2log (problem 3.3.6)

Percentage Accurate: 24.0% → 99.3%

Time: 12.4s

Alternatives: 10

Speedup: 68.3×

Specification

\[N > 1 \land N < 10^{+40}\]

Math

\[\begin{array}{l} \\ \log \left(N + 1\right) - \log N \end{array} \]

FPCore

(FPCore (N) :precision binary64 (- (log (+ N 1.0)) (log N)))

C

double code(double N) {
	return log((N + 1.0)) - log(N);
}

Fortran

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

Java

public static double code(double N) {
	return Math.log((N + 1.0)) - Math.log(N);
}

Python

def code(N):
	return math.log((N + 1.0)) - math.log(N)

Julia

function code(N)
	return Float64(log(Float64(N + 1.0)) - log(N))
end

MATLAB

function tmp = code(N)
	tmp = log((N + 1.0)) - log(N);
end

Wolfram

code[N_] := N[(N[Log[N[(N + 1.0), $MachinePrecision]], $MachinePrecision] - N[Log[N], $MachinePrecision]), $MachinePrecision]

Initial Program: 24.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \log \left(N + 1\right) - \log N \end{array} \]

FPCore

(FPCore (N) :precision binary64 (- (log (+ N 1.0)) (log N)))

C

double code(double N) {
	return log((N + 1.0)) - log(N);
}

Fortran

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

Java

public static double code(double N) {
	return Math.log((N + 1.0)) - Math.log(N);
}

Python

def code(N):
	return math.log((N + 1.0)) - math.log(N)

Julia

function code(N)
	return Float64(log(Float64(N + 1.0)) - log(N))
end

MATLAB

function tmp = code(N)
	tmp = log((N + 1.0)) - log(N);
end

Wolfram

code[N_] := N[(N[Log[N[(N + 1.0), $MachinePrecision]], $MachinePrecision] - N[Log[N], $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 0.0005:\\ \;\;\;\;\frac{0.3333333333333333}{{N}^{3}} + \left(\frac{1}{N} - \left(\frac{0.5}{{N}^{2}} + \frac{0.25}{{N}^{4}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt{\log \left(\frac{N + 1}{N}\right)}\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= (- (log (+ N 1.0)) (log N)) 0.0005)
   (+
    (/ 0.3333333333333333 (pow N 3.0))
    (- (/ 1.0 N) (+ (/ 0.5 (pow N 2.0)) (/ 0.25 (pow N 4.0)))))
   (pow (sqrt (log (/ (+ N 1.0) N))) 2.0)))

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 / N) - ((0.5 / pow(N, 2.0)) + (0.25 / pow(N, 4.0))));
	} else {
		tmp = pow(sqrt(log(((N + 1.0) / N))), 2.0);
	}
	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 = (0.3333333333333333d0 / (n ** 3.0d0)) + ((1.0d0 / n) - ((0.5d0 / (n ** 2.0d0)) + (0.25d0 / (n ** 4.0d0))))
    else
        tmp = sqrt(log(((n + 1.0d0) / n))) ** 2.0d0
    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 = (0.3333333333333333 / Math.pow(N, 3.0)) + ((1.0 / N) - ((0.5 / Math.pow(N, 2.0)) + (0.25 / Math.pow(N, 4.0))));
	} else {
		tmp = Math.pow(Math.sqrt(Math.log(((N + 1.0) / N))), 2.0);
	}
	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 / N) - ((0.5 / math.pow(N, 2.0)) + (0.25 / math.pow(N, 4.0))))
	else:
		tmp = math.pow(math.sqrt(math.log(((N + 1.0) / N))), 2.0)
	return tmp

Julia

function code(N)
	tmp = 0.0
	if (Float64(log(Float64(N + 1.0)) - log(N)) <= 0.0005)
		tmp = Float64(Float64(0.3333333333333333 / (N ^ 3.0)) + Float64(Float64(1.0 / N) - Float64(Float64(0.5 / (N ^ 2.0)) + Float64(0.25 / (N ^ 4.0)))));
	else
		tmp = sqrt(log(Float64(Float64(N + 1.0) / N))) ^ 2.0;
	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 / N) - ((0.5 / (N ^ 2.0)) + (0.25 / (N ^ 4.0))));
	else
		tmp = sqrt(log(((N + 1.0) / N))) ^ 2.0;
	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[(0.3333333333333333 / N[Power[N, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / N), $MachinePrecision] - N[(N[(0.5 / N[Power[N, 2.0], $MachinePrecision]), $MachinePrecision] + N[(0.25 / N[Power[N, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[Sqrt[N[Log[N[(N[(N + 1.0), $MachinePrecision] / N), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], 2.0], $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 19.6%

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

      1. +-commutative 19.6%

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

      2. log1p-def 19.6%

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

   3. Simplified 19.6%

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

   5. Taylor expanded in N around inf 99.8%

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

      1. associate--l+ 99.8%

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

      2. associate-*r/ 99.8%

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

      3. metadata-eval 99.8%

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

      4. +-commutative 99.8%

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

      5. associate-*r/ 99.8%

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

      6. metadata-eval 99.8%

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

      7. associate-*r/ 99.8%

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

      8. metadata-eval 99.8%

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

   7. Simplified 99.8%

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

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

   1. Initial program 93.9%

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

      1. +-commutative 93.9%

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

      2. log1p-def 94.1%

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

   3. Simplified 94.1%

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

      1. add-sqr-sqrt 94.3%

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

      2. pow2 94.3%

         \[\leadsto \color{blue}{{\left(\sqrt{\mathsf{log1p}\left(N\right) - \log N}\right)}^{2}} \]

   6. Applied egg-rr 94.3%

      \[\leadsto \color{blue}{{\left(\sqrt{\mathsf{log1p}\left(N\right) - \log N}\right)}^{2}} \]
   7. Step-by-step derivation

      1. add-log-exp 94.1%

         \[\leadsto {\left(\sqrt{\color{blue}{\log \left(e^{\mathsf{log1p}\left(N\right)}\right)} - \log N}\right)}^{2} \]

      2. log1p-expm1-u 94.1%

         \[\leadsto {\left(\sqrt{\log \left(e^{\mathsf{log1p}\left(N\right)}\right) - \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log N\right)\right)}}\right)}^{2} \]

      3. log1p-udef 94.1%

         \[\leadsto {\left(\sqrt{\log \left(e^{\mathsf{log1p}\left(N\right)}\right) - \color{blue}{\log \left(1 + \mathsf{expm1}\left(\log N\right)\right)}}\right)}^{2} \]

      4. diff-log 93.9%

         \[\leadsto {\left(\sqrt{\color{blue}{\log \left(\frac{e^{\mathsf{log1p}\left(N\right)}}{1 + \mathsf{expm1}\left(\log N\right)}\right)}}\right)}^{2} \]

      5. log1p-udef 93.8%

         \[\leadsto {\left(\sqrt{\log \left(\frac{e^{\color{blue}{\log \left(1 + N\right)}}}{1 + \mathsf{expm1}\left(\log N\right)}\right)}\right)}^{2} \]

      6. rem-exp-log 95.1%

         \[\leadsto {\left(\sqrt{\log \left(\frac{\color{blue}{1 + N}}{1 + \mathsf{expm1}\left(\log N\right)}\right)}\right)}^{2} \]

      7. +-commutative 95.1%

         \[\leadsto {\left(\sqrt{\log \left(\frac{\color{blue}{N + 1}}{1 + \mathsf{expm1}\left(\log N\right)}\right)}\right)}^{2} \]

      8. add-exp-log 95.1%

         \[\leadsto {\left(\sqrt{\log \left(\frac{N + 1}{\color{blue}{e^{\log \left(1 + \mathsf{expm1}\left(\log N\right)\right)}}}\right)}\right)}^{2} \]

      9. log1p-udef 95.1%

         \[\leadsto {\left(\sqrt{\log \left(\frac{N + 1}{e^{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log N\right)\right)}}}\right)}\right)}^{2} \]

      10. log1p-expm1-u 95.1%

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

      11. add-exp-log 96.4%

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

   8. Applied egg-rr 96.4%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 0.0005:\\ \;\;\;\;\frac{0.3333333333333333}{{N}^{3}} + \left(\frac{1}{N} - \left(\frac{0.5}{{N}^{2}} + \frac{0.25}{{N}^{4}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt{\log \left(\frac{N + 1}{N}\right)}\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 0.0005:\\ \;\;\;\;\frac{0.3333333333333333}{{N}^{3}} + \left({N}^{-2} \cdot \left(N + -0.5\right) + -0.25 \cdot {N}^{-4}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt{\log \left(\frac{N + 1}{N}\right)}\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= (- (log (+ N 1.0)) (log N)) 0.0005)
   (+
    (/ 0.3333333333333333 (pow N 3.0))
    (+ (* (pow N -2.0) (+ N -0.5)) (* -0.25 (pow N -4.0))))
   (pow (sqrt (log (/ (+ N 1.0) N))) 2.0)))

C

double code(double N) {
	double tmp;
	if ((log((N + 1.0)) - log(N)) <= 0.0005) {
		tmp = (0.3333333333333333 / pow(N, 3.0)) + ((pow(N, -2.0) * (N + -0.5)) + (-0.25 * pow(N, -4.0)));
	} else {
		tmp = pow(sqrt(log(((N + 1.0) / N))), 2.0);
	}
	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 = (0.3333333333333333d0 / (n ** 3.0d0)) + (((n ** (-2.0d0)) * (n + (-0.5d0))) + ((-0.25d0) * (n ** (-4.0d0))))
    else
        tmp = sqrt(log(((n + 1.0d0) / n))) ** 2.0d0
    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 = (0.3333333333333333 / Math.pow(N, 3.0)) + ((Math.pow(N, -2.0) * (N + -0.5)) + (-0.25 * Math.pow(N, -4.0)));
	} else {
		tmp = Math.pow(Math.sqrt(Math.log(((N + 1.0) / N))), 2.0);
	}
	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)) + ((math.pow(N, -2.0) * (N + -0.5)) + (-0.25 * math.pow(N, -4.0)))
	else:
		tmp = math.pow(math.sqrt(math.log(((N + 1.0) / N))), 2.0)
	return tmp

Julia

function code(N)
	tmp = 0.0
	if (Float64(log(Float64(N + 1.0)) - log(N)) <= 0.0005)
		tmp = Float64(Float64(0.3333333333333333 / (N ^ 3.0)) + Float64(Float64((N ^ -2.0) * Float64(N + -0.5)) + Float64(-0.25 * (N ^ -4.0))));
	else
		tmp = sqrt(log(Float64(Float64(N + 1.0) / N))) ^ 2.0;
	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)) + (((N ^ -2.0) * (N + -0.5)) + (-0.25 * (N ^ -4.0)));
	else
		tmp = sqrt(log(((N + 1.0) / N))) ^ 2.0;
	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[(0.3333333333333333 / N[Power[N, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Power[N, -2.0], $MachinePrecision] * N[(N + -0.5), $MachinePrecision]), $MachinePrecision] + N[(-0.25 * N[Power[N, -4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[Sqrt[N[Log[N[(N[(N + 1.0), $MachinePrecision] / N), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], 2.0], $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 19.6%

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

      1. +-commutative 19.6%

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

      2. log1p-def 19.6%

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

   3. Simplified 19.6%

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

   5. Taylor expanded in N around inf 99.8%

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

      1. associate--l+ 99.8%

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

      2. associate-*r/ 99.8%

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

      3. metadata-eval 99.8%

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

      4. +-commutative 99.8%

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

      5. associate-*r/ 99.8%

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

      6. metadata-eval 99.8%

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

      7. associate-*r/ 99.8%

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

      8. metadata-eval 99.8%

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

   7. Simplified 99.8%

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

      1. associate--r+ 99.8%

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

      2. add-cube-cbrt 97.8%

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

      3. fma-neg 97.8%

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

   9. Applied egg-rr 97.9%

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

       1. fma-udef 97.9%

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

       2. pow-plus 97.9%

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

       3. metadata-eval 97.9%

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

       4. cube-div 98.0%

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

       5. rem-cube-cbrt 99.4%

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

       6. associate-/l* 99.4%

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

       7. sub-neg 99.4%

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

       8. metadata-eval 99.4%

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

       9. distribute-lft-neg-in 99.4%

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

       10. metadata-eval 99.4%

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

   11. Simplified 99.4%

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

       1. associate-/r/ 99.6%

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

       2. pow1 99.6%

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

       3. pow-div 99.7%

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

       4. metadata-eval 99.7%

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

   13. Applied egg-rr 99.7%

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

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

   1. Initial program 93.9%

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

      1. +-commutative 93.9%

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

      2. log1p-def 94.1%

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

   3. Simplified 94.1%

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

      1. add-sqr-sqrt 94.3%

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

      2. pow2 94.3%

         \[\leadsto \color{blue}{{\left(\sqrt{\mathsf{log1p}\left(N\right) - \log N}\right)}^{2}} \]

   6. Applied egg-rr 94.3%

      \[\leadsto \color{blue}{{\left(\sqrt{\mathsf{log1p}\left(N\right) - \log N}\right)}^{2}} \]
   7. Step-by-step derivation

      1. add-log-exp 94.1%

         \[\leadsto {\left(\sqrt{\color{blue}{\log \left(e^{\mathsf{log1p}\left(N\right)}\right)} - \log N}\right)}^{2} \]

      2. log1p-expm1-u 94.1%

         \[\leadsto {\left(\sqrt{\log \left(e^{\mathsf{log1p}\left(N\right)}\right) - \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log N\right)\right)}}\right)}^{2} \]

      3. log1p-udef 94.1%

         \[\leadsto {\left(\sqrt{\log \left(e^{\mathsf{log1p}\left(N\right)}\right) - \color{blue}{\log \left(1 + \mathsf{expm1}\left(\log N\right)\right)}}\right)}^{2} \]

      4. diff-log 93.9%

         \[\leadsto {\left(\sqrt{\color{blue}{\log \left(\frac{e^{\mathsf{log1p}\left(N\right)}}{1 + \mathsf{expm1}\left(\log N\right)}\right)}}\right)}^{2} \]

      5. log1p-udef 93.8%

         \[\leadsto {\left(\sqrt{\log \left(\frac{e^{\color{blue}{\log \left(1 + N\right)}}}{1 + \mathsf{expm1}\left(\log N\right)}\right)}\right)}^{2} \]

      6. rem-exp-log 95.1%

         \[\leadsto {\left(\sqrt{\log \left(\frac{\color{blue}{1 + N}}{1 + \mathsf{expm1}\left(\log N\right)}\right)}\right)}^{2} \]

      7. +-commutative 95.1%

         \[\leadsto {\left(\sqrt{\log \left(\frac{\color{blue}{N + 1}}{1 + \mathsf{expm1}\left(\log N\right)}\right)}\right)}^{2} \]

      8. add-exp-log 95.1%

         \[\leadsto {\left(\sqrt{\log \left(\frac{N + 1}{\color{blue}{e^{\log \left(1 + \mathsf{expm1}\left(\log N\right)\right)}}}\right)}\right)}^{2} \]

      9. log1p-udef 95.1%

         \[\leadsto {\left(\sqrt{\log \left(\frac{N + 1}{e^{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log N\right)\right)}}}\right)}\right)}^{2} \]

      10. log1p-expm1-u 95.1%

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

      11. add-exp-log 96.4%

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

   8. Applied egg-rr 96.4%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 0.0005:\\ \;\;\;\;\frac{0.3333333333333333}{{N}^{3}} + \left({N}^{-2} \cdot \left(N + -0.5\right) + -0.25 \cdot {N}^{-4}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt{\log \left(\frac{N + 1}{N}\right)}\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 6 \cdot 10^{-5}:\\ \;\;\;\;\frac{0.3333333333333333}{{N}^{3}} + \frac{1}{0.5 + \left(N + \frac{0.25}{N}\right)}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= (- (log (+ N 1.0)) (log N)) 6e-5)
   (+ (/ 0.3333333333333333 (pow N 3.0)) (/ 1.0 (+ 0.5 (+ N (/ 0.25 N)))))
   (log (/ (+ N 1.0) N))))

C

double code(double N) {
	double tmp;
	if ((log((N + 1.0)) - log(N)) <= 6e-5) {
		tmp = (0.3333333333333333 / pow(N, 3.0)) + (1.0 / (0.5 + (N + (0.25 / 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)) <= 6d-5) then
        tmp = (0.3333333333333333d0 / (n ** 3.0d0)) + (1.0d0 / (0.5d0 + (n + (0.25d0 / 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)) <= 6e-5) {
		tmp = (0.3333333333333333 / Math.pow(N, 3.0)) + (1.0 / (0.5 + (N + (0.25 / 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)) <= 6e-5:
		tmp = (0.3333333333333333 / math.pow(N, 3.0)) + (1.0 / (0.5 + (N + (0.25 / 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)) <= 6e-5)
		tmp = Float64(Float64(0.3333333333333333 / (N ^ 3.0)) + Float64(1.0 / Float64(0.5 + Float64(N + Float64(0.25 / 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)) <= 6e-5)
		tmp = (0.3333333333333333 / (N ^ 3.0)) + (1.0 / (0.5 + (N + (0.25 / 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], 6e-5], N[(N[(0.3333333333333333 / N[Power[N, 3.0], $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(0.5 + N[(N + N[(0.25 / N), $MachinePrecision]), $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)) < 6.00000000000000015e-5

   1. Initial program 18.6%

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

      1. +-commutative 18.6%

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

      2. log1p-def 18.6%

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

   3. Simplified 18.6%

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

   5. Taylor expanded in N around inf 99.6%

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

      1. associate--l+ 99.6%

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

      2. associate-*r/ 99.6%

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

      3. metadata-eval 99.6%

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

      4. associate-*r/ 99.6%

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

      5. metadata-eval 99.6%

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

   7. Simplified 99.6%

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

      1. frac-sub 99.2%

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

      2. unpow2 99.2%

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

      3. cube-mult 99.1%

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

      4. clear-num 99.3%

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

      5. *-un-lft-identity 99.3%

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

      6. unpow2 99.3%

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

      7. distribute-lft-out-- 99.3%

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

   9. Applied egg-rr 99.3%

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

   10. Taylor expanded in N around inf 99.6%

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

       1. associate-*r/ 99.6%

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

       2. metadata-eval 99.6%

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

   12. Simplified 99.6%

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

   if 6.00000000000000015e-5 < (-.f64 (log.f64 (+.f64 N 1)) (log.f64 N))

   1. Initial program 91.6%

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

      1. +-commutative 91.6%

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

      2. log1p-def 91.8%

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

   3. Simplified 91.8%

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

      1. add-log-exp 91.8%

         \[\leadsto {\left(\sqrt{\color{blue}{\log \left(e^{\mathsf{log1p}\left(N\right)}\right)} - \log N}\right)}^{2} \]

      2. log1p-expm1-u 91.8%

         \[\leadsto {\left(\sqrt{\log \left(e^{\mathsf{log1p}\left(N\right)}\right) - \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log N\right)\right)}}\right)}^{2} \]

      3. log1p-udef 91.8%

         \[\leadsto {\left(\sqrt{\log \left(e^{\mathsf{log1p}\left(N\right)}\right) - \color{blue}{\log \left(1 + \mathsf{expm1}\left(\log N\right)\right)}}\right)}^{2} \]

      4. diff-log 91.6%

         \[\leadsto {\left(\sqrt{\color{blue}{\log \left(\frac{e^{\mathsf{log1p}\left(N\right)}}{1 + \mathsf{expm1}\left(\log N\right)}\right)}}\right)}^{2} \]

      5. log1p-udef 91.4%

         \[\leadsto {\left(\sqrt{\log \left(\frac{e^{\color{blue}{\log \left(1 + N\right)}}}{1 + \mathsf{expm1}\left(\log N\right)}\right)}\right)}^{2} \]

      6. rem-exp-log 92.5%

         \[\leadsto {\left(\sqrt{\log \left(\frac{\color{blue}{1 + N}}{1 + \mathsf{expm1}\left(\log N\right)}\right)}\right)}^{2} \]

      7. +-commutative 92.5%

         \[\leadsto {\left(\sqrt{\log \left(\frac{\color{blue}{N + 1}}{1 + \mathsf{expm1}\left(\log N\right)}\right)}\right)}^{2} \]

      8. add-exp-log 92.5%

         \[\leadsto {\left(\sqrt{\log \left(\frac{N + 1}{\color{blue}{e^{\log \left(1 + \mathsf{expm1}\left(\log N\right)\right)}}}\right)}\right)}^{2} \]

      9. log1p-udef 92.5%

         \[\leadsto {\left(\sqrt{\log \left(\frac{N + 1}{e^{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log N\right)\right)}}}\right)}\right)}^{2} \]

      10. log1p-expm1-u 92.5%

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

      11. add-exp-log 94.6%

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

   6. Applied egg-rr 94.6%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 6 \cdot 10^{-5}:\\ \;\;\;\;\frac{0.3333333333333333}{{N}^{3}} + \frac{1}{0.5 + \left(N + \frac{0.25}{N}\right)}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;N \leq 1100:\\ \;\;\;\;{\left(\sqrt{\log \left(\frac{N + 1}{N}\right)}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{{N}^{3}} + \left(-0.25 \cdot {N}^{-4} + \frac{N}{N \cdot \left(N + 0.5\right) + \left(0.25 + \frac{0.125}{N}\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= N 1100.0)
   (pow (sqrt (log (/ (+ N 1.0) N))) 2.0)
   (+
    (/ 0.3333333333333333 (pow N 3.0))
    (+
     (* -0.25 (pow N -4.0))
     (/ N (+ (* N (+ N 0.5)) (+ 0.25 (/ 0.125 N))))))))

C

double code(double N) {
	double tmp;
	if (N <= 1100.0) {
		tmp = pow(sqrt(log(((N + 1.0) / N))), 2.0);
	} else {
		tmp = (0.3333333333333333 / pow(N, 3.0)) + ((-0.25 * pow(N, -4.0)) + (N / ((N * (N + 0.5)) + (0.25 + (0.125 / N)))));
	}
	return tmp;
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= 1100.0d0) then
        tmp = sqrt(log(((n + 1.0d0) / n))) ** 2.0d0
    else
        tmp = (0.3333333333333333d0 / (n ** 3.0d0)) + (((-0.25d0) * (n ** (-4.0d0))) + (n / ((n * (n + 0.5d0)) + (0.25d0 + (0.125d0 / n)))))
    end if
    code = tmp
end function

Java

public static double code(double N) {
	double tmp;
	if (N <= 1100.0) {
		tmp = Math.pow(Math.sqrt(Math.log(((N + 1.0) / N))), 2.0);
	} else {
		tmp = (0.3333333333333333 / Math.pow(N, 3.0)) + ((-0.25 * Math.pow(N, -4.0)) + (N / ((N * (N + 0.5)) + (0.25 + (0.125 / N)))));
	}
	return tmp;
}

Python

def code(N):
	tmp = 0
	if N <= 1100.0:
		tmp = math.pow(math.sqrt(math.log(((N + 1.0) / N))), 2.0)
	else:
		tmp = (0.3333333333333333 / math.pow(N, 3.0)) + ((-0.25 * math.pow(N, -4.0)) + (N / ((N * (N + 0.5)) + (0.25 + (0.125 / N)))))
	return tmp

Julia

function code(N)
	tmp = 0.0
	if (N <= 1100.0)
		tmp = sqrt(log(Float64(Float64(N + 1.0) / N))) ^ 2.0;
	else
		tmp = Float64(Float64(0.3333333333333333 / (N ^ 3.0)) + Float64(Float64(-0.25 * (N ^ -4.0)) + Float64(N / Float64(Float64(N * Float64(N + 0.5)) + Float64(0.25 + Float64(0.125 / N))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(N)
	tmp = 0.0;
	if (N <= 1100.0)
		tmp = sqrt(log(((N + 1.0) / N))) ^ 2.0;
	else
		tmp = (0.3333333333333333 / (N ^ 3.0)) + ((-0.25 * (N ^ -4.0)) + (N / ((N * (N + 0.5)) + (0.25 + (0.125 / N)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[N_] := If[LessEqual[N, 1100.0], N[Power[N[Sqrt[N[Log[N[(N[(N + 1.0), $MachinePrecision] / N), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision], N[(N[(0.3333333333333333 / N[Power[N, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(-0.25 * N[Power[N, -4.0], $MachinePrecision]), $MachinePrecision] + N[(N / N[(N[(N * N[(N + 0.5), $MachinePrecision]), $MachinePrecision] + N[(0.25 + N[(0.125 / N), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if N < 1100

   1. Initial program 93.9%

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

      1. +-commutative 93.9%

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

      2. log1p-def 94.1%

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

   3. Simplified 94.1%

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

      1. add-sqr-sqrt 94.3%

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

      2. pow2 94.3%

         \[\leadsto \color{blue}{{\left(\sqrt{\mathsf{log1p}\left(N\right) - \log N}\right)}^{2}} \]

   6. Applied egg-rr 94.3%

      \[\leadsto \color{blue}{{\left(\sqrt{\mathsf{log1p}\left(N\right) - \log N}\right)}^{2}} \]
   7. Step-by-step derivation

      1. add-log-exp 94.1%

         \[\leadsto {\left(\sqrt{\color{blue}{\log \left(e^{\mathsf{log1p}\left(N\right)}\right)} - \log N}\right)}^{2} \]

      2. log1p-expm1-u 94.1%

         \[\leadsto {\left(\sqrt{\log \left(e^{\mathsf{log1p}\left(N\right)}\right) - \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log N\right)\right)}}\right)}^{2} \]

      3. log1p-udef 94.1%

         \[\leadsto {\left(\sqrt{\log \left(e^{\mathsf{log1p}\left(N\right)}\right) - \color{blue}{\log \left(1 + \mathsf{expm1}\left(\log N\right)\right)}}\right)}^{2} \]

      4. diff-log 93.9%

         \[\leadsto {\left(\sqrt{\color{blue}{\log \left(\frac{e^{\mathsf{log1p}\left(N\right)}}{1 + \mathsf{expm1}\left(\log N\right)}\right)}}\right)}^{2} \]

      5. log1p-udef 93.8%

         \[\leadsto {\left(\sqrt{\log \left(\frac{e^{\color{blue}{\log \left(1 + N\right)}}}{1 + \mathsf{expm1}\left(\log N\right)}\right)}\right)}^{2} \]

      6. rem-exp-log 95.1%

         \[\leadsto {\left(\sqrt{\log \left(\frac{\color{blue}{1 + N}}{1 + \mathsf{expm1}\left(\log N\right)}\right)}\right)}^{2} \]

      7. +-commutative 95.1%

         \[\leadsto {\left(\sqrt{\log \left(\frac{\color{blue}{N + 1}}{1 + \mathsf{expm1}\left(\log N\right)}\right)}\right)}^{2} \]

      8. add-exp-log 95.1%

         \[\leadsto {\left(\sqrt{\log \left(\frac{N + 1}{\color{blue}{e^{\log \left(1 + \mathsf{expm1}\left(\log N\right)\right)}}}\right)}\right)}^{2} \]

      9. log1p-udef 95.1%

         \[\leadsto {\left(\sqrt{\log \left(\frac{N + 1}{e^{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log N\right)\right)}}}\right)}\right)}^{2} \]

      10. log1p-expm1-u 95.1%

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

      11. add-exp-log 96.4%

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

   8. Applied egg-rr 96.4%

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

   if 1100 < N

   1. Initial program 19.6%

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

      1. +-commutative 19.6%

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

      2. log1p-def 19.6%

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

   3. Simplified 19.6%

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

   5. Taylor expanded in N around inf 99.8%

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

      1. associate--l+ 99.8%

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

      2. associate-*r/ 99.8%

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

      3. metadata-eval 99.8%

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

      4. +-commutative 99.8%

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

      5. associate-*r/ 99.8%

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

      6. metadata-eval 99.8%

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

      7. associate-*r/ 99.8%

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

      8. metadata-eval 99.8%

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

   7. Simplified 99.8%

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

      1. associate--r+ 99.8%

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

      2. add-cube-cbrt 97.8%

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

      3. fma-neg 97.8%

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

   9. Applied egg-rr 97.9%

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

       1. fma-udef 97.9%

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

       2. pow-plus 97.9%

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

       3. metadata-eval 97.9%

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

       4. cube-div 98.0%

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

       5. rem-cube-cbrt 99.4%

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

       6. associate-/l* 99.4%

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

       7. sub-neg 99.4%

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

       8. metadata-eval 99.4%

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

       9. distribute-lft-neg-in 99.4%

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

       10. metadata-eval 99.4%

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

   11. Simplified 99.4%

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

   12. Taylor expanded in N around inf 99.5%

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

       1. +-commutative 99.5%

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

       2. +-commutative 99.5%

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

       3. associate-+l+ 99.5%

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

       4. +-commutative 99.5%

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

       5. unpow2 99.5%

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

       6. distribute-rgt-out 99.6%

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

       7. associate-*r/ 99.6%

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

       8. metadata-eval 99.6%

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

   14. Simplified 99.6%

       \[\leadsto \frac{0.3333333333333333}{{N}^{3}} + \left(\frac{N}{\color{blue}{N \cdot \left(N + 0.5\right) + \left(\frac{0.125}{N} + 0.25\right)}} + -0.25 \cdot {N}^{-4}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;N \leq 1100:\\ \;\;\;\;{\left(\sqrt{\log \left(\frac{N + 1}{N}\right)}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{{N}^{3}} + \left(-0.25 \cdot {N}^{-4} + \frac{N}{N \cdot \left(N + 0.5\right) + \left(0.25 + \frac{0.125}{N}\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;N \leq 1100:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{{N}^{3}} + \left(-0.25 \cdot {N}^{-4} + \frac{N}{N \cdot \left(N + 0.5\right) + \left(0.25 + \frac{0.125}{N}\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= N 1100.0)
   (log (/ (+ N 1.0) N))
   (+
    (/ 0.3333333333333333 (pow N 3.0))
    (+
     (* -0.25 (pow N -4.0))
     (/ N (+ (* N (+ N 0.5)) (+ 0.25 (/ 0.125 N))))))))

C

double code(double N) {
	double tmp;
	if (N <= 1100.0) {
		tmp = log(((N + 1.0) / N));
	} else {
		tmp = (0.3333333333333333 / pow(N, 3.0)) + ((-0.25 * pow(N, -4.0)) + (N / ((N * (N + 0.5)) + (0.25 + (0.125 / N)))));
	}
	return tmp;
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= 1100.0d0) then
        tmp = log(((n + 1.0d0) / n))
    else
        tmp = (0.3333333333333333d0 / (n ** 3.0d0)) + (((-0.25d0) * (n ** (-4.0d0))) + (n / ((n * (n + 0.5d0)) + (0.25d0 + (0.125d0 / n)))))
    end if
    code = tmp
end function

Java

public static double code(double N) {
	double tmp;
	if (N <= 1100.0) {
		tmp = Math.log(((N + 1.0) / N));
	} else {
		tmp = (0.3333333333333333 / Math.pow(N, 3.0)) + ((-0.25 * Math.pow(N, -4.0)) + (N / ((N * (N + 0.5)) + (0.25 + (0.125 / N)))));
	}
	return tmp;
}

Python

def code(N):
	tmp = 0
	if N <= 1100.0:
		tmp = math.log(((N + 1.0) / N))
	else:
		tmp = (0.3333333333333333 / math.pow(N, 3.0)) + ((-0.25 * math.pow(N, -4.0)) + (N / ((N * (N + 0.5)) + (0.25 + (0.125 / N)))))
	return tmp

Julia

function code(N)
	tmp = 0.0
	if (N <= 1100.0)
		tmp = log(Float64(Float64(N + 1.0) / N));
	else
		tmp = Float64(Float64(0.3333333333333333 / (N ^ 3.0)) + Float64(Float64(-0.25 * (N ^ -4.0)) + Float64(N / Float64(Float64(N * Float64(N + 0.5)) + Float64(0.25 + Float64(0.125 / N))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(N)
	tmp = 0.0;
	if (N <= 1100.0)
		tmp = log(((N + 1.0) / N));
	else
		tmp = (0.3333333333333333 / (N ^ 3.0)) + ((-0.25 * (N ^ -4.0)) + (N / ((N * (N + 0.5)) + (0.25 + (0.125 / N)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[N_] := If[LessEqual[N, 1100.0], N[Log[N[(N[(N + 1.0), $MachinePrecision] / N), $MachinePrecision]], $MachinePrecision], N[(N[(0.3333333333333333 / N[Power[N, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(-0.25 * N[Power[N, -4.0], $MachinePrecision]), $MachinePrecision] + N[(N / N[(N[(N * N[(N + 0.5), $MachinePrecision]), $MachinePrecision] + N[(0.25 + N[(0.125 / N), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if N < 1100

   1. Initial program 93.9%

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

      1. +-commutative 93.9%

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

      2. log1p-def 94.1%

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

   3. Simplified 94.1%

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

      1. add-log-exp 94.1%

         \[\leadsto {\left(\sqrt{\color{blue}{\log \left(e^{\mathsf{log1p}\left(N\right)}\right)} - \log N}\right)}^{2} \]

      2. log1p-expm1-u 94.1%

         \[\leadsto {\left(\sqrt{\log \left(e^{\mathsf{log1p}\left(N\right)}\right) - \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log N\right)\right)}}\right)}^{2} \]

      3. log1p-udef 94.1%

         \[\leadsto {\left(\sqrt{\log \left(e^{\mathsf{log1p}\left(N\right)}\right) - \color{blue}{\log \left(1 + \mathsf{expm1}\left(\log N\right)\right)}}\right)}^{2} \]

      4. diff-log 93.9%

         \[\leadsto {\left(\sqrt{\color{blue}{\log \left(\frac{e^{\mathsf{log1p}\left(N\right)}}{1 + \mathsf{expm1}\left(\log N\right)}\right)}}\right)}^{2} \]

      5. log1p-udef 93.8%

         \[\leadsto {\left(\sqrt{\log \left(\frac{e^{\color{blue}{\log \left(1 + N\right)}}}{1 + \mathsf{expm1}\left(\log N\right)}\right)}\right)}^{2} \]

      6. rem-exp-log 95.1%

         \[\leadsto {\left(\sqrt{\log \left(\frac{\color{blue}{1 + N}}{1 + \mathsf{expm1}\left(\log N\right)}\right)}\right)}^{2} \]

      7. +-commutative 95.1%

         \[\leadsto {\left(\sqrt{\log \left(\frac{\color{blue}{N + 1}}{1 + \mathsf{expm1}\left(\log N\right)}\right)}\right)}^{2} \]

      8. add-exp-log 95.1%

         \[\leadsto {\left(\sqrt{\log \left(\frac{N + 1}{\color{blue}{e^{\log \left(1 + \mathsf{expm1}\left(\log N\right)\right)}}}\right)}\right)}^{2} \]

      9. log1p-udef 95.1%

         \[\leadsto {\left(\sqrt{\log \left(\frac{N + 1}{e^{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log N\right)\right)}}}\right)}\right)}^{2} \]

      10. log1p-expm1-u 95.1%

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

      11. add-exp-log 96.4%

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

   6. Applied egg-rr 96.4%

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

   if 1100 < N

   1. Initial program 19.6%

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

      1. +-commutative 19.6%

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

      2. log1p-def 19.6%

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

   3. Simplified 19.6%

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

   5. Taylor expanded in N around inf 99.8%

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

      1. associate--l+ 99.8%

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

      2. associate-*r/ 99.8%

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

      3. metadata-eval 99.8%

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

      4. +-commutative 99.8%

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

      5. associate-*r/ 99.8%

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

      6. metadata-eval 99.8%

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

      7. associate-*r/ 99.8%

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

      8. metadata-eval 99.8%

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

   7. Simplified 99.8%

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

      1. associate--r+ 99.8%

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

      2. add-cube-cbrt 97.8%

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

      3. fma-neg 97.8%

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

   9. Applied egg-rr 97.9%

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

       1. fma-udef 97.9%

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

       2. pow-plus 97.9%

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

       3. metadata-eval 97.9%

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

       4. cube-div 98.0%

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

       5. rem-cube-cbrt 99.4%

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

       6. associate-/l* 99.4%

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

       7. sub-neg 99.4%

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

       8. metadata-eval 99.4%

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

       9. distribute-lft-neg-in 99.4%

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

       10. metadata-eval 99.4%

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

   11. Simplified 99.4%

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

   12. Taylor expanded in N around inf 99.5%

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

       1. +-commutative 99.5%

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

       2. +-commutative 99.5%

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

       3. associate-+l+ 99.5%

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

       4. +-commutative 99.5%

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

       5. unpow2 99.5%

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

       6. distribute-rgt-out 99.6%

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

       7. associate-*r/ 99.6%

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

       8. metadata-eval 99.6%

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

   14. Simplified 99.6%

       \[\leadsto \frac{0.3333333333333333}{{N}^{3}} + \left(\frac{N}{\color{blue}{N \cdot \left(N + 0.5\right) + \left(\frac{0.125}{N} + 0.25\right)}} + -0.25 \cdot {N}^{-4}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;N \leq 1100:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{{N}^{3}} + \left(-0.25 \cdot {N}^{-4} + \frac{N}{N \cdot \left(N + 0.5\right) + \left(0.25 + \frac{0.125}{N}\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;N \leq 160000:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{{N}^{3}} + \frac{1}{N + 0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= N 160000.0)
   (log (/ (+ N 1.0) N))
   (+ (/ 0.3333333333333333 (pow N 3.0)) (/ 1.0 (+ N 0.5)))))

C

double code(double N) {
	double tmp;
	if (N <= 160000.0) {
		tmp = log(((N + 1.0) / N));
	} else {
		tmp = (0.3333333333333333 / pow(N, 3.0)) + (1.0 / (N + 0.5));
	}
	return tmp;
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= 160000.0d0) then
        tmp = log(((n + 1.0d0) / n))
    else
        tmp = (0.3333333333333333d0 / (n ** 3.0d0)) + (1.0d0 / (n + 0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double N) {
	double tmp;
	if (N <= 160000.0) {
		tmp = Math.log(((N + 1.0) / N));
	} else {
		tmp = (0.3333333333333333 / Math.pow(N, 3.0)) + (1.0 / (N + 0.5));
	}
	return tmp;
}

Python

def code(N):
	tmp = 0
	if N <= 160000.0:
		tmp = math.log(((N + 1.0) / N))
	else:
		tmp = (0.3333333333333333 / math.pow(N, 3.0)) + (1.0 / (N + 0.5))
	return tmp

Julia

function code(N)
	tmp = 0.0
	if (N <= 160000.0)
		tmp = log(Float64(Float64(N + 1.0) / N));
	else
		tmp = Float64(Float64(0.3333333333333333 / (N ^ 3.0)) + Float64(1.0 / Float64(N + 0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(N)
	tmp = 0.0;
	if (N <= 160000.0)
		tmp = log(((N + 1.0) / N));
	else
		tmp = (0.3333333333333333 / (N ^ 3.0)) + (1.0 / (N + 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[N_] := If[LessEqual[N, 160000.0], N[Log[N[(N[(N + 1.0), $MachinePrecision] / N), $MachinePrecision]], $MachinePrecision], N[(N[(0.3333333333333333 / N[Power[N, 3.0], $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if N < 1.6e5

   1. Initial program 87.2%

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

      1. +-commutative 87.2%

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

      2. log1p-def 87.3%

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

   3. Simplified 87.3%

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

      1. add-log-exp 87.3%

         \[\leadsto {\left(\sqrt{\color{blue}{\log \left(e^{\mathsf{log1p}\left(N\right)}\right)} - \log N}\right)}^{2} \]

      2. log1p-expm1-u 87.3%

         \[\leadsto {\left(\sqrt{\log \left(e^{\mathsf{log1p}\left(N\right)}\right) - \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log N\right)\right)}}\right)}^{2} \]

      3. log1p-udef 87.3%

         \[\leadsto {\left(\sqrt{\log \left(e^{\mathsf{log1p}\left(N\right)}\right) - \color{blue}{\log \left(1 + \mathsf{expm1}\left(\log N\right)\right)}}\right)}^{2} \]

      4. diff-log 87.0%

         \[\leadsto {\left(\sqrt{\color{blue}{\log \left(\frac{e^{\mathsf{log1p}\left(N\right)}}{1 + \mathsf{expm1}\left(\log N\right)}\right)}}\right)}^{2} \]

      5. log1p-udef 86.9%

         \[\leadsto {\left(\sqrt{\log \left(\frac{e^{\color{blue}{\log \left(1 + N\right)}}}{1 + \mathsf{expm1}\left(\log N\right)}\right)}\right)}^{2} \]

      6. rem-exp-log 88.0%

         \[\leadsto {\left(\sqrt{\log \left(\frac{\color{blue}{1 + N}}{1 + \mathsf{expm1}\left(\log N\right)}\right)}\right)}^{2} \]

      7. +-commutative 88.0%

         \[\leadsto {\left(\sqrt{\log \left(\frac{\color{blue}{N + 1}}{1 + \mathsf{expm1}\left(\log N\right)}\right)}\right)}^{2} \]

      8. add-exp-log 88.1%

         \[\leadsto {\left(\sqrt{\log \left(\frac{N + 1}{\color{blue}{e^{\log \left(1 + \mathsf{expm1}\left(\log N\right)\right)}}}\right)}\right)}^{2} \]

      9. log1p-udef 88.1%

         \[\leadsto {\left(\sqrt{\log \left(\frac{N + 1}{e^{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log N\right)\right)}}}\right)}\right)}^{2} \]

      10. log1p-expm1-u 88.1%

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

      11. add-exp-log 90.8%

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

   6. Applied egg-rr 90.7%

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

   if 1.6e5 < N

   1. Initial program 16.3%

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

      1. +-commutative 16.3%

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

      2. log1p-def 16.4%

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

   3. Simplified 16.4%

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

   5. Taylor expanded in N around inf 99.9%

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

      1. associate--l+ 99.9%

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

      2. associate-*r/ 99.9%

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

      3. metadata-eval 99.9%

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

      4. associate-*r/ 99.9%

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

      5. metadata-eval 99.9%

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

   7. Simplified 99.9%

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

      1. frac-sub 99.5%

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

      2. unpow2 99.5%

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

      3. cube-mult 99.4%

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

      4. clear-num 99.6%

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

      5. *-un-lft-identity 99.6%

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

      6. unpow2 99.6%

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

      7. distribute-lft-out-- 99.6%

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

   9. Applied egg-rr 99.6%

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

   10. Taylor expanded in N around inf 98.9%

       \[\leadsto \frac{0.3333333333333333}{{N}^{3}} + \frac{1}{\color{blue}{0.5 + N}} \]
   11. Step-by-step derivation

       1. +-commutative 98.9%

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

   12. Simplified 98.9%

       \[\leadsto \frac{0.3333333333333333}{{N}^{3}} + \frac{1}{\color{blue}{N + 0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;N \leq 160000:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{{N}^{3}} + \frac{1}{N + 0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;N \leq 160000:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N} - \frac{0.5}{{N}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= N 160000.0) (log (/ (+ N 1.0) N)) (- (/ 1.0 N) (/ 0.5 (pow N 2.0)))))

C

double code(double N) {
	double tmp;
	if (N <= 160000.0) {
		tmp = log(((N + 1.0) / N));
	} else {
		tmp = (1.0 / N) - (0.5 / pow(N, 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= 160000.0d0) then
        tmp = log(((n + 1.0d0) / n))
    else
        tmp = (1.0d0 / n) - (0.5d0 / (n ** 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double N) {
	double tmp;
	if (N <= 160000.0) {
		tmp = Math.log(((N + 1.0) / N));
	} else {
		tmp = (1.0 / N) - (0.5 / Math.pow(N, 2.0));
	}
	return tmp;
}

Python

def code(N):
	tmp = 0
	if N <= 160000.0:
		tmp = math.log(((N + 1.0) / N))
	else:
		tmp = (1.0 / N) - (0.5 / math.pow(N, 2.0))
	return tmp

Julia

function code(N)
	tmp = 0.0
	if (N <= 160000.0)
		tmp = log(Float64(Float64(N + 1.0) / N));
	else
		tmp = Float64(Float64(1.0 / N) - Float64(0.5 / (N ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(N)
	tmp = 0.0;
	if (N <= 160000.0)
		tmp = log(((N + 1.0) / N));
	else
		tmp = (1.0 / N) - (0.5 / (N ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[N_] := If[LessEqual[N, 160000.0], N[Log[N[(N[(N + 1.0), $MachinePrecision] / N), $MachinePrecision]], $MachinePrecision], N[(N[(1.0 / N), $MachinePrecision] - N[(0.5 / N[Power[N, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if N < 1.6e5

   1. Initial program 87.2%

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

      1. +-commutative 87.2%

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

      2. log1p-def 87.3%

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

   3. Simplified 87.3%

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

      1. add-log-exp 87.3%

         \[\leadsto {\left(\sqrt{\color{blue}{\log \left(e^{\mathsf{log1p}\left(N\right)}\right)} - \log N}\right)}^{2} \]

      2. log1p-expm1-u 87.3%

         \[\leadsto {\left(\sqrt{\log \left(e^{\mathsf{log1p}\left(N\right)}\right) - \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log N\right)\right)}}\right)}^{2} \]

      3. log1p-udef 87.3%

         \[\leadsto {\left(\sqrt{\log \left(e^{\mathsf{log1p}\left(N\right)}\right) - \color{blue}{\log \left(1 + \mathsf{expm1}\left(\log N\right)\right)}}\right)}^{2} \]

      4. diff-log 87.0%

         \[\leadsto {\left(\sqrt{\color{blue}{\log \left(\frac{e^{\mathsf{log1p}\left(N\right)}}{1 + \mathsf{expm1}\left(\log N\right)}\right)}}\right)}^{2} \]

      5. log1p-udef 86.9%

         \[\leadsto {\left(\sqrt{\log \left(\frac{e^{\color{blue}{\log \left(1 + N\right)}}}{1 + \mathsf{expm1}\left(\log N\right)}\right)}\right)}^{2} \]

      6. rem-exp-log 88.0%

         \[\leadsto {\left(\sqrt{\log \left(\frac{\color{blue}{1 + N}}{1 + \mathsf{expm1}\left(\log N\right)}\right)}\right)}^{2} \]

      7. +-commutative 88.0%

         \[\leadsto {\left(\sqrt{\log \left(\frac{\color{blue}{N + 1}}{1 + \mathsf{expm1}\left(\log N\right)}\right)}\right)}^{2} \]

      8. add-exp-log 88.1%

         \[\leadsto {\left(\sqrt{\log \left(\frac{N + 1}{\color{blue}{e^{\log \left(1 + \mathsf{expm1}\left(\log N\right)\right)}}}\right)}\right)}^{2} \]

      9. log1p-udef 88.1%

         \[\leadsto {\left(\sqrt{\log \left(\frac{N + 1}{e^{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log N\right)\right)}}}\right)}\right)}^{2} \]

      10. log1p-expm1-u 88.1%

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

      11. add-exp-log 90.8%

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

   6. Applied egg-rr 90.7%

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

   if 1.6e5 < N

   1. Initial program 16.3%

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

      1. +-commutative 16.3%

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

      2. log1p-def 16.4%

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

   3. Simplified 16.4%

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

   5. Taylor expanded in N around inf 98.8%

      \[\leadsto \color{blue}{\frac{1}{N} - 0.5 \cdot \frac{1}{{N}^{2}}} \]
   6. 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%

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

   7. Simplified 98.8%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;N \leq 160000:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N} - \frac{0.5}{{N}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 93.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;N \leq 165000000:\\ \;\;\;\;-\log \left(\frac{N}{N + 1}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N}\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= N 165000000.0) (- (log (/ N (+ N 1.0)))) (/ 1.0 N)))

C

double code(double N) {
	double tmp;
	if (N <= 165000000.0) {
		tmp = -log((N / (N + 1.0)));
	} else {
		tmp = 1.0 / N;
	}
	return tmp;
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= 165000000.0d0) then
        tmp = -log((n / (n + 1.0d0)))
    else
        tmp = 1.0d0 / n
    end if
    code = tmp
end function

Java

public static double code(double N) {
	double tmp;
	if (N <= 165000000.0) {
		tmp = -Math.log((N / (N + 1.0)));
	} else {
		tmp = 1.0 / N;
	}
	return tmp;
}

Python

def code(N):
	tmp = 0
	if N <= 165000000.0:
		tmp = -math.log((N / (N + 1.0)))
	else:
		tmp = 1.0 / N
	return tmp

Julia

function code(N)
	tmp = 0.0
	if (N <= 165000000.0)
		tmp = Float64(-log(Float64(N / Float64(N + 1.0))));
	else
		tmp = Float64(1.0 / N);
	end
	return tmp
end

MATLAB

function tmp_2 = code(N)
	tmp = 0.0;
	if (N <= 165000000.0)
		tmp = -log((N / (N + 1.0)));
	else
		tmp = 1.0 / N;
	end
	tmp_2 = tmp;
end

Wolfram

code[N_] := If[LessEqual[N, 165000000.0], (-N[Log[N[(N / N[(N + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), N[(1.0 / N), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if N < 1.65e8

   1. Initial program 77.9%

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

      1. +-commutative 77.9%

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

      2. log1p-def 78.1%

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

   3. Simplified 78.1%

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

      1. add-log-exp 78.1%

         \[\leadsto {\left(\sqrt{\color{blue}{\log \left(e^{\mathsf{log1p}\left(N\right)}\right)} - \log N}\right)}^{2} \]

      2. log1p-expm1-u 78.1%

         \[\leadsto {\left(\sqrt{\log \left(e^{\mathsf{log1p}\left(N\right)}\right) - \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log N\right)\right)}}\right)}^{2} \]

      3. log1p-udef 78.1%

         \[\leadsto {\left(\sqrt{\log \left(e^{\mathsf{log1p}\left(N\right)}\right) - \color{blue}{\log \left(1 + \mathsf{expm1}\left(\log N\right)\right)}}\right)}^{2} \]

      4. diff-log 78.0%

         \[\leadsto {\left(\sqrt{\color{blue}{\log \left(\frac{e^{\mathsf{log1p}\left(N\right)}}{1 + \mathsf{expm1}\left(\log N\right)}\right)}}\right)}^{2} \]

      5. log1p-udef 77.8%

         \[\leadsto {\left(\sqrt{\log \left(\frac{e^{\color{blue}{\log \left(1 + N\right)}}}{1 + \mathsf{expm1}\left(\log N\right)}\right)}\right)}^{2} \]

      6. rem-exp-log 78.7%

         \[\leadsto {\left(\sqrt{\log \left(\frac{\color{blue}{1 + N}}{1 + \mathsf{expm1}\left(\log N\right)}\right)}\right)}^{2} \]

      7. +-commutative 78.7%

         \[\leadsto {\left(\sqrt{\log \left(\frac{\color{blue}{N + 1}}{1 + \mathsf{expm1}\left(\log N\right)}\right)}\right)}^{2} \]

      8. add-exp-log 78.7%

         \[\leadsto {\left(\sqrt{\log \left(\frac{N + 1}{\color{blue}{e^{\log \left(1 + \mathsf{expm1}\left(\log N\right)\right)}}}\right)}\right)}^{2} \]

      9. log1p-udef 78.7%

         \[\leadsto {\left(\sqrt{\log \left(\frac{N + 1}{e^{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log N\right)\right)}}}\right)}\right)}^{2} \]

      10. log1p-expm1-u 78.7%

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

      11. add-exp-log 82.2%

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

   6. Applied egg-rr 82.2%

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

      1. clear-num 82.1%

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

      2. log-rec 82.7%

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

   8. Applied egg-rr 82.7%

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

   if 1.65e8 < N

   1. Initial program 11.2%

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

      1. +-commutative 11.2%

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

      2. log1p-def 11.2%

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

   3. Simplified 11.2%

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

   5. Taylor expanded in N around inf 95.0%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;N \leq 165000000:\\ \;\;\;\;-\log \left(\frac{N}{N + 1}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 92.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;N \leq 105000000:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N}\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= N 105000000.0) (log (/ (+ N 1.0) N)) (/ 1.0 N)))

C

double code(double N) {
	double tmp;
	if (N <= 105000000.0) {
		tmp = log(((N + 1.0) / N));
	} else {
		tmp = 1.0 / N;
	}
	return tmp;
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= 105000000.0d0) then
        tmp = log(((n + 1.0d0) / n))
    else
        tmp = 1.0d0 / n
    end if
    code = tmp
end function

Java

public static double code(double N) {
	double tmp;
	if (N <= 105000000.0) {
		tmp = Math.log(((N + 1.0) / N));
	} else {
		tmp = 1.0 / N;
	}
	return tmp;
}

Python

def code(N):
	tmp = 0
	if N <= 105000000.0:
		tmp = math.log(((N + 1.0) / N))
	else:
		tmp = 1.0 / N
	return tmp

Julia

function code(N)
	tmp = 0.0
	if (N <= 105000000.0)
		tmp = log(Float64(Float64(N + 1.0) / N));
	else
		tmp = Float64(1.0 / N);
	end
	return tmp
end

MATLAB

function tmp_2 = code(N)
	tmp = 0.0;
	if (N <= 105000000.0)
		tmp = log(((N + 1.0) / N));
	else
		tmp = 1.0 / N;
	end
	tmp_2 = tmp;
end

Wolfram

code[N_] := If[LessEqual[N, 105000000.0], N[Log[N[(N[(N + 1.0), $MachinePrecision] / N), $MachinePrecision]], $MachinePrecision], N[(1.0 / N), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if N < 1.05e8

   1. Initial program 78.7%

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

      1. +-commutative 78.7%

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

      2. log1p-def 79.0%

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

   3. Simplified 79.0%

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

      1. add-log-exp 79.0%

         \[\leadsto {\left(\sqrt{\color{blue}{\log \left(e^{\mathsf{log1p}\left(N\right)}\right)} - \log N}\right)}^{2} \]

      2. log1p-expm1-u 79.0%

         \[\leadsto {\left(\sqrt{\log \left(e^{\mathsf{log1p}\left(N\right)}\right) - \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log N\right)\right)}}\right)}^{2} \]

      3. log1p-udef 79.0%

         \[\leadsto {\left(\sqrt{\log \left(e^{\mathsf{log1p}\left(N\right)}\right) - \color{blue}{\log \left(1 + \mathsf{expm1}\left(\log N\right)\right)}}\right)}^{2} \]

      4. diff-log 78.8%

         \[\leadsto {\left(\sqrt{\color{blue}{\log \left(\frac{e^{\mathsf{log1p}\left(N\right)}}{1 + \mathsf{expm1}\left(\log N\right)}\right)}}\right)}^{2} \]

      5. log1p-udef 78.6%

         \[\leadsto {\left(\sqrt{\log \left(\frac{e^{\color{blue}{\log \left(1 + N\right)}}}{1 + \mathsf{expm1}\left(\log N\right)}\right)}\right)}^{2} \]

      6. rem-exp-log 79.7%

         \[\leadsto {\left(\sqrt{\log \left(\frac{\color{blue}{1 + N}}{1 + \mathsf{expm1}\left(\log N\right)}\right)}\right)}^{2} \]

      7. +-commutative 79.7%

         \[\leadsto {\left(\sqrt{\log \left(\frac{\color{blue}{N + 1}}{1 + \mathsf{expm1}\left(\log N\right)}\right)}\right)}^{2} \]

      8. add-exp-log 79.6%

         \[\leadsto {\left(\sqrt{\log \left(\frac{N + 1}{\color{blue}{e^{\log \left(1 + \mathsf{expm1}\left(\log N\right)\right)}}}\right)}\right)}^{2} \]

      9. log1p-udef 79.6%

         \[\leadsto {\left(\sqrt{\log \left(\frac{N + 1}{e^{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log N\right)\right)}}}\right)}\right)}^{2} \]

      10. log1p-expm1-u 79.6%

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

      11. add-exp-log 83.0%

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

   6. Applied egg-rr 83.0%

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

   if 1.05e8 < N

   1. Initial program 11.6%

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

      1. +-commutative 11.6%

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

      2. log1p-def 11.6%

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

   3. Simplified 11.6%

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

   5. Taylor expanded in N around inf 94.7%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;N \leq 105000000:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 84.3% 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 26.3%

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

   1. +-commutative 26.3%

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

   2. log1p-def 26.3%

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

3. Simplified 26.3%

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

5. Taylor expanded in N around inf 82.6%

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

6. Final simplification 82.6%

   \[\leadsto \frac{1}{N} \]
7. Add Preprocessing

Developer target: 99.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{log1p}\left(\frac{1}{N}\right) \end{array} \]

FPCore

(FPCore (N) :precision binary64 (log1p (/ 1.0 N)))

C

double code(double N) {
	return log1p((1.0 / N));
}

Java

public static double code(double N) {
	return Math.log1p((1.0 / N));
}

Python

def code(N):
	return math.log1p((1.0 / N))

Julia

function code(N)
	return log1p(Float64(1.0 / N))
end

Wolfram

code[N_] := N[Log[1 + N[(1.0 / N), $MachinePrecision]], $MachinePrecision]

Reproduce

herbie shell --seed 2024041 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  :pre (and (> N 1.0) (< N 1e+40))

  :herbie-target
  (log1p (/ 1.0 N))

  (- (log (+ N 1.0)) (log N)))