2log (problem 3.3.6)

Percentage Accurate: 53.5% → 99.8%

Time: 7.8s

Alternatives: 8

Speedup: 2.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 53.5% 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: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\frac{1}{N} + \left(\frac{0.3333333333333333}{{N}^{3}} - \frac{0.5}{N \cdot N}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(-\log \left(\sqrt{\frac{N}{N + 1}}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= (- (log (+ N 1.0)) (log N)) 2e-7)
   (+ (/ 1.0 N) (- (/ 0.3333333333333333 (pow N 3.0)) (/ 0.5 (* N N))))
   (* 2.0 (- (log (sqrt (/ N (+ N 1.0))))))))

C

double code(double N) {
	double tmp;
	if ((log((N + 1.0)) - log(N)) <= 2e-7) {
		tmp = (1.0 / N) + ((0.3333333333333333 / pow(N, 3.0)) - (0.5 / (N * N)));
	} else {
		tmp = 2.0 * -log(sqrt((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-7) then
        tmp = (1.0d0 / n) + ((0.3333333333333333d0 / (n ** 3.0d0)) - (0.5d0 / (n * n)))
    else
        tmp = 2.0d0 * -log(sqrt((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-7) {
		tmp = (1.0 / N) + ((0.3333333333333333 / Math.pow(N, 3.0)) - (0.5 / (N * N)));
	} else {
		tmp = 2.0 * -Math.log(Math.sqrt((N / (N + 1.0))));
	}
	return tmp;
}

Python

def code(N):
	tmp = 0
	if (math.log((N + 1.0)) - math.log(N)) <= 2e-7:
		tmp = (1.0 / N) + ((0.3333333333333333 / math.pow(N, 3.0)) - (0.5 / (N * N)))
	else:
		tmp = 2.0 * -math.log(math.sqrt((N / (N + 1.0))))
	return tmp

Julia

function code(N)
	tmp = 0.0
	if (Float64(log(Float64(N + 1.0)) - log(N)) <= 2e-7)
		tmp = Float64(Float64(1.0 / N) + Float64(Float64(0.3333333333333333 / (N ^ 3.0)) - Float64(0.5 / Float64(N * N))));
	else
		tmp = Float64(2.0 * Float64(-log(sqrt(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-7)
		tmp = (1.0 / N) + ((0.3333333333333333 / (N ^ 3.0)) - (0.5 / (N * N)));
	else
		tmp = 2.0 * -log(sqrt((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-7], N[(N[(1.0 / N), $MachinePrecision] + N[(N[(0.3333333333333333 / N[Power[N, 3.0], $MachinePrecision]), $MachinePrecision] - N[(0.5 / N[(N * N), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * (-N[Log[N[Sqrt[N[(N / N[(N + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (log.f64 (+.f64 N 1)) (log.f64 N)) < 1.9999999999999999e-7

   1. Initial program 6.9%

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

      1. +-commutative 6.9%

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

      2. log1p-def 6.9%

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

   3. Simplified 6.9%

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

   4. Taylor expanded in N around inf 100.0%

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

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

   if 1.9999999999999999e-7 < (-.f64 (log.f64 (+.f64 N 1)) (log.f64 N))

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. log1p-def 99.9%

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

   3. Simplified 99.9%

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

      1. log1p-udef 99.9%

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

      2. diff-log 99.9%

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

      3. +-commutative 99.9%

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

   5. Applied egg-rr 99.9%

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

      1. clear-num 99.9%

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

      2. log-div 99.9%

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

      3. metadata-eval 99.9%

         \[\leadsto \color{blue}{0} - \log \left(\frac{N}{N + 1}\right) \]

   7. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{0 - \log \left(\frac{N}{N + 1}\right)} \]
   8. Step-by-step derivation

      1. neg-sub0 99.9%

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

   9. Simplified 99.9%

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

       1. add-sqr-sqrt 99.9%

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

       2. log-prod 100.0%

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

   11. Applied egg-rr 100.0%

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

       1. count-2 100.0%

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

   13. Simplified 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\frac{1}{N} + \left(\frac{0.3333333333333333}{{N}^{3}} - \frac{0.5}{N \cdot N}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(-\log \left(\sqrt{\frac{N}{N + 1}}\right)\right)\\ \end{array} \]

Alternative 2: 99.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\frac{1}{N} + \left(\frac{0.3333333333333333}{{N}^{3}} - \frac{0.5}{N \cdot N}\right)\\ \mathbf{else}:\\ \;\;\;\;-\log \left(\frac{N}{N + 1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= (- (log (+ N 1.0)) (log N)) 2e-7)
   (+ (/ 1.0 N) (- (/ 0.3333333333333333 (pow N 3.0)) (/ 0.5 (* N N))))
   (- (log (/ N (+ N 1.0))))))

C

double code(double N) {
	double tmp;
	if ((log((N + 1.0)) - log(N)) <= 2e-7) {
		tmp = (1.0 / N) + ((0.3333333333333333 / pow(N, 3.0)) - (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-7) then
        tmp = (1.0d0 / n) + ((0.3333333333333333d0 / (n ** 3.0d0)) - (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-7) {
		tmp = (1.0 / N) + ((0.3333333333333333 / Math.pow(N, 3.0)) - (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-7:
		tmp = (1.0 / N) + ((0.3333333333333333 / math.pow(N, 3.0)) - (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-7)
		tmp = Float64(Float64(1.0 / N) + Float64(Float64(0.3333333333333333 / (N ^ 3.0)) - Float64(0.5 / Float64(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-7)
		tmp = (1.0 / N) + ((0.3333333333333333 / (N ^ 3.0)) - (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-7], N[(N[(1.0 / N), $MachinePrecision] + N[(N[(0.3333333333333333 / N[Power[N, 3.0], $MachinePrecision]), $MachinePrecision] - N[(0.5 / N[(N * N), $MachinePrecision]), $MachinePrecision]), $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)) < 1.9999999999999999e-7

   1. Initial program 6.9%

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

      1. +-commutative 6.9%

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

      2. log1p-def 6.9%

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

   3. Simplified 6.9%

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

   4. Taylor expanded in N around inf 100.0%

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

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

   if 1.9999999999999999e-7 < (-.f64 (log.f64 (+.f64 N 1)) (log.f64 N))

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. log1p-def 99.9%

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

   3. Simplified 99.9%

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

      1. log1p-udef 99.9%

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

      2. diff-log 99.9%

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

      3. +-commutative 99.9%

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

   5. Applied egg-rr 99.9%

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

      1. clear-num 99.9%

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

      2. log-div 99.9%

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

      3. metadata-eval 99.9%

         \[\leadsto \color{blue}{0} - \log \left(\frac{N}{N + 1}\right) \]

   7. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{0 - \log \left(\frac{N}{N + 1}\right)} \]
   8. Step-by-step derivation

      1. neg-sub0 99.9%

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

   9. Simplified 99.9%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\frac{1}{N} + \left(\frac{0.3333333333333333}{{N}^{3}} - \frac{0.5}{N \cdot N}\right)\\ \mathbf{else}:\\ \;\;\;\;-\log \left(\frac{N}{N + 1}\right)\\ \end{array} \]

Alternative 3: 99.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;N \leq 240000:\\ \;\;\;\;-\log \left(\frac{N}{N + 1}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N} - \frac{\frac{0.5}{N}}{N}\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= N 240000.0) (- (log (/ N (+ N 1.0)))) (- (/ 1.0 N) (/ (/ 0.5 N) N))))

C

double code(double N) {
	double tmp;
	if (N <= 240000.0) {
		tmp = -log((N / (N + 1.0)));
	} else {
		tmp = (1.0 / N) - ((0.5 / N) / N);
	}
	return tmp;
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= 240000.0d0) then
        tmp = -log((n / (n + 1.0d0)))
    else
        tmp = (1.0d0 / n) - ((0.5d0 / n) / n)
    end if
    code = tmp
end function

Java

public static double code(double N) {
	double tmp;
	if (N <= 240000.0) {
		tmp = -Math.log((N / (N + 1.0)));
	} else {
		tmp = (1.0 / N) - ((0.5 / N) / N);
	}
	return tmp;
}

Python

def code(N):
	tmp = 0
	if N <= 240000.0:
		tmp = -math.log((N / (N + 1.0)))
	else:
		tmp = (1.0 / N) - ((0.5 / N) / N)
	return tmp

Julia

function code(N)
	tmp = 0.0
	if (N <= 240000.0)
		tmp = Float64(-log(Float64(N / Float64(N + 1.0))));
	else
		tmp = Float64(Float64(1.0 / N) - Float64(Float64(0.5 / N) / N));
	end
	return tmp
end

MATLAB

function tmp_2 = code(N)
	tmp = 0.0;
	if (N <= 240000.0)
		tmp = -log((N / (N + 1.0)));
	else
		tmp = (1.0 / N) - ((0.5 / N) / N);
	end
	tmp_2 = tmp;
end

Wolfram

code[N_] := If[LessEqual[N, 240000.0], (-N[Log[N[(N / N[(N + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), N[(N[(1.0 / N), $MachinePrecision] - N[(N[(0.5 / N), $MachinePrecision] / N), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if N < 2.4e5

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. log1p-def 99.9%

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

   3. Simplified 99.9%

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

      1. log1p-udef 99.9%

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

      2. diff-log 99.9%

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

      3. +-commutative 99.9%

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

   5. Applied egg-rr 99.9%

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

      1. clear-num 99.9%

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

      2. log-div 99.9%

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

      3. metadata-eval 99.9%

         \[\leadsto \color{blue}{0} - \log \left(\frac{N}{N + 1}\right) \]

   7. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{0 - \log \left(\frac{N}{N + 1}\right)} \]
   8. Step-by-step derivation

      1. neg-sub0 99.9%

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

   9. Simplified 99.9%

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

   if 2.4e5 < N

   1. Initial program 6.9%

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

      1. +-commutative 6.9%

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

      2. log1p-def 6.9%

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

   3. Simplified 6.9%

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

   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%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;N \leq 240000:\\ \;\;\;\;-\log \left(\frac{N}{N + 1}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N} - \frac{\frac{0.5}{N}}{N}\\ \end{array} \]

Alternative 4: 99.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;N \leq 205000:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N} - \frac{\frac{0.5}{N}}{N}\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= N 205000.0) (log (/ (+ N 1.0) N)) (- (/ 1.0 N) (/ (/ 0.5 N) N))))

C

double code(double N) {
	double tmp;
	if (N <= 205000.0) {
		tmp = log(((N + 1.0) / N));
	} else {
		tmp = (1.0 / N) - ((0.5 / N) / N);
	}
	return tmp;
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= 205000.0d0) then
        tmp = log(((n + 1.0d0) / n))
    else
        tmp = (1.0d0 / n) - ((0.5d0 / n) / n)
    end if
    code = tmp
end function

Java

public static double code(double N) {
	double tmp;
	if (N <= 205000.0) {
		tmp = Math.log(((N + 1.0) / N));
	} else {
		tmp = (1.0 / N) - ((0.5 / N) / N);
	}
	return tmp;
}

Python

def code(N):
	tmp = 0
	if N <= 205000.0:
		tmp = math.log(((N + 1.0) / N))
	else:
		tmp = (1.0 / N) - ((0.5 / N) / N)
	return tmp

Julia

function code(N)
	tmp = 0.0
	if (N <= 205000.0)
		tmp = log(Float64(Float64(N + 1.0) / N));
	else
		tmp = Float64(Float64(1.0 / N) - Float64(Float64(0.5 / N) / N));
	end
	return tmp
end

MATLAB

function tmp_2 = code(N)
	tmp = 0.0;
	if (N <= 205000.0)
		tmp = log(((N + 1.0) / N));
	else
		tmp = (1.0 / N) - ((0.5 / N) / N);
	end
	tmp_2 = tmp;
end

Wolfram

code[N_] := If[LessEqual[N, 205000.0], N[Log[N[(N[(N + 1.0), $MachinePrecision] / N), $MachinePrecision]], $MachinePrecision], N[(N[(1.0 / N), $MachinePrecision] - N[(N[(0.5 / N), $MachinePrecision] / N), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if N < 205000

   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%

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

   if 205000 < N

   1. Initial program 6.9%

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

      1. +-commutative 6.9%

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

      2. log1p-def 6.9%

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

   3. Simplified 6.9%

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

   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%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;N \leq 205000:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N} - \frac{\frac{0.5}{N}}{N}\\ \end{array} \]

Alternative 5: 99.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if N < 0.900000000000000022

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. log1p-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in N around 0 99.5%

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

      1. neg-mul-1 99.5%

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

      2. unsub-neg 99.5%

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

   6. Simplified 99.5%

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

   if 0.900000000000000022 < N

   1. Initial program 7.6%

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

      1. +-commutative 7.6%

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

      2. log1p-def 7.6%

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

   3. Simplified 7.6%

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

   4. Taylor expanded in N around inf 99.3%

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

      1. associate-*r/ 99.3%

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

      2. metadata-eval 99.3%

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

      3. unpow2 99.3%

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

      4. associate-/r* 99.3%

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

   6. Simplified 99.3%

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

4. Final simplification 99.4%

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

Alternative 6: 98.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;N \leq 0.65:\\ \;\;\;\;-\log N\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N} - \frac{\frac{0.5}{N}}{N}\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= N 0.65) (- (log N)) (- (/ 1.0 N) (/ (/ 0.5 N) N))))

C

double code(double N) {
	double tmp;
	if (N <= 0.65) {
		tmp = -log(N);
	} else {
		tmp = (1.0 / N) - ((0.5 / N) / N);
	}
	return tmp;
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= 0.65d0) then
        tmp = -log(n)
    else
        tmp = (1.0d0 / n) - ((0.5d0 / n) / n)
    end if
    code = tmp
end function

Java

public static double code(double N) {
	double tmp;
	if (N <= 0.65) {
		tmp = -Math.log(N);
	} else {
		tmp = (1.0 / N) - ((0.5 / N) / N);
	}
	return tmp;
}

Python

def code(N):
	tmp = 0
	if N <= 0.65:
		tmp = -math.log(N)
	else:
		tmp = (1.0 / N) - ((0.5 / N) / N)
	return tmp

Julia

function code(N)
	tmp = 0.0
	if (N <= 0.65)
		tmp = Float64(-log(N));
	else
		tmp = Float64(Float64(1.0 / N) - Float64(Float64(0.5 / N) / N));
	end
	return tmp
end

MATLAB

function tmp_2 = code(N)
	tmp = 0.0;
	if (N <= 0.65)
		tmp = -log(N);
	else
		tmp = (1.0 / N) - ((0.5 / N) / N);
	end
	tmp_2 = tmp;
end

Wolfram

code[N_] := If[LessEqual[N, 0.65], (-N[Log[N], $MachinePrecision]), N[(N[(1.0 / N), $MachinePrecision] - N[(N[(0.5 / N), $MachinePrecision] / N), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if N < 0.650000000000000022

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. log1p-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in N around 0 98.1%

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

      1. neg-mul-1 98.1%

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

   6. Simplified 98.1%

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

   if 0.650000000000000022 < N

   1. Initial program 7.6%

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

      1. +-commutative 7.6%

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

      2. log1p-def 7.6%

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

   3. Simplified 7.6%

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

   4. Taylor expanded in N around inf 99.3%

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

      1. associate-*r/ 99.3%

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

      2. metadata-eval 99.3%

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

      3. unpow2 99.3%

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

      4. associate-/r* 99.3%

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

   6. Simplified 99.3%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;N \leq 0.65:\\ \;\;\;\;-\log N\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N} - \frac{\frac{0.5}{N}}{N}\\ \end{array} \]

Alternative 7: 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 59.2%

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

   1. +-commutative 59.2%

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

   2. log1p-def 59.2%

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

3. Simplified 59.2%

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

4. Taylor expanded in N around inf 46.4%

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

5. Final simplification 46.4%

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

Alternative 8: 1.9% accurate, 205.0× speedup

Math

\[\begin{array}{l} \\ -0.5 \end{array} \]

FPCore

(FPCore (N) :precision binary64 -0.5)

C

double code(double N) {
	return -0.5;
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    code = -0.5d0
end function

Java

public static double code(double N) {
	return -0.5;
}

Python

def code(N):
	return -0.5

Julia

function code(N)
	return -0.5
end

MATLAB

function tmp = code(N)
	tmp = -0.5;
end

Wolfram

code[N_] := -0.5

Derivation

1. Initial program 59.2%

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

   1. +-commutative 59.2%

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

   2. log1p-def 59.2%

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

3. Simplified 59.2%

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

4. Taylor expanded in N around inf 44.3%

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

   1. associate-*r/ 44.3%

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

   2. metadata-eval 44.3%

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

   3. unpow2 44.3%

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

   4. associate-/r* 44.3%

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

6. Simplified 44.3%

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

   1. sub-div 44.3%

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

   2. div-inv 44.3%

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

   3. cancel-sign-sub-inv 44.3%

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

   4. metadata-eval 44.3%

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

   5. add-exp-log 44.3%

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

   6. neg-log 44.3%

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

   7. add-sqr-sqrt 0.5%

      \[\leadsto \frac{1 + -0.5 \cdot e^{\color{blue}{\sqrt{-\log N} \cdot \sqrt{-\log N}}}}{N} \]

   8. sqrt-unprod 1.5%

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

   9. sqr-neg 1.5%

      \[\leadsto \frac{1 + -0.5 \cdot e^{\sqrt{\color{blue}{\log N \cdot \log N}}}}{N} \]

   10. unpow2 1.5%

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

   11. unpow2 1.5%

       \[\leadsto \frac{1 + -0.5 \cdot e^{\sqrt{\color{blue}{\log N \cdot \log N}}}}{N} \]

   12. sqrt-prod 1.0%

       \[\leadsto \frac{1 + -0.5 \cdot e^{\color{blue}{\sqrt{\log N} \cdot \sqrt{\log N}}}}{N} \]

   13. add-sqr-sqrt 4.1%

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

   14. add-exp-log 4.1%

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

8. Applied egg-rr 4.1%

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

   1. *-commutative 4.1%

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

10. Simplified 4.1%

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

11. Taylor expanded in N around inf 1.9%

    \[\leadsto \color{blue}{-0.5} \]

12. Final simplification 1.9%

    \[\leadsto -0.5 \]

Reproduce

herbie shell --seed 2023273 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1.0)) (log N)))