2log (problem 3.3.6)

Percentage Accurate: 24.0% → 99.5%

Time: 8.2s

Alternatives: 7

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: 24.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[N_] := If[LessEqual[N[(N[Log[N[(1.0 + N), $MachinePrecision]], $MachinePrecision] - N[Log[N], $MachinePrecision]), $MachinePrecision], 0.001], N[(1.0 / N[(N[(N[(-0.08333333333333333 * N + 0.041666666666666664), $MachinePrecision] / N[(N * N), $MachinePrecision]), $MachinePrecision] + N[(0.5 + N), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(-1.0 / N[Log[N[(N / N[(1.0 + N), $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 19.9%

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

   3. Taylor expanded in N around inf

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

   5. Applied rewrites 99.6%

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

   7. Applied rewrites 99.7%

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

   8. Taylor expanded in N around inf

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

   10. Applied rewrites 99.8%

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

   11. Taylor expanded in N around 0

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

   13. Applied rewrites 99.8%

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

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

   1. Initial program 92.7%

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

      1. 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. div-inv N/A

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

      6. lift-+.f64 N/A

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

      7. flip3-+ N/A

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

      8. clear-num N/A

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

      9. frac-times N/A

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

      10. metadata-eval N/A

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

      11. log-rec N/A

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

      12. lower-neg.f64 N/A

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

      13. lower-log.f64 N/A

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

      14. lower-*.f64 N/A

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

   4. Applied rewrites 94.9%

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

   6. Applied rewrites 95.3%

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

4. Final simplification 99.5%

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

Alternative 2: 99.5% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 19.9%

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

   3. Taylor expanded in N around inf

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

   5. Applied rewrites 99.6%

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

   7. Applied rewrites 99.7%

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

   8. Taylor expanded in N around inf

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

   10. Applied rewrites 99.8%

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

   11. Taylor expanded in N around 0

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

   13. Applied rewrites 99.8%

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

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

   1. Initial program 92.7%

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

      1. 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

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 95.2

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

   4. Applied rewrites 95.2%

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

4. Final simplification 99.5%

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

Alternative 3: 99.5% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 19.9%

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

   3. Taylor expanded in N around inf

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

   5. Applied rewrites 99.6%

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

   7. Applied rewrites 99.7%

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

   8. Taylor expanded in N around inf

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

   10. Applied rewrites 99.8%

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

   11. Taylor expanded in N around 0

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

   13. Applied rewrites 99.8%

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

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

   1. Initial program 92.7%

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

      1. lift--.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. diff-log N/A

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

      5. lower-log.f64 N/A

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

      6. lower-/.f64 94.7

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 94.7

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

   4. Applied rewrites 94.7%

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

4. Final simplification 99.4%

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

Alternative 4: 96.7% accurate, 5.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 25.6%

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

3. Taylor expanded in N around inf

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

5. Applied rewrites 95.9%

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

7. Applied rewrites 96.0%

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

8. Taylor expanded in N around inf

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

10. Applied rewrites 96.4%

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

11. Taylor expanded in N around 0

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

13. Applied rewrites 96.4%

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

14. Final simplification 96.4%

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

Alternative 5: 95.4% accurate, 7.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 25.6%

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

3. Taylor expanded in N around inf

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

5. Applied rewrites 95.9%

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

7. Applied rewrites 96.0%

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

8. Taylor expanded in N around inf

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

10. Applied rewrites 96.4%

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

11. Taylor expanded in N around inf

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

13. Applied rewrites 95.1%

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

14. Final simplification 95.1%

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

Alternative 6: 92.9% accurate, 13.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 25.6%

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

3. Taylor expanded in N around inf

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

5. Applied rewrites 95.9%

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

7. Applied rewrites 96.0%

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

8. Taylor expanded in N around inf

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

10. Applied rewrites 92.5%

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

Alternative 7: 84.3% 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 25.6%

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

3. Taylor expanded in N around inf

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

   1. lower-/.f64 83.3

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

5. Applied rewrites 83.3%

   \[\leadsto \color{blue}{\frac{1}{N}} \]
6. 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: 26.6% 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 2024235 
(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)))