2log (problem 3.3.6)

Percentage Accurate: 24.3% → 99.5%

Time: 11.0s

Alternatives: 13

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: 24.3% 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

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

FPCore

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

C

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

Julia

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

Wolfram

code[N_] := Block[{t$95$0 = N[(N[(N * N), $MachinePrecision] - N), $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[(-0.5 + N[(N[(0.08333333333333333 + N[(-0.041666666666666664 / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision] * (-N) + 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[t$95$0], $MachinePrecision] - N[Log[N[(N[(N * N + N), $MachinePrecision] * t$95$0), $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 99.8%

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

   7. Applied rewrites 99.9%

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

   8. Taylor expanded in N around -inf

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

   10. Applied rewrites 99.8%

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

   12. Applied rewrites 99.9%

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

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

   1. Initial program 91.4%

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

   4. Applied rewrites 94.9%

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

      1. lift-log.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. flip-+ N/A

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

      5. log-div N/A

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

      6. lower--.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. difference-of-squares N/A

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

      9. lift-*.f64 N/A

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

      10. lift-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-log.f64 N/A

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

      14. lower--.f64 95.0

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

   6. Applied rewrites 95.0%

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

4. Final simplification 99.4%

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

Alternative 2: 99.5% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 18.8%

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

   3. Taylor expanded in N around inf

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

   5. Applied rewrites 99.7%

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

   7. Applied rewrites 99.8%

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

   8. Taylor expanded in N around -inf

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

   10. Applied rewrites 99.7%

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

   12. Applied rewrites 99.9%

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

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

   1. Initial program 92.3%

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

   4. Applied rewrites 95.1%

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

      1. lift-log.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. distribute-lft1-in N/A

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

      4. flip-+ N/A

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

      5. associate-*l/ N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

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

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

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

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

      10. lift-fma.f64 N/A

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

      11. log-div N/A

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

      12. sub-neg N/A

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

      13. metadata-eval N/A

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

      14. lift-+.f64 N/A

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

      15. lift-log.f64 N/A

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

      16. lower--.f64 N/A

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

   6. Applied rewrites 95.0%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 0.001:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\frac{-0.5 + \frac{0.08333333333333333 + \frac{-0.041666666666666664}{N}}{N}}{N}, -N, N\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\log \left(\frac{N}{N + 1}\right) \cdot \left(\log \left(N \cdot \mathsf{fma}\left(N, N, -1\right)\right) - \log \left(N + -1\right)\right)\right) \cdot \frac{-1}{\log \left(\mathsf{fma}\left(N, N, N\right)\right)}\\ \end{array} \]
5. 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: 27.0% 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: 96.2% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024226 
(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)))