2log (problem 3.3.6)

Percentage Accurate: 23.4% → 99.4%

Time: 8.3s

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: 23.4% 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: 99.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{0.5 - \frac{0.08333333333333333 - \frac{0.041666666666666664}{N}}{N}}{N}\\ \mathbf{if}\;\log \left(N - -1\right) - \log N \leq 0.002:\\ \;\;\;\;\frac{1}{\frac{\left({t\_0}^{2} - 1\right) \cdot N}{t\_0 + -1}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{N - -1}{N}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (let* ((t_0
         (/
          (- 0.5 (/ (- 0.08333333333333333 (/ 0.041666666666666664 N)) N))
          N)))
   (if (<= (- (log (- N -1.0)) (log N)) 0.002)
     (/ 1.0 (/ (* (- (pow t_0 2.0) 1.0) N) (+ t_0 -1.0)))
     (log (/ (- N -1.0) N)))))

C

double code(double N) {
	double t_0 = (0.5 - ((0.08333333333333333 - (0.041666666666666664 / N)) / N)) / N;
	double tmp;
	if ((log((N - -1.0)) - log(N)) <= 0.002) {
		tmp = 1.0 / (((pow(t_0, 2.0) - 1.0) * N) / (t_0 + -1.0));
	} else {
		tmp = log(((N - -1.0) / N));
	}
	return tmp;
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (0.5d0 - ((0.08333333333333333d0 - (0.041666666666666664d0 / n)) / n)) / n
    if ((log((n - (-1.0d0))) - log(n)) <= 0.002d0) then
        tmp = 1.0d0 / ((((t_0 ** 2.0d0) - 1.0d0) * n) / (t_0 + (-1.0d0)))
    else
        tmp = log(((n - (-1.0d0)) / n))
    end if
    code = tmp
end function

Java

public static double code(double N) {
	double t_0 = (0.5 - ((0.08333333333333333 - (0.041666666666666664 / N)) / N)) / N;
	double tmp;
	if ((Math.log((N - -1.0)) - Math.log(N)) <= 0.002) {
		tmp = 1.0 / (((Math.pow(t_0, 2.0) - 1.0) * N) / (t_0 + -1.0));
	} else {
		tmp = Math.log(((N - -1.0) / N));
	}
	return tmp;
}

Python

def code(N):
	t_0 = (0.5 - ((0.08333333333333333 - (0.041666666666666664 / N)) / N)) / N
	tmp = 0
	if (math.log((N - -1.0)) - math.log(N)) <= 0.002:
		tmp = 1.0 / (((math.pow(t_0, 2.0) - 1.0) * N) / (t_0 + -1.0))
	else:
		tmp = math.log(((N - -1.0) / N))
	return tmp

Julia

function code(N)
	t_0 = Float64(Float64(0.5 - Float64(Float64(0.08333333333333333 - Float64(0.041666666666666664 / N)) / N)) / N)
	tmp = 0.0
	if (Float64(log(Float64(N - -1.0)) - log(N)) <= 0.002)
		tmp = Float64(1.0 / Float64(Float64(Float64((t_0 ^ 2.0) - 1.0) * N) / Float64(t_0 + -1.0)));
	else
		tmp = log(Float64(Float64(N - -1.0) / N));
	end
	return tmp
end

MATLAB

function tmp_2 = code(N)
	t_0 = (0.5 - ((0.08333333333333333 - (0.041666666666666664 / N)) / N)) / N;
	tmp = 0.0;
	if ((log((N - -1.0)) - log(N)) <= 0.002)
		tmp = 1.0 / ((((t_0 ^ 2.0) - 1.0) * N) / (t_0 + -1.0));
	else
		tmp = log(((N - -1.0) / N));
	end
	tmp_2 = tmp;
end

Wolfram

code[N_] := Block[{t$95$0 = N[(N[(0.5 - N[(N[(0.08333333333333333 - N[(0.041666666666666664 / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision]}, If[LessEqual[N[(N[Log[N[(N - -1.0), $MachinePrecision]], $MachinePrecision] - N[Log[N], $MachinePrecision]), $MachinePrecision], 0.002], N[(1.0 / N[(N[(N[(N[Power[t$95$0, 2.0], $MachinePrecision] - 1.0), $MachinePrecision] * N), $MachinePrecision] / N[(t$95$0 + -1.0), $MachinePrecision]), $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)) < 2e-3

   1. Initial program 21.0%

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

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

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

   10. Applied rewrites 99.7%

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

   12. Applied rewrites 99.8%

       \[\leadsto \frac{1}{\frac{\left(1 - {\left(\frac{0.5 - \frac{0.08333333333333333 - \frac{0.041666666666666664}{N}}{N}}{N}\right)}^{2}\right) \cdot \left(-N\right)}{\frac{0.5 - \frac{0.08333333333333333 - \frac{0.041666666666666664}{N}}{N}}{N} + \color{blue}{-1}}} \]

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

   1. Initial program 91.3%

      \[\log \left(N + 1\right) - \log N \]
   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 94.9

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

      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 94.9

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

   4. Applied rewrites 94.9%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(N - -1\right) - \log N \leq 0.002:\\ \;\;\;\;\frac{1}{\frac{\left({\left(\frac{0.5 - \frac{0.08333333333333333 - \frac{0.041666666666666664}{N}}{N}}{N}\right)}^{2} - 1\right) \cdot N}{\frac{0.5 - \frac{0.08333333333333333 - \frac{0.041666666666666664}{N}}{N}}{N} + -1}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{N - -1}{N}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\log \left(N - -1\right) - \log N \leq 0.0005:\\ \;\;\;\;\frac{1}{\frac{N}{\frac{-0.5 - \frac{\frac{0.25}{N} - 0.3333333333333333}{N}}{N} - -1}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{N - -1}{N}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= (- (log (- N -1.0)) (log N)) 0.0005)
   (/
    1.0
    (/ N (- (/ (- -0.5 (/ (- (/ 0.25 N) 0.3333333333333333) N)) N) -1.0)))
   (log (/ (- N -1.0) N))))

C

double code(double N) {
	double tmp;
	if ((log((N - -1.0)) - log(N)) <= 0.0005) {
		tmp = 1.0 / (N / (((-0.5 - (((0.25 / N) - 0.3333333333333333) / N)) / N) - -1.0));
	} else {
		tmp = log(((N - -1.0) / N));
	}
	return tmp;
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((log((n - (-1.0d0))) - log(n)) <= 0.0005d0) then
        tmp = 1.0d0 / (n / ((((-0.5d0) - (((0.25d0 / n) - 0.3333333333333333d0) / n)) / n) - (-1.0d0)))
    else
        tmp = log(((n - (-1.0d0)) / n))
    end if
    code = tmp
end function

Java

public static double code(double N) {
	double tmp;
	if ((Math.log((N - -1.0)) - Math.log(N)) <= 0.0005) {
		tmp = 1.0 / (N / (((-0.5 - (((0.25 / N) - 0.3333333333333333) / N)) / N) - -1.0));
	} else {
		tmp = Math.log(((N - -1.0) / N));
	}
	return tmp;
}

Python

def code(N):
	tmp = 0
	if (math.log((N - -1.0)) - math.log(N)) <= 0.0005:
		tmp = 1.0 / (N / (((-0.5 - (((0.25 / N) - 0.3333333333333333) / N)) / N) - -1.0))
	else:
		tmp = math.log(((N - -1.0) / N))
	return tmp

Julia

function code(N)
	tmp = 0.0
	if (Float64(log(Float64(N - -1.0)) - log(N)) <= 0.0005)
		tmp = Float64(1.0 / Float64(N / Float64(Float64(Float64(-0.5 - Float64(Float64(Float64(0.25 / N) - 0.3333333333333333) / N)) / N) - -1.0)));
	else
		tmp = log(Float64(Float64(N - -1.0) / N));
	end
	return tmp
end

MATLAB

function tmp_2 = code(N)
	tmp = 0.0;
	if ((log((N - -1.0)) - log(N)) <= 0.0005)
		tmp = 1.0 / (N / (((-0.5 - (((0.25 / N) - 0.3333333333333333) / N)) / N) - -1.0));
	else
		tmp = log(((N - -1.0) / N));
	end
	tmp_2 = tmp;
end

Wolfram

code[N_] := If[LessEqual[N[(N[Log[N[(N - -1.0), $MachinePrecision]], $MachinePrecision] - N[Log[N], $MachinePrecision]), $MachinePrecision], 0.0005], N[(1.0 / N[(N / N[(N[(N[(-0.5 - N[(N[(N[(0.25 / N), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision] - -1.0), $MachinePrecision]), $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)) < 5.0000000000000001e-4

   1. Initial program 20.5%

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

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

   7. Applied rewrites 99.8%

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

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

   1. Initial program 90.5%

      \[\log \left(N + 1\right) - \log N \]
   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 94.1

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

      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 94.1

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

   4. Applied rewrites 94.1%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(N - -1\right) - \log N \leq 0.0005:\\ \;\;\;\;\frac{1}{\frac{N}{\frac{-0.5 - \frac{\frac{0.25}{N} - 0.3333333333333333}{N}}{N} - -1}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{N - -1}{N}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.7% accurate, 3.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 25.1%

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

3. Taylor expanded in N around inf

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

5. Applied rewrites 97.0%

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

7. Applied rewrites 97.1%

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

8. Taylor expanded in N around -inf

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

10. Applied rewrites 97.3%

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

11. Taylor expanded in N around 0

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

13. Applied rewrites 97.3%

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

14. Final simplification 97.3%

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

Alternative 4: 96.6% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 25.1%

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

3. Taylor expanded in N around inf

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

5. Applied rewrites 97.0%

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

7. Applied rewrites 97.1%

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

8. Taylor expanded in N around -inf

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

10. Applied rewrites 97.3%

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

11. Taylor expanded in N around 0

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

13. Applied rewrites 97.2%

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

Alternative 5: 95.1% accurate, 5.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 25.1%

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

3. Taylor expanded in N around inf

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

   1. lower-/.f64 N/A

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

   2. associate--l+ N/A

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

   3. +-commutative N/A

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

   4. unpow2 N/A

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

   5. associate-/r* N/A

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

   6. metadata-eval N/A

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

   7. associate-*r/ N/A

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

   8. associate-*r/ N/A

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

   9. metadata-eval N/A

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

   10. div-sub N/A

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

   11. metadata-eval N/A

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

   12. sub-neg N/A

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

   13. lower--.f64 N/A

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

   14. lower-/.f64 N/A

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

   15. lower--.f64 N/A

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

   16. associate-*r/ N/A

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

   17. metadata-eval N/A

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

   18. lower-/.f64 95.7

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

5. Applied rewrites 95.7%

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

Alternative 6: 94.8% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \frac{N - \left(0.5 - \frac{0.3333333333333333}{N}\right)}{N \cdot N} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(N):
	return (N - (0.5 - (0.3333333333333333 / N))) / (N * N)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 25.1%

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

3. Taylor expanded in N around inf

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

5. Applied rewrites 97.0%

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

7. Applied rewrites 96.7%

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

8. Taylor expanded in N around inf

   \[\leadsto \frac{\left(\frac{1}{2} - \frac{1}{3} \cdot \frac{1}{N}\right) - N}{N \cdot \left(-N\right)} \]
9. Step-by-step derivation

10. Applied rewrites 95.4%

    \[\leadsto \frac{\left(0.5 - \frac{0.3333333333333333}{N}\right) - N}{N \cdot \left(-N\right)} \]

11. Final simplification 95.4%

    \[\leadsto \frac{N - \left(0.5 - \frac{0.3333333333333333}{N}\right)}{N \cdot N} \]
12. Add Preprocessing

Alternative 7: 93.2% 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 25.1%

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

3. Taylor expanded in N around inf

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

5. Applied rewrites 97.0%

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

7. Applied rewrites 97.1%

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

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 93.4%

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

Alternative 8: 84.7% 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 25.1%

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

3. Taylor expanded in N around inf

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

   1. lower-/.f64 83.7

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

5. Applied rewrites 83.7%

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

Alternative 9: 3.3% accurate, 207.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (N) :precision binary64 0.0)

C

double code(double N) {
	return 0.0;
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    code = 0.0d0
end function

Java

public static double code(double N) {
	return 0.0;
}

Python

def code(N):
	return 0.0

Julia

function code(N)
	return 0.0
end

MATLAB

function tmp = code(N)
	tmp = 0.0;
end

Wolfram

code[N_] := 0.0

Derivation

1. Initial program 25.1%

   \[\log \left(N + 1\right) - \log N \]
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. clear-num N/A

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

   7. log-rec N/A

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

   8. lower-neg.f64 N/A

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

   9. lower-log.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lower-/.f64 28.2

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

   12. lift-+.f64 N/A

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

   13. +-commutative N/A

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

   14. lower-+.f64 28.2

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

4. Applied rewrites 28.2%

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

6. Applied rewrites 26.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left({\left(\mathsf{log1p}\left(N\right) + \log N\right)}^{-2}, {\left(\mathsf{log1p}\left(N\right)\right)}^{2} \cdot \left(\mathsf{log1p}\left(N\right) + \log N\right), \frac{-1}{\frac{\mathsf{log1p}\left(N\right) + \log N}{{\log N}^{2}}}\right)} \]

7. Taylor expanded in N around inf

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

   1. distribute-rgt-out N/A

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

   2. metadata-eval N/A

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

   3. mul0-rgt 3.3

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

9. Applied rewrites 3.3%

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

Developer Target 1: 96.3% 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 2024242 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  :pre (and (> N 1.0) (< N 1e+40))

  :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)))