2log (problem 3.3.6)

Percentage Accurate: 24.3% → 99.5%

Time: 9.6s

Alternatives: 9

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, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\frac{N}{N + 1}\right)\\ t_1 := \log \left(N \cdot \left(N + 1\right)\right)\\ \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 0.001:\\ \;\;\;\;\frac{1}{N - \left(-0.5 + \frac{\mathsf{fma}\left(N, 0.08333333333333333, -0.041666666666666664\right)}{N \cdot N}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{t\_0}^{3}}{0 \cdot t\_0 - {\left(\frac{1}{\frac{t\_1}{t\_0 \cdot t\_1}}\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (let* ((t_0 (log (/ N (+ N 1.0)))) (t_1 (log (* N (+ N 1.0)))))
   (if (<= (- (log (+ N 1.0)) (log N)) 0.001)
     (/
      1.0
      (-
       N
       (+ -0.5 (/ (fma N 0.08333333333333333 -0.041666666666666664) (* N N)))))
     (/ (pow t_0 3.0) (- (* 0.0 t_0) (pow (/ 1.0 (/ t_1 (* t_0 t_1))) 2.0))))))

C

double code(double N) {
	double t_0 = log((N / (N + 1.0)));
	double t_1 = log((N * (N + 1.0)));
	double tmp;
	if ((log((N + 1.0)) - log(N)) <= 0.001) {
		tmp = 1.0 / (N - (-0.5 + (fma(N, 0.08333333333333333, -0.041666666666666664) / (N * N))));
	} else {
		tmp = pow(t_0, 3.0) / ((0.0 * t_0) - pow((1.0 / (t_1 / (t_0 * t_1))), 2.0));
	}
	return tmp;
}

Julia

function code(N)
	t_0 = log(Float64(N / Float64(N + 1.0)))
	t_1 = log(Float64(N * Float64(N + 1.0)))
	tmp = 0.0
	if (Float64(log(Float64(N + 1.0)) - log(N)) <= 0.001)
		tmp = Float64(1.0 / Float64(N - Float64(-0.5 + Float64(fma(N, 0.08333333333333333, -0.041666666666666664) / Float64(N * N)))));
	else
		tmp = Float64((t_0 ^ 3.0) / Float64(Float64(0.0 * t_0) - (Float64(1.0 / Float64(t_1 / Float64(t_0 * t_1))) ^ 2.0)));
	end
	return tmp
end

Wolfram

code[N_] := Block[{t$95$0 = N[Log[N[(N / N[(N + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Log[N[(N * N[(N + 1.0), $MachinePrecision]), $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[(-0.5 + N[(N[(N * 0.08333333333333333 + -0.041666666666666664), $MachinePrecision] / N[(N * N), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[t$95$0, 3.0], $MachinePrecision] / N[(N[(0.0 * t$95$0), $MachinePrecision] - N[Power[N[(1.0 / N[(t$95$1 / N[(t$95$0 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $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 19.2%

      \[\log \left(N + 1\right) - \log N \]
   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 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 99.7%

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

   8. Taylor expanded in N around -inf

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

   10. Applied rewrites 99.8%

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

   11. Taylor expanded in N around inf

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

   13. Applied rewrites 99.8%

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

   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
   3. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. diff-log N/A

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

      5. clear-num N/A

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

      6. neg-log N/A

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

      7. diff-log N/A

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

      8. lift-log.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lift-log.f64 N/A

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

      12. lift-log.f64 N/A

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

      13. diff-log N/A

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

      14. lower-log.f64 N/A

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

      15. lower-/.f64 95.2

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

   4. Applied rewrites 95.2%

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

      1. lift-neg.f64 N/A

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

      2. neg-sub0 N/A

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

      3. metadata-eval N/A

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

      4. flip3-- N/A

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

      5. lower-/.f64 N/A

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

      6. metadata-eval N/A

         \[\leadsto \frac{{\color{blue}{0}}^{3} - {\log \left(\frac{N}{N + 1}\right)}^{3}}{\log 1 \cdot \log 1 + \left(\log \left(\frac{N}{N + 1}\right) \cdot \log \left(\frac{N}{N + 1}\right) + \log 1 \cdot \log \left(\frac{N}{N + 1}\right)\right)} \]

      7. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{0} - {\log \left(\frac{N}{N + 1}\right)}^{3}}{\log 1 \cdot \log 1 + \left(\log \left(\frac{N}{N + 1}\right) \cdot \log \left(\frac{N}{N + 1}\right) + \log 1 \cdot \log \left(\frac{N}{N + 1}\right)\right)} \]

      8. metadata-eval N/A

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

      9. lower--.f64 N/A

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

      10. metadata-eval N/A

          \[\leadsto \frac{\color{blue}{0} - {\log \left(\frac{N}{N + 1}\right)}^{3}}{\log 1 \cdot \log 1 + \left(\log \left(\frac{N}{N + 1}\right) \cdot \log \left(\frac{N}{N + 1}\right) + \log 1 \cdot \log \left(\frac{N}{N + 1}\right)\right)} \]

      11. lower-pow.f64 N/A

          \[\leadsto \frac{0 - \color{blue}{{\log \left(\frac{N}{N + 1}\right)}^{3}}}{\log 1 \cdot \log 1 + \left(\log \left(\frac{N}{N + 1}\right) \cdot \log \left(\frac{N}{N + 1}\right) + \log 1 \cdot \log \left(\frac{N}{N + 1}\right)\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{0 - {\log \left(\frac{N}{N + 1}\right)}^{3}}{\color{blue}{0} \cdot \log 1 + \left(\log \left(\frac{N}{N + 1}\right) \cdot \log \left(\frac{N}{N + 1}\right) + \log 1 \cdot \log \left(\frac{N}{N + 1}\right)\right)} \]

      13. metadata-eval N/A

          \[\leadsto \frac{0 - {\log \left(\frac{N}{N + 1}\right)}^{3}}{0 \cdot \color{blue}{0} + \left(\log \left(\frac{N}{N + 1}\right) \cdot \log \left(\frac{N}{N + 1}\right) + \log 1 \cdot \log \left(\frac{N}{N + 1}\right)\right)} \]

      14. metadata-eval N/A

          \[\leadsto \frac{0 - {\log \left(\frac{N}{N + 1}\right)}^{3}}{\color{blue}{0} + \left(\log \left(\frac{N}{N + 1}\right) \cdot \log \left(\frac{N}{N + 1}\right) + \log 1 \cdot \log \left(\frac{N}{N + 1}\right)\right)} \]

      15. metadata-eval N/A

          \[\leadsto \frac{0 - {\log \left(\frac{N}{N + 1}\right)}^{3}}{\color{blue}{\log 1} + \left(\log \left(\frac{N}{N + 1}\right) \cdot \log \left(\frac{N}{N + 1}\right) + \log 1 \cdot \log \left(\frac{N}{N + 1}\right)\right)} \]

      16. lower-+.f64 N/A

          \[\leadsto \frac{0 - {\log \left(\frac{N}{N + 1}\right)}^{3}}{\color{blue}{\log 1 + \left(\log \left(\frac{N}{N + 1}\right) \cdot \log \left(\frac{N}{N + 1}\right) + \log 1 \cdot \log \left(\frac{N}{N + 1}\right)\right)}} \]

   6. Applied rewrites 95.2%

      \[\leadsto \color{blue}{\frac{0 - {\log \left(\frac{N}{N + 1}\right)}^{3}}{0 + \left({\log \left(\frac{N}{N + 1}\right)}^{2} + 0 \cdot \log \left(\frac{N}{N + 1}\right)\right)}} \]

   8. Applied rewrites 95.3%

      \[\leadsto \frac{0 - {\log \left(\frac{N}{N + 1}\right)}^{3}}{0 + \left({\color{blue}{\left(\frac{1}{\frac{\log \left(\left(1 + N\right) \cdot N\right)}{\log \left(\left(1 + N\right) \cdot N\right) \cdot \log \left(\frac{N}{1 + N}\right)}}\right)}}^{2} + 0 \cdot \log \left(\frac{N}{N + 1}\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 0.001:\\ \;\;\;\;\frac{1}{N - \left(-0.5 + \frac{\mathsf{fma}\left(N, 0.08333333333333333, -0.041666666666666664\right)}{N \cdot N}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\log \left(\frac{N}{N + 1}\right)}^{3}}{0 \cdot \log \left(\frac{N}{N + 1}\right) - {\left(\frac{1}{\frac{\log \left(N \cdot \left(N + 1\right)\right)}{\log \left(\frac{N}{N + 1}\right) \cdot \log \left(N \cdot \left(N + 1\right)\right)}}\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\frac{N}{N + 1}\right)\\ \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 0.001:\\ \;\;\;\;\frac{1}{N - \left(-0.5 + \frac{\mathsf{fma}\left(N, 0.08333333333333333, -0.041666666666666664\right)}{N \cdot N}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{-1}{\frac{-1}{t\_0}}\right)}^{3}}{0 \cdot t\_0 - {t\_0}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (let* ((t_0 (log (/ N (+ N 1.0)))))
   (if (<= (- (log (+ N 1.0)) (log N)) 0.001)
     (/
      1.0
      (-
       N
       (+ -0.5 (/ (fma N 0.08333333333333333 -0.041666666666666664) (* N N)))))
     (/ (pow (/ -1.0 (/ -1.0 t_0)) 3.0) (- (* 0.0 t_0) (pow t_0 2.0))))))

C

double code(double N) {
	double t_0 = log((N / (N + 1.0)));
	double tmp;
	if ((log((N + 1.0)) - log(N)) <= 0.001) {
		tmp = 1.0 / (N - (-0.5 + (fma(N, 0.08333333333333333, -0.041666666666666664) / (N * N))));
	} else {
		tmp = pow((-1.0 / (-1.0 / t_0)), 3.0) / ((0.0 * t_0) - pow(t_0, 2.0));
	}
	return tmp;
}

Julia

function code(N)
	t_0 = log(Float64(N / Float64(N + 1.0)))
	tmp = 0.0
	if (Float64(log(Float64(N + 1.0)) - log(N)) <= 0.001)
		tmp = Float64(1.0 / Float64(N - Float64(-0.5 + Float64(fma(N, 0.08333333333333333, -0.041666666666666664) / Float64(N * N)))));
	else
		tmp = Float64((Float64(-1.0 / Float64(-1.0 / t_0)) ^ 3.0) / Float64(Float64(0.0 * t_0) - (t_0 ^ 2.0)));
	end
	return tmp
end

Wolfram

code[N_] := Block[{t$95$0 = N[Log[N[(N / N[(N + 1.0), $MachinePrecision]), $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[(-0.5 + N[(N[(N * 0.08333333333333333 + -0.041666666666666664), $MachinePrecision] / N[(N * N), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(-1.0 / N[(-1.0 / t$95$0), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] / N[(N[(0.0 * t$95$0), $MachinePrecision] - N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 18.5%

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

   3. Taylor expanded in N around inf

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

   5. 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

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

   10. Applied rewrites 99.9%

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

   11. Taylor expanded in N around inf

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

   13. Applied rewrites 99.9%

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

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

   1. Initial program 92.2%

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

      1. lift--.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. diff-log N/A

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

      5. clear-num N/A

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

      6. neg-log N/A

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

      7. diff-log N/A

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

      8. lift-log.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lift-log.f64 N/A

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

      12. lift-log.f64 N/A

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

      13. diff-log N/A

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

      14. lower-log.f64 N/A

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

      15. lower-/.f64 95.2

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

   4. Applied rewrites 95.2%

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

      1. lift-neg.f64 N/A

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

      2. neg-sub0 N/A

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

      3. metadata-eval N/A

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

      4. flip3-- N/A

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

      5. lower-/.f64 N/A

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

      6. metadata-eval N/A

         \[\leadsto \frac{{\color{blue}{0}}^{3} - {\log \left(\frac{N}{N + 1}\right)}^{3}}{\log 1 \cdot \log 1 + \left(\log \left(\frac{N}{N + 1}\right) \cdot \log \left(\frac{N}{N + 1}\right) + \log 1 \cdot \log \left(\frac{N}{N + 1}\right)\right)} \]

      7. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{0} - {\log \left(\frac{N}{N + 1}\right)}^{3}}{\log 1 \cdot \log 1 + \left(\log \left(\frac{N}{N + 1}\right) \cdot \log \left(\frac{N}{N + 1}\right) + \log 1 \cdot \log \left(\frac{N}{N + 1}\right)\right)} \]

      8. metadata-eval N/A

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

      9. lower--.f64 N/A

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

      10. metadata-eval N/A

          \[\leadsto \frac{\color{blue}{0} - {\log \left(\frac{N}{N + 1}\right)}^{3}}{\log 1 \cdot \log 1 + \left(\log \left(\frac{N}{N + 1}\right) \cdot \log \left(\frac{N}{N + 1}\right) + \log 1 \cdot \log \left(\frac{N}{N + 1}\right)\right)} \]

      11. lower-pow.f64 N/A

          \[\leadsto \frac{0 - \color{blue}{{\log \left(\frac{N}{N + 1}\right)}^{3}}}{\log 1 \cdot \log 1 + \left(\log \left(\frac{N}{N + 1}\right) \cdot \log \left(\frac{N}{N + 1}\right) + \log 1 \cdot \log \left(\frac{N}{N + 1}\right)\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{0 - {\log \left(\frac{N}{N + 1}\right)}^{3}}{\color{blue}{0} \cdot \log 1 + \left(\log \left(\frac{N}{N + 1}\right) \cdot \log \left(\frac{N}{N + 1}\right) + \log 1 \cdot \log \left(\frac{N}{N + 1}\right)\right)} \]

      13. metadata-eval N/A

          \[\leadsto \frac{0 - {\log \left(\frac{N}{N + 1}\right)}^{3}}{0 \cdot \color{blue}{0} + \left(\log \left(\frac{N}{N + 1}\right) \cdot \log \left(\frac{N}{N + 1}\right) + \log 1 \cdot \log \left(\frac{N}{N + 1}\right)\right)} \]

      14. metadata-eval N/A

          \[\leadsto \frac{0 - {\log \left(\frac{N}{N + 1}\right)}^{3}}{\color{blue}{0} + \left(\log \left(\frac{N}{N + 1}\right) \cdot \log \left(\frac{N}{N + 1}\right) + \log 1 \cdot \log \left(\frac{N}{N + 1}\right)\right)} \]

      15. metadata-eval N/A

          \[\leadsto \frac{0 - {\log \left(\frac{N}{N + 1}\right)}^{3}}{\color{blue}{\log 1} + \left(\log \left(\frac{N}{N + 1}\right) \cdot \log \left(\frac{N}{N + 1}\right) + \log 1 \cdot \log \left(\frac{N}{N + 1}\right)\right)} \]

      16. lower-+.f64 N/A

          \[\leadsto \frac{0 - {\log \left(\frac{N}{N + 1}\right)}^{3}}{\color{blue}{\log 1 + \left(\log \left(\frac{N}{N + 1}\right) \cdot \log \left(\frac{N}{N + 1}\right) + \log 1 \cdot \log \left(\frac{N}{N + 1}\right)\right)}} \]

   6. Applied rewrites 95.2%

      \[\leadsto \color{blue}{\frac{0 - {\log \left(\frac{N}{N + 1}\right)}^{3}}{0 + \left({\log \left(\frac{N}{N + 1}\right)}^{2} + 0 \cdot \log \left(\frac{N}{N + 1}\right)\right)}} \]
   7. Step-by-step derivation

      1. unpow1 N/A

         \[\leadsto \frac{0 - {\color{blue}{\left({\log \left(\frac{N}{N + 1}\right)}^{1}\right)}}^{3}}{0 + \left({\log \left(\frac{N}{N + 1}\right)}^{2} + 0 \cdot \log \left(\frac{N}{N + 1}\right)\right)} \]

      2. metadata-eval N/A

         \[\leadsto \frac{0 - {\left({\log \left(\frac{N}{N + 1}\right)}^{\color{blue}{\left(3 - 2\right)}}\right)}^{3}}{0 + \left({\log \left(\frac{N}{N + 1}\right)}^{2} + 0 \cdot \log \left(\frac{N}{N + 1}\right)\right)} \]

      3. pow-div N/A

         \[\leadsto \frac{0 - {\color{blue}{\left(\frac{{\log \left(\frac{N}{N + 1}\right)}^{3}}{{\log \left(\frac{N}{N + 1}\right)}^{2}}\right)}}^{3}}{0 + \left({\log \left(\frac{N}{N + 1}\right)}^{2} + 0 \cdot \log \left(\frac{N}{N + 1}\right)\right)} \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{0 - {\left(\frac{\color{blue}{{\log \left(\frac{N}{N + 1}\right)}^{3}}}{{\log \left(\frac{N}{N + 1}\right)}^{2}}\right)}^{3}}{0 + \left({\log \left(\frac{N}{N + 1}\right)}^{2} + 0 \cdot \log \left(\frac{N}{N + 1}\right)\right)} \]

      5. lift-pow.f64 N/A

         \[\leadsto \frac{0 - {\left(\frac{{\log \left(\frac{N}{N + 1}\right)}^{3}}{\color{blue}{{\log \left(\frac{N}{N + 1}\right)}^{2}}}\right)}^{3}}{0 + \left({\log \left(\frac{N}{N + 1}\right)}^{2} + 0 \cdot \log \left(\frac{N}{N + 1}\right)\right)} \]

      6. +-lft-identity N/A

         \[\leadsto \frac{0 - {\left(\frac{{\log \left(\frac{N}{N + 1}\right)}^{3}}{\color{blue}{0 + {\log \left(\frac{N}{N + 1}\right)}^{2}}}\right)}^{3}}{0 + \left({\log \left(\frac{N}{N + 1}\right)}^{2} + 0 \cdot \log \left(\frac{N}{N + 1}\right)\right)} \]

      7. +-lft-identity N/A

         \[\leadsto \frac{0 - {\left(\frac{{\log \left(\frac{N}{N + 1}\right)}^{3}}{0 + \color{blue}{\left(0 + {\log \left(\frac{N}{N + 1}\right)}^{2}\right)}}\right)}^{3}}{0 + \left({\log \left(\frac{N}{N + 1}\right)}^{2} + 0 \cdot \log \left(\frac{N}{N + 1}\right)\right)} \]

      8. +-commutative N/A

         \[\leadsto \frac{0 - {\left(\frac{{\log \left(\frac{N}{N + 1}\right)}^{3}}{0 + \color{blue}{\left({\log \left(\frac{N}{N + 1}\right)}^{2} + 0\right)}}\right)}^{3}}{0 + \left({\log \left(\frac{N}{N + 1}\right)}^{2} + 0 \cdot \log \left(\frac{N}{N + 1}\right)\right)} \]

      9. mul0-lft N/A

         \[\leadsto \frac{0 - {\left(\frac{{\log \left(\frac{N}{N + 1}\right)}^{3}}{0 + \left({\log \left(\frac{N}{N + 1}\right)}^{2} + \color{blue}{0 \cdot \log \left(\frac{N}{N + 1}\right)}\right)}\right)}^{3}}{0 + \left({\log \left(\frac{N}{N + 1}\right)}^{2} + 0 \cdot \log \left(\frac{N}{N + 1}\right)\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{0 - {\left(\frac{{\log \left(\frac{N}{N + 1}\right)}^{3}}{0 + \left({\log \left(\frac{N}{N + 1}\right)}^{2} + \color{blue}{0 \cdot \log \left(\frac{N}{N + 1}\right)}\right)}\right)}^{3}}{0 + \left({\log \left(\frac{N}{N + 1}\right)}^{2} + 0 \cdot \log \left(\frac{N}{N + 1}\right)\right)} \]

      11. lift-+.f64 N/A

          \[\leadsto \frac{0 - {\left(\frac{{\log \left(\frac{N}{N + 1}\right)}^{3}}{0 + \color{blue}{\left({\log \left(\frac{N}{N + 1}\right)}^{2} + 0 \cdot \log \left(\frac{N}{N + 1}\right)\right)}}\right)}^{3}}{0 + \left({\log \left(\frac{N}{N + 1}\right)}^{2} + 0 \cdot \log \left(\frac{N}{N + 1}\right)\right)} \]

      12. lift-+.f64 N/A

          \[\leadsto \frac{0 - {\left(\frac{{\log \left(\frac{N}{N + 1}\right)}^{3}}{\color{blue}{0 + \left({\log \left(\frac{N}{N + 1}\right)}^{2} + 0 \cdot \log \left(\frac{N}{N + 1}\right)\right)}}\right)}^{3}}{0 + \left({\log \left(\frac{N}{N + 1}\right)}^{2} + 0 \cdot \log \left(\frac{N}{N + 1}\right)\right)} \]

      13. clear-num N/A

          \[\leadsto \frac{0 - {\color{blue}{\left(\frac{1}{\frac{0 + \left({\log \left(\frac{N}{N + 1}\right)}^{2} + 0 \cdot \log \left(\frac{N}{N + 1}\right)\right)}{{\log \left(\frac{N}{N + 1}\right)}^{3}}}\right)}}^{3}}{0 + \left({\log \left(\frac{N}{N + 1}\right)}^{2} + 0 \cdot \log \left(\frac{N}{N + 1}\right)\right)} \]

   8. Applied rewrites 95.2%

      \[\leadsto \frac{0 - {\color{blue}{\left(\frac{1}{\frac{1}{\log \left(\frac{N}{1 + N}\right)}}\right)}}^{3}}{0 + \left({\log \left(\frac{N}{N + 1}\right)}^{2} + 0 \cdot \log \left(\frac{N}{N + 1}\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 0.001:\\ \;\;\;\;\frac{1}{N - \left(-0.5 + \frac{\mathsf{fma}\left(N, 0.08333333333333333, -0.041666666666666664\right)}{N \cdot N}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{-1}{\frac{-1}{\log \left(\frac{N}{N + 1}\right)}}\right)}^{3}}{0 \cdot \log \left(\frac{N}{N + 1}\right) - {\log \left(\frac{N}{N + 1}\right)}^{2}}\\ \end{array} \]
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: 27.1% 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: 96.0% 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 2024228 
(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)))