N/A

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

   10. lift-*.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 96.1

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

10. Applied rewrites 96.1%

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

Alternative 10: 94.9% accurate, 5.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 24.2%

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

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

Alternative 11: 93.0% accurate, 7.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 24.2%

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

3. Taylor expanded in N around inf

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

5. Applied rewrites 96.5%

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

   1. lift-/.f64 N/A

      \[\leadsto \frac{1 + \frac{\frac{-1}{2} - \frac{\color{blue}{\frac{\frac{1}{4}}{N}} + \frac{-1}{3}}{N}}{N}}{N} \]

   2. lift-+.f64 N/A

      \[\leadsto \frac{1 + \frac{\frac{-1}{2} - \frac{\color{blue}{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}}{N}}{N}}{N} \]

   3. lift-/.f64 N/A

      \[\leadsto \frac{1 + \frac{\frac{-1}{2} - \color{blue}{\frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}}{N}}{N} \]

   4. lift--.f64 N/A

      \[\leadsto \frac{1 + \frac{\color{blue}{\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}}{N}}{N} \]

   5. lift-/.f64 N/A

      \[\leadsto \frac{1 + \color{blue}{\frac{\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}{N}}}{N} \]

   6. lift-+.f64 N/A

      \[\leadsto \frac{\color{blue}{1 + \frac{\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}{N}}}{N} \]

   7. clear-num N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{N}{1 + \frac{\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}{N}}}} \]

   8. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{N}{1 + \frac{\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}{N}}}} \]

   9. lower-/.f64 96.5

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

7. Applied rewrites 96.5%

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

8. Taylor expanded in N around inf

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. *-rgt-identity N/A

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

   4. lower-fma.f64 N/A

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

   5. associate-*r/ N/A

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

   6. metadata-eval N/A

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

   7. lower-/.f64 93.1

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

10. Applied rewrites 93.1%

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

Alternative 12: 92.3% 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 24.2%

   \[\log \left(N + 1\right) - \log N \]
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 92.5

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

5. Applied rewrites 92.5%

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

Alternative 13: 92.0% accurate, 10.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 24.2%

   \[\log \left(N + 1\right) - \log N \]
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 92.5

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

5. Applied rewrites 92.5%

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

6. Taylor expanded in N around 0

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

   1. lower-/.f64 N/A

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

   2. sub-neg N/A

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

   3. metadata-eval N/A

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

   4. lower-+.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 92.2

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

8. Applied rewrites 92.2%

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

Alternative 14: 84.2% accurate, 17.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(N):
	return 1.0 / N

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 24.2%

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

3. Taylor expanded in N around inf

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

   1. lower-/.f64 84.3

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

5. Applied rewrites 84.3%

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

Alternative 15: 7.4% accurate, 18.8× speedup

Math

\[\begin{array}{l} \\ \left(N \cdot N\right) \cdot 24 \end{array} \]

FPCore

(FPCore (N) :precision binary64 (* (* N N) 24.0))

C

double code(double N) {
	return (N * N) * 24.0;
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    code = (n * n) * 24.0d0
end function

Java

public static double code(double N) {
	return (N * N) * 24.0;
}

Python

def code(N):
	return (N * N) * 24.0

Julia

function code(N)
	return Float64(Float64(N * N) * 24.0)
end

MATLAB

function tmp = code(N)
	tmp = (N * N) * 24.0;
end

Wolfram

code[N_] := N[(N[(N * N), $MachinePrecision] * 24.0), $MachinePrecision]

Derivation

1. Initial program 24.2%

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

3. Taylor expanded in N around inf

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

5. Applied rewrites 96.5%

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

   1. lift-/.f64 N/A

      \[\leadsto \frac{1 + \frac{\frac{-1}{2} - \frac{\color{blue}{\frac{\frac{1}{4}}{N}} + \frac{-1}{3}}{N}}{N}}{N} \]

   2. lift-+.f64 N/A

      \[\leadsto \frac{1 + \frac{\frac{-1}{2} - \frac{\color{blue}{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}}{N}}{N}}{N} \]

   3. lift-/.f64 N/A

      \[\leadsto \frac{1 + \frac{\frac{-1}{2} - \color{blue}{\frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}}{N}}{N} \]

   4. lift--.f64 N/A

      \[\leadsto \frac{1 + \frac{\color{blue}{\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}}{N}}{N} \]

   5. lift-/.f64 N/A

      \[\leadsto \frac{1 + \color{blue}{\frac{\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}{N}}}{N} \]

   6. lift-+.f64 N/A

      \[\leadsto \frac{\color{blue}{1 + \frac{\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}{N}}}{N} \]

   7. clear-num N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{N}{1 + \frac{\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}{N}}}} \]

   8. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{N}{1 + \frac{\frac{-1}{2} - \frac{\frac{\frac{1}{4}}{N} + \frac{-1}{3}}{N}}{N}}}} \]

   9. lower-/.f64 96.5

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

7. Applied rewrites 96.5%

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

8. Taylor expanded in N around -inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. sub-neg N/A

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

   4. metadata-eval N/A

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

   5. distribute-lft-in N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

10. Applied rewrites 97.0%

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

11. Taylor expanded in N around 0

    \[\leadsto \color{blue}{24 \cdot {N}^{2}} \]
12. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \color{blue}{{N}^{2} \cdot 24} \]

    2. lower-*.f64 N/A

       \[\leadsto \color{blue}{{N}^{2} \cdot 24} \]

    3. unpow2 N/A

       \[\leadsto \color{blue}{\left(N \cdot N\right)} \cdot 24 \]

    4. lower-*.f64 7.4

       \[\leadsto \color{blue}{\left(N \cdot N\right)} \cdot 24 \]

13. Applied rewrites 7.4%

    \[\leadsto \color{blue}{\left(N \cdot N\right) \cdot 24} \]
14. 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: 26.8% 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 2024219 
(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)))