2log (problem 3.3.6)

Percentage Accurate: 53.4% → 99.9%

Time: 8.5s

Alternatives: 9

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: 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

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 0.0005:\\ \;\;\;\;\left(\frac{0.3333333333333333}{{N}^{3}} + \frac{1 + \frac{-0.5}{N}}{N}\right) + \frac{-0.25}{{N}^{4}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= (- (log (+ N 1.0)) (log N)) 0.0005)
   (+
    (+ (/ 0.3333333333333333 (pow N 3.0)) (/ (+ 1.0 (/ -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)) <= 0.0005) {
		tmp = ((0.3333333333333333 / pow(N, 3.0)) + ((1.0 + (-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)) <= 0.0005d0) then
        tmp = ((0.3333333333333333d0 / (n ** 3.0d0)) + ((1.0d0 + ((-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)) <= 0.0005) {
		tmp = ((0.3333333333333333 / Math.pow(N, 3.0)) + ((1.0 + (-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)) <= 0.0005:
		tmp = ((0.3333333333333333 / math.pow(N, 3.0)) + ((1.0 + (-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)) <= 0.0005)
		tmp = Float64(Float64(Float64(0.3333333333333333 / (N ^ 3.0)) + Float64(Float64(1.0 + Float64(-0.5 / 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)) <= 0.0005)
		tmp = ((0.3333333333333333 / (N ^ 3.0)) + ((1.0 + (-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], 0.0005], N[(N[(N[(0.3333333333333333 / N[Power[N, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 + N[(-0.5 / N), $MachinePrecision]), $MachinePrecision] / N), $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)) < 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. 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%

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

      2. metadata-eval 99.9%

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

      3. +-commutative 99.9%

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

      4. associate-*r/ 99.9%

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

      5. metadata-eval 99.9%

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

      6. unpow2 99.9%

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

      7. associate-*r/ 99.9%

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

      8. metadata-eval 99.9%

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

   6. Simplified 99.9%

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

   7. Taylor expanded in N around 0 99.9%

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

   9. Simplified 99.9%

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

   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%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 0.0005:\\ \;\;\;\;\left(\frac{0.3333333333333333}{{N}^{3}} + \frac{1 + \frac{-0.5}{N}}{N}\right) + \frac{-0.25}{{N}^{4}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \end{array} \]

Alternative 2: 99.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\frac{0.3333333333333333}{{N}^{3}} + \frac{1 + \frac{-0.5}{N}}{N}\\ \mathbf{else}:\\ \;\;\;\;-\log \left(\frac{N}{N + 1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= (- (log (+ N 1.0)) (log N)) 2e-6)
   (+ (/ 0.3333333333333333 (pow N 3.0)) (/ (+ 1.0 (/ -0.5 N)) N))
   (- (log (/ N (+ N 1.0))))))

C

double code(double N) {
	double tmp;
	if ((log((N + 1.0)) - log(N)) <= 2e-6) {
		tmp = (0.3333333333333333 / pow(N, 3.0)) + ((1.0 + (-0.5 / N)) / 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)) <= 2d-6) then
        tmp = (0.3333333333333333d0 / (n ** 3.0d0)) + ((1.0d0 + ((-0.5d0) / n)) / 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)) <= 2e-6) {
		tmp = (0.3333333333333333 / Math.pow(N, 3.0)) + ((1.0 + (-0.5 / N)) / 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)) <= 2e-6:
		tmp = (0.3333333333333333 / math.pow(N, 3.0)) + ((1.0 + (-0.5 / N)) / 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)) <= 2e-6)
		tmp = Float64(Float64(0.3333333333333333 / (N ^ 3.0)) + Float64(Float64(1.0 + Float64(-0.5 / N)) / 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)) <= 2e-6)
		tmp = (0.3333333333333333 / (N ^ 3.0)) + ((1.0 + (-0.5 / N)) / 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], 2e-6], N[(N[(0.3333333333333333 / N[Power[N, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 + N[(-0.5 / N), $MachinePrecision]), $MachinePrecision] / N), $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)) < 1.99999999999999991e-6

   1. Initial program 6.7%

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

      1. +-commutative 6.7%

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

      2. log1p-def 6.7%

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

   3. Simplified 6.7%

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

   4. Taylor expanded in N around inf 100.0%

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

      1. associate--l+ 100.0%

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

      2. associate-*r/ 100.0%

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

      3. metadata-eval 100.0%

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

      4. associate-*r/ 100.0%

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

      5. metadata-eval 100.0%

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

      6. unpow2 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in N around 0 100.0%

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

      1. associate--l+ 100.0%

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

      2. associate-*r/ 100.0%

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

      3. metadata-eval 100.0%

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

      4. sub-neg 100.0%

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

      5. distribute-lft-neg-in 100.0%

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

      6. metadata-eval 100.0%

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

      7. unpow2 100.0%

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

      8. associate-*r/ 100.0%

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

      9. metadata-eval 100.0%

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

      10. associate-/r* 100.0%

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

      11. *-rgt-identity 100.0%

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

      12. associate-*r/ 100.0%

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

      13. distribute-rgt1-in 100.0%

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

      14. +-commutative 100.0%

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

      15. associate-*r/ 100.0%

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

      16. *-rgt-identity 100.0%

          \[\leadsto \frac{0.3333333333333333}{{N}^{3}} + \frac{\color{blue}{1 + \frac{-0.5}{N}}}{N} \]

   9. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{0.3333333333333333}{{N}^{3}} + \frac{1 + \frac{-0.5}{N}}{N}} \]

   if 1.99999999999999991e-6 < (-.f64 (log.f64 (+.f64 N 1)) (log.f64 N))

   1. Initial program 99.7%

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

      1. +-commutative 99.7%

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

      2. log1p-def 99.7%

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

   3. Simplified 99.7%

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

      1. log1p-udef 99.7%

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

      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 99.8%

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

   7. Applied egg-rr 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\frac{0.3333333333333333}{{N}^{3}} + \frac{1 + \frac{-0.5}{N}}{N}\\ \mathbf{else}:\\ \;\;\;\;-\log \left(\frac{N}{N + 1}\right)\\ \end{array} \]

Alternative 3: 99.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;N \leq 126000:\\ \;\;\;\;-\log \left(\frac{N}{N + 1}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N + 0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= N 126000.0) (- (log (/ N (+ N 1.0)))) (/ 1.0 (+ N 0.5))))

C

double code(double N) {
	double tmp;
	if (N <= 126000.0) {
		tmp = -log((N / (N + 1.0)));
	} 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 <= 126000.0d0) then
        tmp = -log((n / (n + 1.0d0)))
    else
        tmp = 1.0d0 / (n + 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double N) {
	double tmp;
	if (N <= 126000.0) {
		tmp = -Math.log((N / (N + 1.0)));
	} else {
		tmp = 1.0 / (N + 0.5);
	}
	return tmp;
}

Python

def code(N):
	tmp = 0
	if N <= 126000.0:
		tmp = -math.log((N / (N + 1.0)))
	else:
		tmp = 1.0 / (N + 0.5)
	return tmp

Julia

function code(N)
	tmp = 0.0
	if (N <= 126000.0)
		tmp = Float64(-log(Float64(N / Float64(N + 1.0))));
	else
		tmp = Float64(1.0 / Float64(N + 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(N)
	tmp = 0.0;
	if (N <= 126000.0)
		tmp = -log((N / (N + 1.0)));
	else
		tmp = 1.0 / (N + 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[N_] := If[LessEqual[N, 126000.0], (-N[Log[N[(N / N[(N + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), N[(1.0 / N[(N + 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if N < 126000

   1. Initial program 99.7%

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

      1. +-commutative 99.7%

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

      2. log1p-def 99.7%

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

   3. Simplified 99.7%

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

      1. log1p-udef 99.7%

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

      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 99.8%

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

   7. Applied egg-rr 99.8%

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

   if 126000 < N

   1. Initial program 6.7%

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

      1. +-commutative 6.7%

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

      2. log1p-def 6.7%

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

   3. Simplified 6.7%

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

   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. clear-num 99.8%

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

      3. sub-neg 99.8%

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

      4. distribute-neg-frac 99.8%

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

      5. metadata-eval 99.8%

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

   8. Applied egg-rr 99.8%

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

   9. Taylor expanded in N around inf 99.8%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;N \leq 126000:\\ \;\;\;\;-\log \left(\frac{N}{N + 1}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N + 0.5}\\ \end{array} \]

Alternative 4: 99.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;N \leq 195000:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N + 0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= N 195000.0) (log (/ (+ N 1.0) N)) (/ 1.0 (+ N 0.5))))

C

double code(double N) {
	double tmp;
	if (N <= 195000.0) {
		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 <= 195000.0d0) 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 <= 195000.0) {
		tmp = Math.log(((N + 1.0) / N));
	} else {
		tmp = 1.0 / (N + 0.5);
	}
	return tmp;
}

Python

def code(N):
	tmp = 0
	if N <= 195000.0:
		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 <= 195000.0)
		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 <= 195000.0)
		tmp = log(((N + 1.0) / N));
	else
		tmp = 1.0 / (N + 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[N_] := If[LessEqual[N, 195000.0], 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 < 195000

   1. Initial program 99.7%

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

      1. +-commutative 99.7%

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

      2. log1p-def 99.7%

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

   3. Simplified 99.7%

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

      1. log1p-udef 99.7%

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

      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 195000 < N

   1. Initial program 6.7%

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

      1. +-commutative 6.7%

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

      2. log1p-def 6.7%

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

   3. Simplified 6.7%

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

   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. clear-num 99.8%

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

      3. sub-neg 99.8%

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

      4. distribute-neg-frac 99.8%

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

      5. metadata-eval 99.8%

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

   8. Applied egg-rr 99.8%

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

   9. Taylor expanded in N around inf 99.8%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;N \leq 195000:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N + 0.5}\\ \end{array} \]

Alternative 5: 99.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;N \leq 0.58:\\ \;\;\;\;N - \log N\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N + 0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= N 0.58) (- N (log N)) (/ 1.0 (+ N 0.5))))

C

double code(double N) {
	double tmp;
	if (N <= 0.58) {
		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 <= 0.58d0) 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 <= 0.58) {
		tmp = N - Math.log(N);
	} else {
		tmp = 1.0 / (N + 0.5);
	}
	return tmp;
}

Python

def code(N):
	tmp = 0
	if N <= 0.58:
		tmp = N - math.log(N)
	else:
		tmp = 1.0 / (N + 0.5)
	return tmp

Julia

function code(N)
	tmp = 0.0
	if (N <= 0.58)
		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 <= 0.58)
		tmp = N - log(N);
	else
		tmp = 1.0 / (N + 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[N_] := If[LessEqual[N, 0.58], 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 < 0.57999999999999996

   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 99.8%

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

      1. neg-mul-1 99.8%

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

      2. unsub-neg 99.8%

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

   6. Simplified 99.8%

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

   if 0.57999999999999996 < N

   1. Initial program 8.4%

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

      1. +-commutative 8.4%

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

      2. log1p-def 8.4%

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

   3. Simplified 8.4%

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

   4. Taylor expanded in N around inf 98.7%

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

      1. associate-*r/ 98.7%

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

      2. metadata-eval 98.7%

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

      3. unpow2 98.7%

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

      4. associate-/r* 98.7%

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

   6. Simplified 98.7%

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

      1. sub-div 98.7%

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

      2. clear-num 98.7%

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

      3. sub-neg 98.7%

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

      4. distribute-neg-frac 98.7%

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

      5. metadata-eval 98.7%

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

   8. Applied egg-rr 98.7%

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

   9. Taylor expanded in N around inf 98.8%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;N \leq 0.58:\\ \;\;\;\;N - \log N\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N + 0.5}\\ \end{array} \]

Alternative 6: 98.8% 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 99.0%

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

      1. neg-mul-1 99.0%

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

   6. Simplified 99.0%

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

   if 0.28000000000000003 < N

   1. Initial program 8.4%

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

      1. +-commutative 8.4%

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

      2. log1p-def 8.4%

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

   3. Simplified 8.4%

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

   4. Taylor expanded in N around inf 98.7%

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

      1. associate-*r/ 98.7%

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

      2. metadata-eval 98.7%

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

      3. unpow2 98.7%

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

      4. associate-/r* 98.7%

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

   6. Simplified 98.7%

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

      1. sub-div 98.7%

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

      2. clear-num 98.7%

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

      3. sub-neg 98.7%

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

      4. distribute-neg-frac 98.7%

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

      5. metadata-eval 98.7%

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

   8. Applied egg-rr 98.7%

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

   9. Taylor expanded in N around inf 98.8%

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

4. Final simplification 98.9%

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

Alternative 7: 57.2% 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 46.7%

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

   1. +-commutative 46.7%

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

   2. log1p-def 46.7%

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

3. Simplified 46.7%

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

4. Taylor expanded in N around inf 57.8%

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

   1. associate-*r/ 57.8%

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

   2. metadata-eval 57.8%

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

   3. unpow2 57.8%

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

   4. associate-/r* 57.8%

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

6. Simplified 57.8%

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

   1. sub-div 57.8%

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

   2. clear-num 57.8%

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

   3. sub-neg 57.8%

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

   4. distribute-neg-frac 57.8%

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

   5. metadata-eval 57.8%

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

8. Applied egg-rr 57.8%

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

9. Taylor expanded in N around inf 63.5%

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

10. Final simplification 63.5%

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

Alternative 8: 52.3% 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 46.7%

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

   1. +-commutative 46.7%

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

   2. log1p-def 46.7%

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

3. Simplified 46.7%

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

4. Taylor expanded in N around inf 59.2%

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

5. Final simplification 59.2%

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

Alternative 9: 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 46.7%

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

   1. +-commutative 46.7%

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

   2. log1p-def 46.7%

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

3. Simplified 46.7%

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

4. Taylor expanded in N around 0 43.9%

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

   1. neg-mul-1 43.9%

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

   2. unsub-neg 43.9%

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

6. Simplified 43.9%

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

7. Taylor expanded in N around inf 4.4%

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

8. Final simplification 4.4%

   \[\leadsto N \]

Reproduce

herbie shell --seed 2023240 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1.0)) (log N)))