2log (problem 3.3.6)

Percentage Accurate: 54.7% → 100.0%

Time: 7.0s

Alternatives: 8

Speedup: 41.0×

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: 54.7% 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: 100.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 0.0005:\\ \;\;\;\;\frac{e^{\left(\frac{\frac{0.20833333333333334}{N}}{N} + \frac{-0.5}{N}\right) + \frac{-0.125}{{N}^{3}}}}{N}\\ \mathbf{else}:\\ \;\;\;\;-\log \left(\frac{N}{N + 1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= (- (log (+ N 1.0)) (log N)) 0.0005)
   (/
    (exp
     (+ (+ (/ (/ 0.20833333333333334 N) N) (/ -0.5 N)) (/ -0.125 (pow N 3.0))))
    N)
   (- (log (/ N (+ N 1.0))))))

C

double code(double N) {
	double tmp;
	if ((log((N + 1.0)) - log(N)) <= 0.0005) {
		tmp = exp(((((0.20833333333333334 / N) / N) + (-0.5 / N)) + (-0.125 / pow(N, 3.0)))) / N;
	} else {
		tmp = -log((N / (N + 1.0)));
	}
	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 = exp(((((0.20833333333333334d0 / n) / n) + ((-0.5d0) / n)) + ((-0.125d0) / (n ** 3.0d0)))) / n
    else
        tmp = -log((n / (n + 1.0d0)))
    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 = Math.exp(((((0.20833333333333334 / N) / N) + (-0.5 / N)) + (-0.125 / Math.pow(N, 3.0)))) / N;
	} else {
		tmp = -Math.log((N / (N + 1.0)));
	}
	return tmp;
}

Python

def code(N):
	tmp = 0
	if (math.log((N + 1.0)) - math.log(N)) <= 0.0005:
		tmp = math.exp(((((0.20833333333333334 / N) / N) + (-0.5 / N)) + (-0.125 / math.pow(N, 3.0)))) / N
	else:
		tmp = -math.log((N / (N + 1.0)))
	return tmp

Julia

function code(N)
	tmp = 0.0
	if (Float64(log(Float64(N + 1.0)) - log(N)) <= 0.0005)
		tmp = Float64(exp(Float64(Float64(Float64(Float64(0.20833333333333334 / N) / N) + Float64(-0.5 / N)) + Float64(-0.125 / (N ^ 3.0)))) / N);
	else
		tmp = Float64(-log(Float64(N / Float64(N + 1.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(N)
	tmp = 0.0;
	if ((log((N + 1.0)) - log(N)) <= 0.0005)
		tmp = exp(((((0.20833333333333334 / N) / N) + (-0.5 / N)) + (-0.125 / (N ^ 3.0)))) / N;
	else
		tmp = -log((N / (N + 1.0)));
	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[(N[Exp[N[(N[(N[(N[(0.20833333333333334 / N), $MachinePrecision] / N), $MachinePrecision] + N[(-0.5 / N), $MachinePrecision]), $MachinePrecision] + N[(-0.125 / N[Power[N, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N), $MachinePrecision], (-N[Log[N[(N / N[(N + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (log.f64 (+.f64 N 1)) (log.f64 N)) < 5.0000000000000001e-4

   1. Initial program 7.2%

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

      1. +-commutative 7.2%

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

      2. log1p-def 7.2%

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

   3. Simplified 7.2%

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

      1. add-exp-log 7.2%

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

   5. Applied egg-rr 7.2%

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

   6. Taylor expanded in N around inf 90.7%

      \[\leadsto e^{\color{blue}{\left(\log \left(\frac{1}{N}\right) + 0.20833333333333334 \cdot \frac{1}{{N}^{2}}\right) - \left(0.125 \cdot \frac{1}{{N}^{3}} + 0.5 \cdot \frac{1}{N}\right)}} \]
   7. Step-by-step derivation

      1. log-rec 90.7%

         \[\leadsto e^{\left(\color{blue}{\left(-\log N\right)} + 0.20833333333333334 \cdot \frac{1}{{N}^{2}}\right) - \left(0.125 \cdot \frac{1}{{N}^{3}} + 0.5 \cdot \frac{1}{N}\right)} \]

      2. +-commutative 90.7%

         \[\leadsto e^{\color{blue}{\left(0.20833333333333334 \cdot \frac{1}{{N}^{2}} + \left(-\log N\right)\right)} - \left(0.125 \cdot \frac{1}{{N}^{3}} + 0.5 \cdot \frac{1}{N}\right)} \]

      3. unsub-neg 90.7%

         \[\leadsto e^{\color{blue}{\left(0.20833333333333334 \cdot \frac{1}{{N}^{2}} - \log N\right)} - \left(0.125 \cdot \frac{1}{{N}^{3}} + 0.5 \cdot \frac{1}{N}\right)} \]

      4. unpow2 90.7%

         \[\leadsto e^{\left(0.20833333333333334 \cdot \frac{1}{\color{blue}{N \cdot N}} - \log N\right) - \left(0.125 \cdot \frac{1}{{N}^{3}} + 0.5 \cdot \frac{1}{N}\right)} \]

      5. associate-*r/ 90.7%

         \[\leadsto e^{\left(\color{blue}{\frac{0.20833333333333334 \cdot 1}{N \cdot N}} - \log N\right) - \left(0.125 \cdot \frac{1}{{N}^{3}} + 0.5 \cdot \frac{1}{N}\right)} \]

      6. metadata-eval 90.7%

         \[\leadsto e^{\left(\frac{\color{blue}{0.20833333333333334}}{N \cdot N} - \log N\right) - \left(0.125 \cdot \frac{1}{{N}^{3}} + 0.5 \cdot \frac{1}{N}\right)} \]

      7. associate-*r/ 90.7%

         \[\leadsto e^{\left(\frac{0.20833333333333334}{N \cdot N} - \log N\right) - \left(\color{blue}{\frac{0.125 \cdot 1}{{N}^{3}}} + 0.5 \cdot \frac{1}{N}\right)} \]

      8. metadata-eval 90.7%

         \[\leadsto e^{\left(\frac{0.20833333333333334}{N \cdot N} - \log N\right) - \left(\frac{\color{blue}{0.125}}{{N}^{3}} + 0.5 \cdot \frac{1}{N}\right)} \]

      9. associate-*r/ 90.7%

         \[\leadsto e^{\left(\frac{0.20833333333333334}{N \cdot N} - \log N\right) - \left(\frac{0.125}{{N}^{3}} + \color{blue}{\frac{0.5 \cdot 1}{N}}\right)} \]

      10. metadata-eval 90.7%

          \[\leadsto e^{\left(\frac{0.20833333333333334}{N \cdot N} - \log N\right) - \left(\frac{0.125}{{N}^{3}} + \frac{\color{blue}{0.5}}{N}\right)} \]

   8. Simplified 90.7%

      \[\leadsto e^{\color{blue}{\left(\frac{0.20833333333333334}{N \cdot N} - \log N\right) - \left(\frac{0.125}{{N}^{3}} + \frac{0.5}{N}\right)}} \]

   9. Taylor expanded in N around 0 90.7%

      \[\leadsto \color{blue}{e^{0.20833333333333334 \cdot \frac{1}{{N}^{2}} - \left(\log N + \left(0.125 \cdot \frac{1}{{N}^{3}} + 0.5 \cdot \frac{1}{N}\right)\right)}} \]

   11. Simplified 99.9%

       \[\leadsto \color{blue}{\frac{e^{\left(\frac{\frac{0.20833333333333334}{N}}{N} + \frac{-0.5}{N}\right) + \frac{-0.125}{{N}^{3}}}}{N}} \]

   if 5.0000000000000001e-4 < (-.f64 (log.f64 (+.f64 N 1)) (log.f64 N))

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. log1p-def 99.9%

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

   3. Simplified 99.9%

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

      1. log1p-udef 99.9%

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

      2. diff-log 99.9%

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

      3. +-commutative 99.9%

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

   5. Applied egg-rr 99.9%

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

      1. clear-num 99.9%

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

      2. log-rec 100.0%

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

      3. +-commutative 100.0%

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

   7. 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%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 0.0005:\\ \;\;\;\;\frac{e^{\left(\frac{\frac{0.20833333333333334}{N}}{N} + \frac{-0.5}{N}\right) + \frac{-0.125}{{N}^{3}}}}{N}\\ \mathbf{else}:\\ \;\;\;\;-\log \left(\frac{N}{N + 1}\right)\\ \end{array} \]

Alternative 2: 99.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\frac{1}{N + 0.5}\\ \mathbf{else}:\\ \;\;\;\;-\log \left(\frac{N}{N + 1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= (- (log (+ N 1.0)) (log N)) 2e-8)
   (/ 1.0 (+ N 0.5))
   (- (log (/ N (+ N 1.0))))))

C

double code(double N) {
	double tmp;
	if ((log((N + 1.0)) - log(N)) <= 2e-8) {
		tmp = 1.0 / (N + 0.5);
	} else {
		tmp = -log((N / (N + 1.0)));
	}
	return tmp;
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((log((n + 1.0d0)) - log(n)) <= 2d-8) then
        tmp = 1.0d0 / (n + 0.5d0)
    else
        tmp = -log((n / (n + 1.0d0)))
    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-8) {
		tmp = 1.0 / (N + 0.5);
	} else {
		tmp = -Math.log((N / (N + 1.0)));
	}
	return tmp;
}

Python

def code(N):
	tmp = 0
	if (math.log((N + 1.0)) - math.log(N)) <= 2e-8:
		tmp = 1.0 / (N + 0.5)
	else:
		tmp = -math.log((N / (N + 1.0)))
	return tmp

Julia

function code(N)
	tmp = 0.0
	if (Float64(log(Float64(N + 1.0)) - log(N)) <= 2e-8)
		tmp = Float64(1.0 / Float64(N + 0.5));
	else
		tmp = Float64(-log(Float64(N / Float64(N + 1.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(N)
	tmp = 0.0;
	if ((log((N + 1.0)) - log(N)) <= 2e-8)
		tmp = 1.0 / (N + 0.5);
	else
		tmp = -log((N / (N + 1.0)));
	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-8], N[(1.0 / N[(N + 0.5), $MachinePrecision]), $MachinePrecision], (-N[Log[N[(N / N[(N + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (log.f64 (+.f64 N 1)) (log.f64 N)) < 2e-8

   1. Initial program 6.6%

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

      1. +-commutative 6.6%

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

      2. log1p-def 6.6%

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

   3. Simplified 6.6%

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

   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. associate-*r/ 100.0%

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

      2. metadata-eval 100.0%

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

      3. unpow2 100.0%

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

      4. associate-/r* 100.0%

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

   6. Simplified 100.0%

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

      1. sub-div 100.0%

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

      2. clear-num 100.0%

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

      3. sub-neg 100.0%

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

      4. distribute-neg-frac 100.0%

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

      5. metadata-eval 100.0%

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

   8. Applied egg-rr 100.0%

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

   9. Taylor expanded in N around inf 100.0%

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

       1. +-commutative 100.0%

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

   11. Simplified 100.0%

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

   if 2e-8 < (-.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. +-commutative 99.8%

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

      2. log1p-def 99.8%

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

   3. Simplified 99.8%

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

      1. log1p-udef 99.8%

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

      2. diff-log 99.8%

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

      3. +-commutative 99.8%

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

   5. Applied egg-rr 99.8%

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

      1. clear-num 99.8%

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

      2. log-rec 99.9%

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

      3. +-commutative 99.9%

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

   7. 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%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\frac{1}{N + 0.5}\\ \mathbf{else}:\\ \;\;\;\;-\log \left(\frac{N}{N + 1}\right)\\ \end{array} \]

Alternative 3: 99.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;N \leq 100000:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N + 0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= N 100000.0) (log (/ (+ N 1.0) N)) (/ 1.0 (+ N 0.5))))

C

double code(double N) {
	double tmp;
	if (N <= 100000.0) {
		tmp = log(((N + 1.0) / N));
	} else {
		tmp = 1.0 / (N + 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= 100000.0d0) then
        tmp = log(((n + 1.0d0) / n))
    else
        tmp = 1.0d0 / (n + 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double N) {
	double tmp;
	if (N <= 100000.0) {
		tmp = Math.log(((N + 1.0) / N));
	} else {
		tmp = 1.0 / (N + 0.5);
	}
	return tmp;
}

Python

def code(N):
	tmp = 0
	if N <= 100000.0:
		tmp = math.log(((N + 1.0) / N))
	else:
		tmp = 1.0 / (N + 0.5)
	return tmp

Julia

function code(N)
	tmp = 0.0
	if (N <= 100000.0)
		tmp = log(Float64(Float64(N + 1.0) / N));
	else
		tmp = Float64(1.0 / Float64(N + 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(N)
	tmp = 0.0;
	if (N <= 100000.0)
		tmp = log(((N + 1.0) / N));
	else
		tmp = 1.0 / (N + 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[N_] := If[LessEqual[N, 100000.0], N[Log[N[(N[(N + 1.0), $MachinePrecision] / N), $MachinePrecision]], $MachinePrecision], N[(1.0 / N[(N + 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if N < 1e5

   1. Initial program 99.8%

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

      1. +-commutative 99.8%

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

      2. log1p-def 99.8%

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

   3. Simplified 99.8%

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

      1. log1p-udef 99.8%

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

      2. diff-log 99.8%

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

      3. +-commutative 99.8%

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

   5. Applied egg-rr 99.8%

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

   if 1e5 < N

   1. Initial program 6.6%

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

      1. +-commutative 6.6%

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

      2. log1p-def 6.6%

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

   3. Simplified 6.6%

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

   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. associate-*r/ 100.0%

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

      2. metadata-eval 100.0%

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

      3. unpow2 100.0%

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

      4. associate-/r* 100.0%

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

   6. Simplified 100.0%

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

      1. sub-div 100.0%

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

      2. clear-num 100.0%

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

      3. sub-neg 100.0%

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

      4. distribute-neg-frac 100.0%

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

      5. metadata-eval 100.0%

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

   8. Applied egg-rr 100.0%

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

   9. Taylor expanded in N around inf 100.0%

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

       1. +-commutative 100.0%

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

   11. Simplified 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;N \leq 100000:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N + 0.5}\\ \end{array} \]

Alternative 4: 99.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;N \leq 0.6:\\ \;\;\;\;N - \log N\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N + 0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= N 0.6) (- N (log N)) (/ 1.0 (+ N 0.5))))

C

double code(double N) {
	double tmp;
	if (N <= 0.6) {
		tmp = N - log(N);
	} else {
		tmp = 1.0 / (N + 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= 0.6d0) then
        tmp = n - log(n)
    else
        tmp = 1.0d0 / (n + 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double N) {
	double tmp;
	if (N <= 0.6) {
		tmp = N - Math.log(N);
	} else {
		tmp = 1.0 / (N + 0.5);
	}
	return tmp;
}

Python

def code(N):
	tmp = 0
	if N <= 0.6:
		tmp = N - math.log(N)
	else:
		tmp = 1.0 / (N + 0.5)
	return tmp

Julia

function code(N)
	tmp = 0.0
	if (N <= 0.6)
		tmp = Float64(N - log(N));
	else
		tmp = Float64(1.0 / Float64(N + 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(N)
	tmp = 0.0;
	if (N <= 0.6)
		tmp = N - log(N);
	else
		tmp = 1.0 / (N + 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[N_] := If[LessEqual[N, 0.6], N[(N - N[Log[N], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N + 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if N < 0.599999999999999978

   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 99.4%

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

      1. neg-mul-1 99.4%

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

      2. unsub-neg 99.4%

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

   6. Simplified 99.4%

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

   if 0.599999999999999978 < N

   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 99.2%

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

      1. associate-*r/ 99.2%

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

      2. metadata-eval 99.2%

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

      3. unpow2 99.2%

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

      4. associate-/r* 99.2%

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

   6. Simplified 99.2%

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

      1. sub-div 99.2%

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

      2. clear-num 99.2%

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

      3. sub-neg 99.2%

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

      4. distribute-neg-frac 99.2%

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

      5. metadata-eval 99.2%

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

   8. Applied egg-rr 99.2%

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

   9. Taylor expanded in N around inf 99.3%

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

       1. +-commutative 99.3%

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

   11. Simplified 99.3%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;N \leq 0.6:\\ \;\;\;\;N - \log N\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N + 0.5}\\ \end{array} \]

Alternative 5: 98.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;N \leq 0.28:\\ \;\;\;\;-\log N\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N + 0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (N) :precision binary64 (if (<= N 0.28) (- (log N)) (/ 1.0 (+ N 0.5))))

C

double code(double N) {
	double tmp;
	if (N <= 0.28) {
		tmp = -log(N);
	} else {
		tmp = 1.0 / (N + 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= 0.28d0) then
        tmp = -log(n)
    else
        tmp = 1.0d0 / (n + 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double N) {
	double tmp;
	if (N <= 0.28) {
		tmp = -Math.log(N);
	} else {
		tmp = 1.0 / (N + 0.5);
	}
	return tmp;
}

Python

def code(N):
	tmp = 0
	if N <= 0.28:
		tmp = -math.log(N)
	else:
		tmp = 1.0 / (N + 0.5)
	return tmp

Julia

function code(N)
	tmp = 0.0
	if (N <= 0.28)
		tmp = Float64(-log(N));
	else
		tmp = Float64(1.0 / Float64(N + 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(N)
	tmp = 0.0;
	if (N <= 0.28)
		tmp = -log(N);
	else
		tmp = 1.0 / (N + 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[N_] := If[LessEqual[N, 0.28], (-N[Log[N], $MachinePrecision]), N[(1.0 / N[(N + 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if N < 0.28000000000000003

   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%

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

      1. neg-mul-1 98.7%

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

   6. Simplified 98.7%

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

   if 0.28000000000000003 < N

   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 99.2%

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

      1. associate-*r/ 99.2%

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

      2. metadata-eval 99.2%

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

      3. unpow2 99.2%

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

      4. associate-/r* 99.2%

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

   6. Simplified 99.2%

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

      1. sub-div 99.2%

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

      2. clear-num 99.2%

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

      3. sub-neg 99.2%

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

      4. distribute-neg-frac 99.2%

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

      5. metadata-eval 99.2%

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

   8. Applied egg-rr 99.2%

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

   9. Taylor expanded in N around inf 99.3%

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

       1. +-commutative 99.3%

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

   11. Simplified 99.3%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;N \leq 0.28:\\ \;\;\;\;-\log N\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N + 0.5}\\ \end{array} \]

Alternative 6: 55.6% accurate, 40.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;N \leq 0.5:\\ \;\;\;\;2\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N}\\ \end{array} \end{array} \]

FPCore

(FPCore (N) :precision binary64 (if (<= N 0.5) 2.0 (/ 1.0 N)))

C

double code(double N) {
	double tmp;
	if (N <= 0.5) {
		tmp = 2.0;
	} else {
		tmp = 1.0 / N;
	}
	return tmp;
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= 0.5d0) then
        tmp = 2.0d0
    else
        tmp = 1.0d0 / n
    end if
    code = tmp
end function

Java

public static double code(double N) {
	double tmp;
	if (N <= 0.5) {
		tmp = 2.0;
	} else {
		tmp = 1.0 / N;
	}
	return tmp;
}

Python

def code(N):
	tmp = 0
	if N <= 0.5:
		tmp = 2.0
	else:
		tmp = 1.0 / N
	return tmp

Julia

function code(N)
	tmp = 0.0
	if (N <= 0.5)
		tmp = 2.0;
	else
		tmp = Float64(1.0 / N);
	end
	return tmp
end

MATLAB

function tmp_2 = code(N)
	tmp = 0.0;
	if (N <= 0.5)
		tmp = 2.0;
	else
		tmp = 1.0 / N;
	end
	tmp_2 = tmp;
end

Wolfram

code[N_] := If[LessEqual[N, 0.5], 2.0, N[(1.0 / N), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if N < 0.5

   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 inf 0.9%

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

      1. associate-*r/ 0.9%

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

      2. metadata-eval 0.9%

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

      3. unpow2 0.9%

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

      4. associate-/r* 0.9%

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

   6. Simplified 0.9%

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

      1. sub-div 0.9%

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

      2. clear-num 0.9%

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

      3. sub-neg 0.9%

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

      4. distribute-neg-frac 0.9%

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

      5. metadata-eval 0.9%

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

   8. Applied egg-rr 0.9%

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

   9. Taylor expanded in N around inf 14.4%

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

       1. +-commutative 14.4%

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

   11. Simplified 14.4%

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

   12. Taylor expanded in N around 0 14.4%

       \[\leadsto \color{blue}{2} \]

   if 0.5 < N

   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 98.3%

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

4. Final simplification 58.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;N \leq 0.5:\\ \;\;\;\;2\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N}\\ \end{array} \]

Alternative 7: 56.2% accurate, 41.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.4%

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

   1. +-commutative 51.4%

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

   2. log1p-def 51.4%

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

3. Simplified 51.4%

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

4. Taylor expanded in N around inf 52.7%

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

   1. associate-*r/ 52.7%

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

   2. metadata-eval 52.7%

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

   3. unpow2 52.7%

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

   4. associate-/r* 52.7%

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

6. Simplified 52.7%

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

   1. sub-div 52.7%

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

   2. clear-num 52.7%

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

   3. sub-neg 52.7%

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

   4. distribute-neg-frac 52.7%

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

   5. metadata-eval 52.7%

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

8. Applied egg-rr 52.7%

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

9. Taylor expanded in N around inf 59.2%

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

    1. +-commutative 59.2%

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

11. Simplified 59.2%

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

12. Final simplification 59.2%

    \[\leadsto \frac{1}{N + 0.5} \]

Alternative 8: 10.0% accurate, 205.0× speedup

Math

\[\begin{array}{l} \\ 2 \end{array} \]

FPCore

(FPCore (N) :precision binary64 2.0)

C

double code(double N) {
	return 2.0;
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    code = 2.0d0
end function

Java

public static double code(double N) {
	return 2.0;
}

Python

def code(N):
	return 2.0

Julia

function code(N)
	return 2.0
end

MATLAB

function tmp = code(N)
	tmp = 2.0;
end

Wolfram

code[N_] := 2.0

Derivation

1. Initial program 51.4%

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

   1. +-commutative 51.4%

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

   2. log1p-def 51.4%

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

3. Simplified 51.4%

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

4. Taylor expanded in N around inf 52.7%

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

   1. associate-*r/ 52.7%

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

   2. metadata-eval 52.7%

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

   3. unpow2 52.7%

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

   4. associate-/r* 52.7%

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

6. Simplified 52.7%

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

   1. sub-div 52.7%

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

   2. clear-num 52.7%

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

   3. sub-neg 52.7%

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

   4. distribute-neg-frac 52.7%

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

   5. metadata-eval 52.7%

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

8. Applied egg-rr 52.7%

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

9. Taylor expanded in N around inf 59.2%

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

    1. +-commutative 59.2%

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

11. Simplified 59.2%

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

12. Taylor expanded in N around 0 9.6%

    \[\leadsto \color{blue}{2} \]

13. Final simplification 9.6%

    \[\leadsto 2 \]

Reproduce

herbie shell --seed 2023271 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1.0)) (log N)))