2log (problem 3.3.6)

Percentage Accurate: 53.4% → 99.9%

Time: 6.1s

Alternatives: 8

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: 53.4% 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: 99.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\left(\frac{1}{N} + \frac{0.3333333333333333}{{N}^{3}}\right) - \left(\frac{0.5}{N \cdot N} + \frac{0.25}{{N}^{4}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= (- (log (+ N 1.0)) (log N)) 2e-5)
   (-
    (+ (/ 1.0 N) (/ 0.3333333333333333 (pow N 3.0)))
    (+ (/ 0.5 (* N N)) (/ 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)) <= 2e-5) {
		tmp = ((1.0 / N) + (0.3333333333333333 / pow(N, 3.0))) - ((0.5 / (N * N)) + (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)) <= 2d-5) then
        tmp = ((1.0d0 / n) + (0.3333333333333333d0 / (n ** 3.0d0))) - ((0.5d0 / (n * n)) + (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)) <= 2e-5) {
		tmp = ((1.0 / N) + (0.3333333333333333 / Math.pow(N, 3.0))) - ((0.5 / (N * N)) + (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)) <= 2e-5:
		tmp = ((1.0 / N) + (0.3333333333333333 / math.pow(N, 3.0))) - ((0.5 / (N * N)) + (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)) <= 2e-5)
		tmp = Float64(Float64(Float64(1.0 / N) + Float64(0.3333333333333333 / (N ^ 3.0))) - Float64(Float64(0.5 / Float64(N * N)) + 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)) <= 2e-5)
		tmp = ((1.0 / N) + (0.3333333333333333 / (N ^ 3.0))) - ((0.5 / (N * N)) + (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], 2e-5], N[(N[(N[(1.0 / N), $MachinePrecision] + N[(0.3333333333333333 / N[Power[N, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(0.5 / N[(N * N), $MachinePrecision]), $MachinePrecision] + N[(0.25 / N[Power[N, 4.0], $MachinePrecision]), $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)) < 2.00000000000000016e-5

   1. Initial program 7.9%

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

      1. +-commutative 7.9%

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

      2. log1p-def 7.9%

         \[\leadsto \color{blue}{\mathsf{log1p}\left(N\right)} - \log N \]

   3. Simplified 7.9%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(N\right) - \log N} \]

   4. Taylor expanded in N around inf 100.0%

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

      1. associate-*r/ 100.0%

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

      2. metadata-eval 100.0%

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

      3. +-commutative 100.0%

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

      4. associate-*r/ 100.0%

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

      5. metadata-eval 100.0%

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

      6. unpow2 100.0%

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

      7. associate-*r/ 100.0%

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

      8. metadata-eval 100.0%

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

   6. Simplified 100.0%

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

   if 2.00000000000000016e-5 < (-.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. +-commutative 100.0%

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

      2. log1p-def 100.0%

         \[\leadsto \color{blue}{\mathsf{log1p}\left(N\right)} - \log N \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(N\right) - \log N} \]
   4. Step-by-step derivation

      1. log1p-udef 100.0%

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

      2. diff-log 100.0%

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

      3. +-commutative 100.0%

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

   5. 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 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\left(\frac{1}{N} + \frac{0.3333333333333333}{{N}^{3}}\right) - \left(\frac{0.5}{N \cdot N} + \frac{0.25}{{N}^{4}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \end{array} \]

Alternative 2: 99.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\frac{0.3333333333333333}{{N}^{3}} + \left(\frac{1}{N} + \frac{-0.5}{N \cdot N}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= (- (log (+ N 1.0)) (log N)) 2e-5)
   (+ (/ 0.3333333333333333 (pow N 3.0)) (+ (/ 1.0 N) (/ -0.5 (* N N))))
   (log (/ (+ N 1.0) N))))

C

double code(double N) {
	double tmp;
	if ((log((N + 1.0)) - log(N)) <= 2e-5) {
		tmp = (0.3333333333333333 / pow(N, 3.0)) + ((1.0 / N) + (-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)) <= 2d-5) then
        tmp = (0.3333333333333333d0 / (n ** 3.0d0)) + ((1.0d0 / n) + ((-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)) <= 2e-5) {
		tmp = (0.3333333333333333 / Math.pow(N, 3.0)) + ((1.0 / N) + (-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)) <= 2e-5:
		tmp = (0.3333333333333333 / math.pow(N, 3.0)) + ((1.0 / N) + (-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)) <= 2e-5)
		tmp = Float64(Float64(0.3333333333333333 / (N ^ 3.0)) + Float64(Float64(1.0 / N) + Float64(-0.5 / Float64(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)) <= 2e-5)
		tmp = (0.3333333333333333 / (N ^ 3.0)) + ((1.0 / N) + (-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], 2e-5], N[(N[(0.3333333333333333 / N[Power[N, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / N), $MachinePrecision] + N[(-0.5 / N[(N * N), $MachinePrecision]), $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)) < 2.00000000000000016e-5

   1. Initial program 7.9%

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

      1. +-commutative 7.9%

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

      2. log1p-def 7.9%

         \[\leadsto \color{blue}{\mathsf{log1p}\left(N\right)} - \log N \]

   3. Simplified 7.9%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(N\right) - \log N} \]

   4. Taylor expanded in N around inf 99.9%

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

      1. +-commutative 99.9%

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

      2. associate--l+ 99.9%

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

      3. associate-*r/ 99.9%

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

      4. metadata-eval 99.9%

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

      5. sub-neg 99.9%

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

      6. associate-*r/ 99.9%

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

      7. metadata-eval 99.9%

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

      8. distribute-neg-frac 99.9%

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

      9. metadata-eval 99.9%

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

      10. unpow2 99.9%

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

   6. Simplified 99.9%

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

   if 2.00000000000000016e-5 < (-.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. +-commutative 100.0%

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

      2. log1p-def 100.0%

         \[\leadsto \color{blue}{\mathsf{log1p}\left(N\right)} - \log N \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(N\right) - \log N} \]
   4. Step-by-step derivation

      1. log1p-udef 100.0%

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

      2. diff-log 100.0%

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

      3. +-commutative 100.0%

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

   5. 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 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\frac{0.3333333333333333}{{N}^{3}} + \left(\frac{1}{N} + \frac{-0.5}{N \cdot N}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \end{array} \]

Alternative 3: 99.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;N \leq 210000:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{\left(1 + \frac{-0.5}{N}\right) \cdot \left(1 + \frac{-1}{N}\right)}}{N}\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= N 210000.0)
   (log (/ (+ N 1.0) N))
   (/ (cbrt (* (+ 1.0 (/ -0.5 N)) (+ 1.0 (/ -1.0 N)))) N)))

C

double code(double N) {
	double tmp;
	if (N <= 210000.0) {
		tmp = log(((N + 1.0) / N));
	} else {
		tmp = cbrt(((1.0 + (-0.5 / N)) * (1.0 + (-1.0 / N)))) / N;
	}
	return tmp;
}

Java

public static double code(double N) {
	double tmp;
	if (N <= 210000.0) {
		tmp = Math.log(((N + 1.0) / N));
	} else {
		tmp = Math.cbrt(((1.0 + (-0.5 / N)) * (1.0 + (-1.0 / N)))) / N;
	}
	return tmp;
}

Julia

function code(N)
	tmp = 0.0
	if (N <= 210000.0)
		tmp = log(Float64(Float64(N + 1.0) / N));
	else
		tmp = Float64(cbrt(Float64(Float64(1.0 + Float64(-0.5 / N)) * Float64(1.0 + Float64(-1.0 / N)))) / N);
	end
	return tmp
end

Wolfram

code[N_] := If[LessEqual[N, 210000.0], N[Log[N[(N[(N + 1.0), $MachinePrecision] / N), $MachinePrecision]], $MachinePrecision], N[(N[Power[N[(N[(1.0 + N[(-0.5 / N), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(-1.0 / N), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] / N), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if N < 2.1e5

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

      2. log1p-def 99.6%

         \[\leadsto \color{blue}{\mathsf{log1p}\left(N\right)} - \log N \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(N\right) - \log N} \]
   4. Step-by-step derivation

      1. log1p-udef 99.6%

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

      2. diff-log 99.7%

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

      3. +-commutative 99.7%

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

   5. Applied egg-rr 99.7%

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

   if 2.1e5 < N

   1. Initial program 6.8%

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

      1. +-commutative 6.8%

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

      2. log1p-def 6.8%

         \[\leadsto \color{blue}{\mathsf{log1p}\left(N\right)} - \log N \]

   3. Simplified 6.8%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(N\right) - \log N} \]

   4. Taylor expanded in N around inf 99.8%

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

      1. associate-*r/ 99.8%

         \[\leadsto \frac{1}{N} - \color{blue}{\frac{0.5 \cdot 1}{{N}^{2}}} \]

      2. metadata-eval 99.8%

         \[\leadsto \frac{1}{N} - \frac{\color{blue}{0.5}}{{N}^{2}} \]

      3. unpow2 99.8%

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

      4. associate-/r* 99.8%

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

   6. Simplified 99.8%

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

      1. sub-div 99.8%

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

      2. sub-neg 99.8%

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

      3. distribute-neg-frac 99.8%

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

      4. metadata-eval 99.8%

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

   8. Applied egg-rr 99.8%

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

      1. add-cbrt-cube 99.8%

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

   10. Applied egg-rr 99.8%

       \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(\left(1 + \frac{-0.5}{N}\right) \cdot \left(1 + \frac{-0.5}{N}\right)\right) \cdot \left(1 + \frac{-0.5}{N}\right)}}}{N} \]
   11. Step-by-step derivation

       1. associate-*l* 99.8%

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

   12. Simplified 99.8%

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

   13. Taylor expanded in N around inf 99.8%

       \[\leadsto \frac{\sqrt[3]{\left(1 + \frac{-0.5}{N}\right) \cdot \color{blue}{\left(1 - \frac{1}{N}\right)}}}{N} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;N \leq 210000:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{\left(1 + \frac{-0.5}{N}\right) \cdot \left(1 + \frac{-1}{N}\right)}}{N}\\ \end{array} \]

Alternative 4: 99.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;N \leq 165000:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{0.25}{N \cdot N}}{N \cdot \left(1 + \frac{0.5}{N}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= N 165000.0)
   (log (/ (+ N 1.0) N))
   (/ (- 1.0 (/ 0.25 (* N N))) (* N (+ 1.0 (/ 0.5 N))))))

C

double code(double N) {
	double tmp;
	if (N <= 165000.0) {
		tmp = log(((N + 1.0) / N));
	} else {
		tmp = (1.0 - (0.25 / (N * N))) / (N * (1.0 + (0.5 / N)));
	}
	return tmp;
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= 165000.0d0) then
        tmp = log(((n + 1.0d0) / n))
    else
        tmp = (1.0d0 - (0.25d0 / (n * n))) / (n * (1.0d0 + (0.5d0 / n)))
    end if
    code = tmp
end function

Java

public static double code(double N) {
	double tmp;
	if (N <= 165000.0) {
		tmp = Math.log(((N + 1.0) / N));
	} else {
		tmp = (1.0 - (0.25 / (N * N))) / (N * (1.0 + (0.5 / N)));
	}
	return tmp;
}

Python

def code(N):
	tmp = 0
	if N <= 165000.0:
		tmp = math.log(((N + 1.0) / N))
	else:
		tmp = (1.0 - (0.25 / (N * N))) / (N * (1.0 + (0.5 / N)))
	return tmp

Julia

function code(N)
	tmp = 0.0
	if (N <= 165000.0)
		tmp = log(Float64(Float64(N + 1.0) / N));
	else
		tmp = Float64(Float64(1.0 - Float64(0.25 / Float64(N * N))) / Float64(N * Float64(1.0 + Float64(0.5 / N))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(N)
	tmp = 0.0;
	if (N <= 165000.0)
		tmp = log(((N + 1.0) / N));
	else
		tmp = (1.0 - (0.25 / (N * N))) / (N * (1.0 + (0.5 / N)));
	end
	tmp_2 = tmp;
end

Wolfram

code[N_] := If[LessEqual[N, 165000.0], N[Log[N[(N[(N + 1.0), $MachinePrecision] / N), $MachinePrecision]], $MachinePrecision], N[(N[(1.0 - N[(0.25 / N[(N * N), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N * N[(1.0 + N[(0.5 / N), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if N < 165000

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

      2. log1p-def 99.6%

         \[\leadsto \color{blue}{\mathsf{log1p}\left(N\right)} - \log N \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(N\right) - \log N} \]
   4. Step-by-step derivation

      1. log1p-udef 99.6%

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

      2. diff-log 99.7%

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

      3. +-commutative 99.7%

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

   5. Applied egg-rr 99.7%

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

   if 165000 < N

   1. Initial program 6.8%

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

      1. +-commutative 6.8%

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

      2. log1p-def 6.8%

         \[\leadsto \color{blue}{\mathsf{log1p}\left(N\right)} - \log N \]

   3. Simplified 6.8%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(N\right) - \log N} \]

   4. Taylor expanded in N around inf 99.8%

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

      1. associate-*r/ 99.8%

         \[\leadsto \frac{1}{N} - \color{blue}{\frac{0.5 \cdot 1}{{N}^{2}}} \]

      2. metadata-eval 99.8%

         \[\leadsto \frac{1}{N} - \frac{\color{blue}{0.5}}{{N}^{2}} \]

      3. unpow2 99.8%

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

      4. associate-/r* 99.8%

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

   6. Simplified 99.8%

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

      1. sub-div 99.8%

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

      2. sub-neg 99.8%

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

      3. distribute-neg-frac 99.8%

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

      4. metadata-eval 99.8%

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

   8. Applied egg-rr 99.8%

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

      1. flip-+ 99.8%

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

      2. metadata-eval 99.8%

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

   10. Applied egg-rr 99.8%

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

       1. associate-*l/ 99.8%

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

       2. associate-*r/ 99.8%

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

       3. metadata-eval 99.8%

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

   12. Simplified 99.8%

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

       1. expm1-log1p-u 99.8%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{1 - \frac{\frac{0.25}{N}}{N}}{1 - \frac{-0.5}{N}}}{N}\right)\right)} \]

       2. expm1-udef 7.1%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{1 - \frac{\frac{0.25}{N}}{N}}{1 - \frac{-0.5}{N}}}{N}\right)} - 1} \]

       3. associate-/l/ 7.1%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{1 - \frac{\frac{0.25}{N}}{N}}{N \cdot \left(1 - \frac{-0.5}{N}\right)}}\right)} - 1 \]

   14. Applied egg-rr 7.1%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1 - \frac{\frac{0.25}{N}}{N}}{N \cdot \left(1 - \frac{-0.5}{N}\right)}\right)} - 1} \]
   15. Step-by-step derivation

       1. expm1-def 99.8%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1 - \frac{\frac{0.25}{N}}{N}}{N \cdot \left(1 - \frac{-0.5}{N}\right)}\right)\right)} \]

       2. expm1-log1p 99.8%

          \[\leadsto \color{blue}{\frac{1 - \frac{\frac{0.25}{N}}{N}}{N \cdot \left(1 - \frac{-0.5}{N}\right)}} \]

       3. associate-/l/ 99.8%

          \[\leadsto \frac{1 - \color{blue}{\frac{0.25}{N \cdot N}}}{N \cdot \left(1 - \frac{-0.5}{N}\right)} \]

       4. sub-neg 99.8%

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

       5. distribute-neg-frac 99.8%

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

       6. metadata-eval 99.8%

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

   16. Simplified 99.8%

       \[\leadsto \color{blue}{\frac{1 - \frac{0.25}{N \cdot N}}{N \cdot \left(1 + \frac{0.5}{N}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;N \leq 165000:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{0.25}{N \cdot N}}{N \cdot \left(1 + \frac{0.5}{N}\right)}\\ \end{array} \]

Alternative 5: 99.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;N \leq 0.9:\\ \;\;\;\;N - \log N\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{0.25}{N \cdot N}}{N \cdot \left(1 + \frac{0.5}{N}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= N 0.9)
   (- N (log N))
   (/ (- 1.0 (/ 0.25 (* N N))) (* N (+ 1.0 (/ 0.5 N))))))

C

double code(double N) {
	double tmp;
	if (N <= 0.9) {
		tmp = N - log(N);
	} else {
		tmp = (1.0 - (0.25 / (N * N))) / (N * (1.0 + (0.5 / 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 - (0.25d0 / (n * n))) / (n * (1.0d0 + (0.5d0 / 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 - (0.25 / (N * N))) / (N * (1.0 + (0.5 / N)));
	}
	return tmp;
}

Python

def code(N):
	tmp = 0
	if N <= 0.9:
		tmp = N - math.log(N)
	else:
		tmp = (1.0 - (0.25 / (N * N))) / (N * (1.0 + (0.5 / 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(0.25 / Float64(N * N))) / Float64(N * Float64(1.0 + Float64(0.5 / 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 - (0.25 / (N * N))) / (N * (1.0 + (0.5 / 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[(0.25 / N[(N * N), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N * N[(1.0 + N[(0.5 / N), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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. Step-by-step derivation

      1. +-commutative 100.0%

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

      2. log1p-def 100.0%

         \[\leadsto \color{blue}{\mathsf{log1p}\left(N\right)} - \log N \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(N\right) - \log N} \]

   4. Taylor expanded in N around 0 98.7%

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

      1. neg-mul-1 98.7%

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

      2. unsub-neg 98.7%

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

   6. Simplified 98.7%

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

   if 0.900000000000000022 < N

   1. Initial program 8.6%

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

      1. +-commutative 8.6%

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

      2. log1p-def 8.6%

         \[\leadsto \color{blue}{\mathsf{log1p}\left(N\right)} - \log N \]

   3. Simplified 8.6%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(N\right) - \log N} \]

   4. Taylor expanded in N around inf 98.8%

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

      1. associate-*r/ 98.8%

         \[\leadsto \frac{1}{N} - \color{blue}{\frac{0.5 \cdot 1}{{N}^{2}}} \]

      2. metadata-eval 98.8%

         \[\leadsto \frac{1}{N} - \frac{\color{blue}{0.5}}{{N}^{2}} \]

      3. unpow2 98.8%

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

      4. associate-/r* 98.8%

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

   6. Simplified 98.8%

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

      1. sub-div 98.8%

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

      2. sub-neg 98.8%

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

      3. distribute-neg-frac 98.8%

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

      4. metadata-eval 98.8%

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

   8. Applied egg-rr 98.8%

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

      1. flip-+ 98.8%

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

      2. metadata-eval 98.8%

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

   10. Applied egg-rr 98.8%

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

       1. associate-*l/ 98.8%

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

       2. associate-*r/ 98.8%

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

       3. metadata-eval 98.8%

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

   12. Simplified 98.8%

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

       1. expm1-log1p-u 98.8%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{1 - \frac{\frac{0.25}{N}}{N}}{1 - \frac{-0.5}{N}}}{N}\right)\right)} \]

       2. expm1-udef 8.4%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{1 - \frac{\frac{0.25}{N}}{N}}{1 - \frac{-0.5}{N}}}{N}\right)} - 1} \]

       3. associate-/l/ 8.4%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{1 - \frac{\frac{0.25}{N}}{N}}{N \cdot \left(1 - \frac{-0.5}{N}\right)}}\right)} - 1 \]

   14. Applied egg-rr 8.4%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1 - \frac{\frac{0.25}{N}}{N}}{N \cdot \left(1 - \frac{-0.5}{N}\right)}\right)} - 1} \]
   15. Step-by-step derivation

       1. expm1-def 98.8%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1 - \frac{\frac{0.25}{N}}{N}}{N \cdot \left(1 - \frac{-0.5}{N}\right)}\right)\right)} \]

       2. expm1-log1p 98.8%

          \[\leadsto \color{blue}{\frac{1 - \frac{\frac{0.25}{N}}{N}}{N \cdot \left(1 - \frac{-0.5}{N}\right)}} \]

       3. associate-/l/ 98.8%

          \[\leadsto \frac{1 - \color{blue}{\frac{0.25}{N \cdot N}}}{N \cdot \left(1 - \frac{-0.5}{N}\right)} \]

       4. sub-neg 98.8%

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

       5. distribute-neg-frac 98.8%

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

       6. metadata-eval 98.8%

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

   16. Simplified 98.8%

       \[\leadsto \color{blue}{\frac{1 - \frac{0.25}{N \cdot N}}{N \cdot \left(1 + \frac{0.5}{N}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;N \leq 0.9:\\ \;\;\;\;N - \log N\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{0.25}{N \cdot N}}{N \cdot \left(1 + \frac{0.5}{N}\right)}\\ \end{array} \]

Alternative 6: 98.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;N \leq 0.68:\\ \;\;\;\;-\log N\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{0.25}{N \cdot N}}{N \cdot \left(1 + \frac{0.5}{N}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= N 0.68)
   (- (log N))
   (/ (- 1.0 (/ 0.25 (* N N))) (* N (+ 1.0 (/ 0.5 N))))))

C

double code(double N) {
	double tmp;
	if (N <= 0.68) {
		tmp = -log(N);
	} else {
		tmp = (1.0 - (0.25 / (N * N))) / (N * (1.0 + (0.5 / 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 - (0.25d0 / (n * n))) / (n * (1.0d0 + (0.5d0 / 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 - (0.25 / (N * N))) / (N * (1.0 + (0.5 / N)));
	}
	return tmp;
}

Python

def code(N):
	tmp = 0
	if N <= 0.68:
		tmp = -math.log(N)
	else:
		tmp = (1.0 - (0.25 / (N * N))) / (N * (1.0 + (0.5 / 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(0.25 / Float64(N * N))) / Float64(N * Float64(1.0 + Float64(0.5 / 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 - (0.25 / (N * N))) / (N * (1.0 + (0.5 / N)));
	end
	tmp_2 = tmp;
end

Wolfram

code[N_] := If[LessEqual[N, 0.68], (-N[Log[N], $MachinePrecision]), N[(N[(1.0 - N[(0.25 / N[(N * N), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N * N[(1.0 + N[(0.5 / N), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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. Step-by-step derivation

      1. +-commutative 100.0%

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

      2. log1p-def 100.0%

         \[\leadsto \color{blue}{\mathsf{log1p}\left(N\right)} - \log N \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(N\right) - \log N} \]

   4. Taylor expanded in N around 0 98.0%

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

      1. neg-mul-1 98.0%

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

   6. Simplified 98.0%

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

   if 0.680000000000000049 < N

   1. Initial program 8.6%

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

      1. +-commutative 8.6%

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

      2. log1p-def 8.6%

         \[\leadsto \color{blue}{\mathsf{log1p}\left(N\right)} - \log N \]

   3. Simplified 8.6%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(N\right) - \log N} \]

   4. Taylor expanded in N around inf 98.8%

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

      1. associate-*r/ 98.8%

         \[\leadsto \frac{1}{N} - \color{blue}{\frac{0.5 \cdot 1}{{N}^{2}}} \]

      2. metadata-eval 98.8%

         \[\leadsto \frac{1}{N} - \frac{\color{blue}{0.5}}{{N}^{2}} \]

      3. unpow2 98.8%

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

      4. associate-/r* 98.8%

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

   6. Simplified 98.8%

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

      1. sub-div 98.8%

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

      2. sub-neg 98.8%

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

      3. distribute-neg-frac 98.8%

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

      4. metadata-eval 98.8%

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

   8. Applied egg-rr 98.8%

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

      1. flip-+ 98.8%

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

      2. metadata-eval 98.8%

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

   10. Applied egg-rr 98.8%

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

       1. associate-*l/ 98.8%

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

       2. associate-*r/ 98.8%

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

       3. metadata-eval 98.8%

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

   12. Simplified 98.8%

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

       1. expm1-log1p-u 98.8%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{1 - \frac{\frac{0.25}{N}}{N}}{1 - \frac{-0.5}{N}}}{N}\right)\right)} \]

       2. expm1-udef 8.4%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{1 - \frac{\frac{0.25}{N}}{N}}{1 - \frac{-0.5}{N}}}{N}\right)} - 1} \]

       3. associate-/l/ 8.4%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{1 - \frac{\frac{0.25}{N}}{N}}{N \cdot \left(1 - \frac{-0.5}{N}\right)}}\right)} - 1 \]

   14. Applied egg-rr 8.4%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1 - \frac{\frac{0.25}{N}}{N}}{N \cdot \left(1 - \frac{-0.5}{N}\right)}\right)} - 1} \]
   15. Step-by-step derivation

       1. expm1-def 98.8%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1 - \frac{\frac{0.25}{N}}{N}}{N \cdot \left(1 - \frac{-0.5}{N}\right)}\right)\right)} \]

       2. expm1-log1p 98.8%

          \[\leadsto \color{blue}{\frac{1 - \frac{\frac{0.25}{N}}{N}}{N \cdot \left(1 - \frac{-0.5}{N}\right)}} \]

       3. associate-/l/ 98.8%

          \[\leadsto \frac{1 - \color{blue}{\frac{0.25}{N \cdot N}}}{N \cdot \left(1 - \frac{-0.5}{N}\right)} \]

       4. sub-neg 98.8%

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

       5. distribute-neg-frac 98.8%

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

       6. metadata-eval 98.8%

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

   16. Simplified 98.8%

       \[\leadsto \color{blue}{\frac{1 - \frac{0.25}{N \cdot N}}{N \cdot \left(1 + \frac{0.5}{N}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;N \leq 0.68:\\ \;\;\;\;-\log N\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{0.25}{N \cdot N}}{N \cdot \left(1 + \frac{0.5}{N}\right)}\\ \end{array} \]

Alternative 7: 52.4% 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 55.4%

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

   1. +-commutative 55.4%

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

   2. log1p-def 55.4%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(N\right)} - \log N \]

3. Simplified 55.4%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(N\right) - \log N} \]

4. Taylor expanded in N around inf 50.4%

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

5. Final simplification 50.4%

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

Alternative 8: 4.5% 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 55.4%

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

   1. +-commutative 55.4%

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

   2. log1p-def 55.4%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(N\right)} - \log N \]

3. Simplified 55.4%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(N\right) - \log N} \]

4. Taylor expanded in N around 0 52.4%

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

   1. neg-mul-1 52.4%

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

   2. unsub-neg 52.4%

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

6. Simplified 52.4%

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

7. Taylor expanded in N around inf 4.6%

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

8. Final simplification 4.6%

   \[\leadsto N \]

Reproduce

herbie shell --seed 2023230 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1.0)) (log N)))