2log (problem 3.3.6)

Percentage Accurate: 54.0% → 99.9%

Time: 6.6s

Alternatives: 7

Speedup: 2.0×

Specification

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: 54.0% 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: 99.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 0.001:\\ \;\;\;\;\left(\frac{0.3333333333333333}{{N}^{3}} + {\left(N + \left(0.5 + \left(\frac{0.25}{N} + \frac{0.125}{N \cdot N}\right)\right)\right)}^{-1}\right) - \frac{0.25}{{N}^{4}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= (- (log (+ N 1.0)) (log N)) 0.001)
   (-
    (+
     (/ 0.3333333333333333 (pow N 3.0))
     (pow (+ N (+ 0.5 (+ (/ 0.25 N) (/ 0.125 (* N N))))) -1.0))
    (/ 0.25 (pow N 4.0)))
   (log (/ (+ N 1.0) N))))

C

double code(double N) {
	double tmp;
	if ((log((N + 1.0)) - log(N)) <= 0.001) {
		tmp = ((0.3333333333333333 / pow(N, 3.0)) + pow((N + (0.5 + ((0.25 / N) + (0.125 / (N * N))))), -1.0)) - (0.25 / pow(N, 4.0));
	} 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)) <= 0.001d0) then
        tmp = ((0.3333333333333333d0 / (n ** 3.0d0)) + ((n + (0.5d0 + ((0.25d0 / n) + (0.125d0 / (n * n))))) ** (-1.0d0))) - (0.25d0 / (n ** 4.0d0))
    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)) <= 0.001) {
		tmp = ((0.3333333333333333 / Math.pow(N, 3.0)) + Math.pow((N + (0.5 + ((0.25 / N) + (0.125 / (N * N))))), -1.0)) - (0.25 / Math.pow(N, 4.0));
	} 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)) <= 0.001:
		tmp = ((0.3333333333333333 / math.pow(N, 3.0)) + math.pow((N + (0.5 + ((0.25 / N) + (0.125 / (N * N))))), -1.0)) - (0.25 / math.pow(N, 4.0))
	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)) <= 0.001)
		tmp = Float64(Float64(Float64(0.3333333333333333 / (N ^ 3.0)) + (Float64(N + Float64(0.5 + Float64(Float64(0.25 / N) + Float64(0.125 / Float64(N * N))))) ^ -1.0)) - Float64(0.25 / (N ^ 4.0)));
	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)) <= 0.001)
		tmp = ((0.3333333333333333 / (N ^ 3.0)) + ((N + (0.5 + ((0.25 / N) + (0.125 / (N * N))))) ^ -1.0)) - (0.25 / (N ^ 4.0));
	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], 0.001], N[(N[(N[(0.3333333333333333 / N[Power[N, 3.0], $MachinePrecision]), $MachinePrecision] + N[Power[N[(N + N[(0.5 + N[(N[(0.25 / N), $MachinePrecision] + N[(0.125 / N[(N * N), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision] - N[(0.25 / N[Power[N, 4.0], $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 1)) (log.f64 N)) < 1e-3

   1. Initial program 6.7%

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

      1. diff-log 7.0%

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

   3. Applied egg-rr 7.0%

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

   4. Taylor expanded in N around inf 99.9%

      \[\leadsto \color{blue}{\left(\frac{1}{N} + 0.3333333333333333 \cdot \frac{1}{{N}^{3}}\right) - \left(0.25 \cdot \frac{1}{{N}^{4}} + 0.5 \cdot \frac{1}{{N}^{2}}\right)} \]
   5. Step-by-step derivation

      1. associate-*r/ 99.9%

         \[\leadsto \left(\frac{1}{N} + 0.3333333333333333 \cdot \frac{1}{{N}^{3}}\right) - \left(\color{blue}{\frac{0.25 \cdot 1}{{N}^{4}}} + 0.5 \cdot \frac{1}{{N}^{2}}\right) \]

      2. metadata-eval 99.9%

         \[\leadsto \left(\frac{1}{N} + 0.3333333333333333 \cdot \frac{1}{{N}^{3}}\right) - \left(\frac{\color{blue}{0.25}}{{N}^{4}} + 0.5 \cdot \frac{1}{{N}^{2}}\right) \]

      3. +-commutative 99.9%

         \[\leadsto \left(\frac{1}{N} + 0.3333333333333333 \cdot \frac{1}{{N}^{3}}\right) - \color{blue}{\left(0.5 \cdot \frac{1}{{N}^{2}} + \frac{0.25}{{N}^{4}}\right)} \]

      4. unpow2 99.9%

         \[\leadsto \left(\frac{1}{N} + 0.3333333333333333 \cdot \frac{1}{{N}^{3}}\right) - \left(0.5 \cdot \frac{1}{\color{blue}{N \cdot N}} + \frac{0.25}{{N}^{4}}\right) \]

      5. associate-*r/ 99.9%

         \[\leadsto \left(\frac{1}{N} + 0.3333333333333333 \cdot \frac{1}{{N}^{3}}\right) - \left(\color{blue}{\frac{0.5 \cdot 1}{N \cdot N}} + \frac{0.25}{{N}^{4}}\right) \]

      6. metadata-eval 99.9%

         \[\leadsto \left(\frac{1}{N} + 0.3333333333333333 \cdot \frac{1}{{N}^{3}}\right) - \left(\frac{\color{blue}{0.5}}{N \cdot N} + \frac{0.25}{{N}^{4}}\right) \]

      7. associate-/l/ 99.9%

         \[\leadsto \left(\frac{1}{N} + 0.3333333333333333 \cdot \frac{1}{{N}^{3}}\right) - \left(\color{blue}{\frac{\frac{0.5}{N}}{N}} + \frac{0.25}{{N}^{4}}\right) \]

      8. associate--r+ 99.9%

         \[\leadsto \color{blue}{\left(\left(\frac{1}{N} + 0.3333333333333333 \cdot \frac{1}{{N}^{3}}\right) - \frac{\frac{0.5}{N}}{N}\right) - \frac{0.25}{{N}^{4}}} \]

   6. Simplified 50.9%

      \[\leadsto \color{blue}{\left(\frac{0.3333333333333333}{{N}^{3}} + \frac{N + -0.5}{N \cdot N}\right) - \frac{0.25}{{N}^{4}}} \]
   7. Step-by-step derivation

      1. clear-num 51.1%

         \[\leadsto \left(\frac{0.3333333333333333}{{N}^{3}} + \color{blue}{\frac{1}{\frac{N \cdot N}{N + -0.5}}}\right) - \frac{0.25}{{N}^{4}} \]

      2. inv-pow 51.1%

         \[\leadsto \left(\frac{0.3333333333333333}{{N}^{3}} + \color{blue}{{\left(\frac{N \cdot N}{N + -0.5}\right)}^{-1}}\right) - \frac{0.25}{{N}^{4}} \]

   8. Applied egg-rr 51.1%

      \[\leadsto \left(\frac{0.3333333333333333}{{N}^{3}} + \color{blue}{{\left(\frac{N \cdot N}{N + -0.5}\right)}^{-1}}\right) - \frac{0.25}{{N}^{4}} \]

   9. Taylor expanded in N around inf 99.9%

      \[\leadsto \left(\frac{0.3333333333333333}{{N}^{3}} + {\color{blue}{\left(N + \left(0.5 + \left(0.25 \cdot \frac{1}{N} + 0.125 \cdot \frac{1}{{N}^{2}}\right)\right)\right)}}^{-1}\right) - \frac{0.25}{{N}^{4}} \]
   10. Step-by-step derivation

       1. associate-*r/ 99.9%

          \[\leadsto \left(\frac{0.3333333333333333}{{N}^{3}} + {\left(N + \left(0.5 + \left(\color{blue}{\frac{0.25 \cdot 1}{N}} + 0.125 \cdot \frac{1}{{N}^{2}}\right)\right)\right)}^{-1}\right) - \frac{0.25}{{N}^{4}} \]

       2. metadata-eval 99.9%

          \[\leadsto \left(\frac{0.3333333333333333}{{N}^{3}} + {\left(N + \left(0.5 + \left(\frac{\color{blue}{0.25}}{N} + 0.125 \cdot \frac{1}{{N}^{2}}\right)\right)\right)}^{-1}\right) - \frac{0.25}{{N}^{4}} \]

       3. unpow2 99.9%

          \[\leadsto \left(\frac{0.3333333333333333}{{N}^{3}} + {\left(N + \left(0.5 + \left(\frac{0.25}{N} + 0.125 \cdot \frac{1}{\color{blue}{N \cdot N}}\right)\right)\right)}^{-1}\right) - \frac{0.25}{{N}^{4}} \]

       4. associate-*r/ 99.9%

          \[\leadsto \left(\frac{0.3333333333333333}{{N}^{3}} + {\left(N + \left(0.5 + \left(\frac{0.25}{N} + \color{blue}{\frac{0.125 \cdot 1}{N \cdot N}}\right)\right)\right)}^{-1}\right) - \frac{0.25}{{N}^{4}} \]

       5. metadata-eval 99.9%

          \[\leadsto \left(\frac{0.3333333333333333}{{N}^{3}} + {\left(N + \left(0.5 + \left(\frac{0.25}{N} + \frac{\color{blue}{0.125}}{N \cdot N}\right)\right)\right)}^{-1}\right) - \frac{0.25}{{N}^{4}} \]

   11. Simplified 99.9%

       \[\leadsto \left(\frac{0.3333333333333333}{{N}^{3}} + {\color{blue}{\left(N + \left(0.5 + \left(\frac{0.25}{N} + \frac{0.125}{N \cdot N}\right)\right)\right)}}^{-1}\right) - \frac{0.25}{{N}^{4}} \]

   if 1e-3 < (-.f64 (log.f64 (+.f64 N 1)) (log.f64 N))

   1. Initial program 100.0%

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

      1. diff-log 100.0%

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

   3. Applied egg-rr 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 0.001:\\ \;\;\;\;\left(\frac{0.3333333333333333}{{N}^{3}} + {\left(N + \left(0.5 + \left(\frac{0.25}{N} + \frac{0.125}{N \cdot N}\right)\right)\right)}^{-1}\right) - \frac{0.25}{{N}^{4}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \end{array} \]

Alternative 2: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 10^{-11}:\\ \;\;\;\;\frac{1 - \frac{0.5}{N}}{N}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \log \left(\sqrt{\frac{N + 1}{N}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= (- (log (+ N 1.0)) (log N)) 1e-11)
   (/ (- 1.0 (/ 0.5 N)) N)
   (* 2.0 (log (sqrt (/ (+ N 1.0) N))))))

C

double code(double N) {
	double tmp;
	if ((log((N + 1.0)) - log(N)) <= 1e-11) {
		tmp = (1.0 - (0.5 / N)) / N;
	} else {
		tmp = 2.0 * log(sqrt(((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)) <= 1d-11) then
        tmp = (1.0d0 - (0.5d0 / n)) / n
    else
        tmp = 2.0d0 * log(sqrt(((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)) <= 1e-11) {
		tmp = (1.0 - (0.5 / N)) / N;
	} else {
		tmp = 2.0 * Math.log(Math.sqrt(((N + 1.0) / N)));
	}
	return tmp;
}

Python

def code(N):
	tmp = 0
	if (math.log((N + 1.0)) - math.log(N)) <= 1e-11:
		tmp = (1.0 - (0.5 / N)) / N
	else:
		tmp = 2.0 * math.log(math.sqrt(((N + 1.0) / N)))
	return tmp

Julia

function code(N)
	tmp = 0.0
	if (Float64(log(Float64(N + 1.0)) - log(N)) <= 1e-11)
		tmp = Float64(Float64(1.0 - Float64(0.5 / N)) / N);
	else
		tmp = Float64(2.0 * log(sqrt(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)) <= 1e-11)
		tmp = (1.0 - (0.5 / N)) / N;
	else
		tmp = 2.0 * log(sqrt(((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], 1e-11], N[(N[(1.0 - N[(0.5 / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision], N[(2.0 * N[Log[N[Sqrt[N[(N[(N + 1.0), $MachinePrecision] / N), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (log.f64 (+.f64 N 1)) (log.f64 N)) < 9.99999999999999939e-12

   1. Initial program 6.1%

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

      1. diff-log 6.4%

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

   3. Applied egg-rr 6.4%

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

   4. Taylor expanded in N around inf 100.0%

      \[\leadsto \color{blue}{\frac{1}{N} - 0.5 \cdot \frac{1}{{N}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 100.0%

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

      2. associate-*r/ 100.0%

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

      3. metadata-eval 100.0%

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

      4. associate-/l/ 100.0%

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

      5. div-sub 100.0%

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

   6. Simplified 100.0%

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

   if 9.99999999999999939e-12 < (-.f64 (log.f64 (+.f64 N 1)) (log.f64 N))

   1. Initial program 99.8%

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

      1. diff-log 99.9%

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

   3. Applied egg-rr 99.9%

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

      1. add-sqr-sqrt 99.9%

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

      2. log-prod 99.9%

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

   5. Applied egg-rr 99.9%

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

      1. count-2 99.9%

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

   7. Simplified 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 10^{-11}:\\ \;\;\;\;\frac{1 - \frac{0.5}{N}}{N}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \log \left(\sqrt{\frac{N + 1}{N}}\right)\\ \end{array} \]

Alternative 3: 99.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 10^{-11}:\\ \;\;\;\;\frac{1 - \frac{0.5}{N}}{N}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= (- (log (+ N 1.0)) (log N)) 1e-11)
   (/ (- 1.0 (/ 0.5 N)) N)
   (log (/ (+ N 1.0) N))))

C

double code(double N) {
	double tmp;
	if ((log((N + 1.0)) - log(N)) <= 1e-11) {
		tmp = (1.0 - (0.5 / N)) / N;
	} 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)) <= 1d-11) then
        tmp = (1.0d0 - (0.5d0 / n)) / n
    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)) <= 1e-11) {
		tmp = (1.0 - (0.5 / N)) / N;
	} 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)) <= 1e-11:
		tmp = (1.0 - (0.5 / N)) / N
	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)) <= 1e-11)
		tmp = Float64(Float64(1.0 - Float64(0.5 / N)) / N);
	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)) <= 1e-11)
		tmp = (1.0 - (0.5 / N)) / N;
	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], 1e-11], N[(N[(1.0 - N[(0.5 / N), $MachinePrecision]), $MachinePrecision] / N), $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 1)) (log.f64 N)) < 9.99999999999999939e-12

   1. Initial program 6.1%

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

      1. diff-log 6.4%

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

   3. Applied egg-rr 6.4%

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

   4. Taylor expanded in N around inf 100.0%

      \[\leadsto \color{blue}{\frac{1}{N} - 0.5 \cdot \frac{1}{{N}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 100.0%

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

      2. associate-*r/ 100.0%

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

      3. metadata-eval 100.0%

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

      4. associate-/l/ 100.0%

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

      5. div-sub 100.0%

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

   6. Simplified 100.0%

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

   if 9.99999999999999939e-12 < (-.f64 (log.f64 (+.f64 N 1)) (log.f64 N))

   1. Initial program 99.8%

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

      1. diff-log 99.9%

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

   3. Applied egg-rr 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 10^{-11}:\\ \;\;\;\;\frac{1 - \frac{0.5}{N}}{N}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \end{array} \]

Alternative 4: 99.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;N \leq 0.9:\\ \;\;\;\;N - \log N\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{0.5}{N}}{N}\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= N 0.9) (- N (log N)) (/ (- 1.0 (/ 0.5 N)) N)))

C

double code(double N) {
	double tmp;
	if (N <= 0.9) {
		tmp = N - log(N);
	} else {
		tmp = (1.0 - (0.5 / N)) / N;
	}
	return tmp;
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= 0.9d0) then
        tmp = n - log(n)
    else
        tmp = (1.0d0 - (0.5d0 / n)) / n
    end if
    code = tmp
end function

Java

public static double code(double N) {
	double tmp;
	if (N <= 0.9) {
		tmp = N - Math.log(N);
	} else {
		tmp = (1.0 - (0.5 / N)) / N;
	}
	return tmp;
}

Python

def code(N):
	tmp = 0
	if N <= 0.9:
		tmp = N - math.log(N)
	else:
		tmp = (1.0 - (0.5 / N)) / N
	return tmp

Julia

function code(N)
	tmp = 0.0
	if (N <= 0.9)
		tmp = Float64(N - log(N));
	else
		tmp = Float64(Float64(1.0 - Float64(0.5 / N)) / N);
	end
	return tmp
end

MATLAB

function tmp_2 = code(N)
	tmp = 0.0;
	if (N <= 0.9)
		tmp = N - log(N);
	else
		tmp = (1.0 - (0.5 / N)) / N;
	end
	tmp_2 = tmp;
end

Wolfram

code[N_] := If[LessEqual[N, 0.9], N[(N - N[Log[N], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(0.5 / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if N < 0.900000000000000022

   1. Initial program 100.0%

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

   2. Taylor expanded in N around 0 99.4%

      \[\leadsto \color{blue}{N} - \log N \]

   if 0.900000000000000022 < N

   1. Initial program 6.7%

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

      1. diff-log 7.0%

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

   3. Applied egg-rr 7.0%

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

   4. Taylor expanded in N around inf 99.6%

      \[\leadsto \color{blue}{\frac{1}{N} - 0.5 \cdot \frac{1}{{N}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 99.6%

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

      2. associate-*r/ 99.6%

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

      3. metadata-eval 99.6%

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

      4. associate-/l/ 99.6%

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

      5. div-sub 99.6%

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

   6. Simplified 99.6%

      \[\leadsto \color{blue}{\frac{1 - \frac{0.5}{N}}{N}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;N \leq 0.9:\\ \;\;\;\;N - \log N\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{0.5}{N}}{N}\\ \end{array} \]

Alternative 5: 98.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;N \leq 0.68:\\ \;\;\;\;-\log N\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{0.5}{N}}{N}\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= N 0.68) (- (log N)) (/ (- 1.0 (/ 0.5 N)) N)))

C

double code(double N) {
	double tmp;
	if (N <= 0.68) {
		tmp = -log(N);
	} else {
		tmp = (1.0 - (0.5 / N)) / N;
	}
	return tmp;
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= 0.68d0) then
        tmp = -log(n)
    else
        tmp = (1.0d0 - (0.5d0 / n)) / n
    end if
    code = tmp
end function

Java

public static double code(double N) {
	double tmp;
	if (N <= 0.68) {
		tmp = -Math.log(N);
	} else {
		tmp = (1.0 - (0.5 / N)) / N;
	}
	return tmp;
}

Python

def code(N):
	tmp = 0
	if N <= 0.68:
		tmp = -math.log(N)
	else:
		tmp = (1.0 - (0.5 / N)) / N
	return tmp

Julia

function code(N)
	tmp = 0.0
	if (N <= 0.68)
		tmp = Float64(-log(N));
	else
		tmp = Float64(Float64(1.0 - Float64(0.5 / N)) / N);
	end
	return tmp
end

MATLAB

function tmp_2 = code(N)
	tmp = 0.0;
	if (N <= 0.68)
		tmp = -log(N);
	else
		tmp = (1.0 - (0.5 / N)) / N;
	end
	tmp_2 = tmp;
end

Wolfram

code[N_] := If[LessEqual[N, 0.68], (-N[Log[N], $MachinePrecision]), N[(N[(1.0 - N[(0.5 / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if N < 0.680000000000000049

   1. Initial program 100.0%

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

   2. Taylor expanded in N around 0 98.6%

      \[\leadsto \color{blue}{-1 \cdot \log N} \]
   3. Step-by-step derivation

      1. neg-mul-1 98.6%

         \[\leadsto \color{blue}{-\log N} \]

   4. Simplified 98.6%

      \[\leadsto \color{blue}{-\log N} \]

   if 0.680000000000000049 < N

   1. Initial program 6.7%

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

      1. diff-log 7.0%

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

   3. Applied egg-rr 7.0%

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

   4. Taylor expanded in N around inf 99.6%

      \[\leadsto \color{blue}{\frac{1}{N} - 0.5 \cdot \frac{1}{{N}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 99.6%

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

      2. associate-*r/ 99.6%

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

      3. metadata-eval 99.6%

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

      4. associate-/l/ 99.6%

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

      5. div-sub 99.6%

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

   6. Simplified 99.6%

      \[\leadsto \color{blue}{\frac{1 - \frac{0.5}{N}}{N}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;N \leq 0.68:\\ \;\;\;\;-\log N\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{0.5}{N}}{N}\\ \end{array} \]

Alternative 6: 51.8% accurate, 68.3× 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 54.8%

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

2. Taylor expanded in N around inf 50.8%

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

3. Final simplification 50.8%

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

Alternative 7: 4.6% accurate, 205.0× speedup

Math

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

FPCore

(FPCore (N) :precision binary64 N)

C

double code(double N) {
	return N;
}

Fortran

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

Java

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

Python

def code(N):
	return N

Julia

function code(N)
	return N
end

MATLAB

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

Wolfram

code[N_] := N

Derivation

1. Initial program 54.8%

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

2. Taylor expanded in N around 0 53.0%

   \[\leadsto \color{blue}{N} - \log N \]

3. Taylor expanded in N around inf 4.5%

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

4. Final simplification 4.5%

   \[\leadsto N \]

Reproduce

herbie shell --seed 2023242 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1.0)) (log N)))