2log (problem 3.3.6)

Percentage Accurate: 24.3% → 99.4%

Time: 18.3s

Alternatives: 9

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: 24.3% 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: 99.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 0.0006:\\ \;\;\;\;\frac{-1}{\frac{N}{-1 - \frac{-0.5 + \frac{0.3333333333333333 + \frac{-0.25}{N}}{N}}{N}}}\\ \mathbf{else}:\\ \;\;\;\;-\log \left(\frac{N}{N + 1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= (- (log (+ N 1.0)) (log N)) 0.0006)
   (/
    -1.0
    (/ N (- -1.0 (/ (+ -0.5 (/ (+ 0.3333333333333333 (/ -0.25 N)) N)) N))))
   (- (log (/ N (+ N 1.0))))))

C

double code(double N) {
	double tmp;
	if ((log((N + 1.0)) - log(N)) <= 0.0006) {
		tmp = -1.0 / (N / (-1.0 - ((-0.5 + ((0.3333333333333333 + (-0.25 / N)) / 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)) <= 0.0006d0) then
        tmp = (-1.0d0) / (n / ((-1.0d0) - (((-0.5d0) + ((0.3333333333333333d0 + ((-0.25d0) / n)) / 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)) <= 0.0006) {
		tmp = -1.0 / (N / (-1.0 - ((-0.5 + ((0.3333333333333333 + (-0.25 / N)) / 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)) <= 0.0006:
		tmp = -1.0 / (N / (-1.0 - ((-0.5 + ((0.3333333333333333 + (-0.25 / N)) / 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)) <= 0.0006)
		tmp = Float64(-1.0 / Float64(N / Float64(-1.0 - Float64(Float64(-0.5 + Float64(Float64(0.3333333333333333 + Float64(-0.25 / N)) / 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)) <= 0.0006)
		tmp = -1.0 / (N / (-1.0 - ((-0.5 + ((0.3333333333333333 + (-0.25 / N)) / 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], 0.0006], N[(-1.0 / N[(N / N[(-1.0 - N[(N[(-0.5 + N[(N[(0.3333333333333333 + N[(-0.25 / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision]), $MachinePrecision] / 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 #s(literal 1 binary64))) (log.f64 N)) < 5.99999999999999947e-4

   1. Initial program 17.1%

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

      1. +-commutative 17.1%

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

      2. log1p-define 17.2%

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

   3. Simplified 17.2%

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

   5. Taylor expanded in N around inf 99.8%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in N around -inf 99.8%

      \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{N} - 0.3333333333333333}{N} - 0.5}{N} - 1}{N}} \]

   10. Simplified 99.8%

       \[\leadsto \color{blue}{\frac{1 - \frac{0.5 - \frac{0.3333333333333333 + \frac{-0.25}{N}}{N}}{N}}{N}} \]
   11. Step-by-step derivation

       1. div-sub 99.8%

          \[\leadsto \color{blue}{\frac{1}{N} - \frac{\frac{0.5 - \frac{0.3333333333333333 + \frac{-0.25}{N}}{N}}{N}}{N}} \]

   12. Applied egg-rr 99.8%

       \[\leadsto \color{blue}{\frac{1}{N} - \frac{\frac{0.5 - \frac{0.3333333333333333 + \frac{-0.25}{N}}{N}}{N}}{N}} \]
   13. Step-by-step derivation

       1. sub-div 99.8%

          \[\leadsto \color{blue}{\frac{1 - \frac{0.5 - \frac{0.3333333333333333 + \frac{-0.25}{N}}{N}}{N}}{N}} \]

       2. clear-num 99.8%

          \[\leadsto \color{blue}{\frac{1}{\frac{N}{1 - \frac{0.5 - \frac{0.3333333333333333 + \frac{-0.25}{N}}{N}}{N}}}} \]

       3. sub-neg 99.8%

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

       4. distribute-neg-frac 99.8%

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

       5. sub-neg 99.8%

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

       6. distribute-neg-in 99.8%

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

       7. metadata-eval 99.8%

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

       8. distribute-neg-frac2 99.8%

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

       9. distribute-frac-neg 99.8%

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

       10. frac-2neg 99.8%

           \[\leadsto \frac{1}{\frac{N}{1 + \frac{-0.5 + \color{blue}{\frac{0.3333333333333333 + \frac{-0.25}{N}}{N}}}{N}}} \]

   14. Applied egg-rr 99.8%

       \[\leadsto \color{blue}{\frac{1}{\frac{N}{1 + \frac{-0.5 + \frac{0.3333333333333333 + \frac{-0.25}{N}}{N}}{N}}}} \]

   if 5.99999999999999947e-4 < (-.f64 (log.f64 (+.f64 N #s(literal 1 binary64))) (log.f64 N))

   1. Initial program 89.8%

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

      1. +-commutative 89.8%

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

      2. log1p-define 89.8%

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

   3. Simplified 89.8%

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

      1. add-log-exp 90.0%

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

      2. add-sqr-sqrt 89.4%

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

      3. log-prod 89.4%

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

      4. exp-diff 89.3%

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

      5. log1p-undefine 89.5%

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

      6. rem-exp-log 89.9%

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

      7. add-exp-log 90.2%

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

      8. +-commutative 90.2%

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

      9. exp-diff 90.0%

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

      10. log1p-undefine 89.9%

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

      11. rem-exp-log 90.5%

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

      12. add-exp-log 90.6%

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

      13. +-commutative 90.6%

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

   6. Applied egg-rr 90.6%

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

      1. count-2 90.6%

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

   8. Simplified 90.6%

      \[\leadsto \color{blue}{2 \cdot \log \left(\sqrt{\frac{N + 1}{N}}\right)} \]
   9. Step-by-step derivation

      1. *-commutative 90.6%

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

      2. add-log-exp 90.8%

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

      3. exp-to-pow 90.6%

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

      4. pow2 90.6%

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

      5. add-sqr-sqrt 92.5%

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

      6. clear-num 92.2%

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

      7. log-div 93.7%

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

      8. metadata-eval 93.7%

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

      9. +-commutative 93.7%

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

   10. Applied egg-rr 93.7%

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

       1. neg-sub0 93.7%

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

       2. +-commutative 93.7%

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

   12. Simplified 93.7%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 0.0006:\\ \;\;\;\;\frac{-1}{\frac{N}{-1 - \frac{-0.5 + \frac{0.3333333333333333 + \frac{-0.25}{N}}{N}}{N}}}\\ \mathbf{else}:\\ \;\;\;\;-\log \left(\frac{N}{N + 1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;N \leq 1720:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{N}{-1 - \frac{-0.5 + \frac{0.3333333333333333 + \frac{-0.25}{N}}{N}}{N}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= N 1720.0)
   (log (/ (+ N 1.0) N))
   (/
    -1.0
    (/ N (- -1.0 (/ (+ -0.5 (/ (+ 0.3333333333333333 (/ -0.25 N)) N)) N))))))

C

double code(double N) {
	double tmp;
	if (N <= 1720.0) {
		tmp = log(((N + 1.0) / N));
	} else {
		tmp = -1.0 / (N / (-1.0 - ((-0.5 + ((0.3333333333333333 + (-0.25 / N)) / N)) / N)));
	}
	return tmp;
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= 1720.0d0) then
        tmp = log(((n + 1.0d0) / n))
    else
        tmp = (-1.0d0) / (n / ((-1.0d0) - (((-0.5d0) + ((0.3333333333333333d0 + ((-0.25d0) / n)) / n)) / n)))
    end if
    code = tmp
end function

Java

public static double code(double N) {
	double tmp;
	if (N <= 1720.0) {
		tmp = Math.log(((N + 1.0) / N));
	} else {
		tmp = -1.0 / (N / (-1.0 - ((-0.5 + ((0.3333333333333333 + (-0.25 / N)) / N)) / N)));
	}
	return tmp;
}

Python

def code(N):
	tmp = 0
	if N <= 1720.0:
		tmp = math.log(((N + 1.0) / N))
	else:
		tmp = -1.0 / (N / (-1.0 - ((-0.5 + ((0.3333333333333333 + (-0.25 / N)) / N)) / N)))
	return tmp

Julia

function code(N)
	tmp = 0.0
	if (N <= 1720.0)
		tmp = log(Float64(Float64(N + 1.0) / N));
	else
		tmp = Float64(-1.0 / Float64(N / Float64(-1.0 - Float64(Float64(-0.5 + Float64(Float64(0.3333333333333333 + Float64(-0.25 / N)) / N)) / N))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(N)
	tmp = 0.0;
	if (N <= 1720.0)
		tmp = log(((N + 1.0) / N));
	else
		tmp = -1.0 / (N / (-1.0 - ((-0.5 + ((0.3333333333333333 + (-0.25 / N)) / N)) / N)));
	end
	tmp_2 = tmp;
end

Wolfram

code[N_] := If[LessEqual[N, 1720.0], N[Log[N[(N[(N + 1.0), $MachinePrecision] / N), $MachinePrecision]], $MachinePrecision], N[(-1.0 / N[(N / N[(-1.0 - N[(N[(-0.5 + N[(N[(0.3333333333333333 + N[(-0.25 / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if N < 1720

   1. Initial program 89.5%

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

      1. +-commutative 89.5%

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

      2. log1p-define 89.5%

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

   3. Simplified 89.5%

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

      1. add-log-exp 89.5%

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

      2. log1p-expm1-u 89.5%

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

      3. log1p-undefine 89.5%

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

      4. diff-log 89.4%

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

      5. log1p-undefine 89.4%

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

      6. rem-exp-log 89.9%

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

      7. +-commutative 89.9%

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

      8. add-exp-log 89.6%

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

      9. log1p-undefine 89.6%

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

      10. log1p-expm1-u 89.6%

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

      11. add-exp-log 92.4%

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

   6. Applied egg-rr 92.4%

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

   if 1720 < N

   1. Initial program 16.8%

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

      1. +-commutative 16.8%

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

      2. log1p-define 16.9%

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

   3. Simplified 16.9%

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

   5. Taylor expanded in N around inf 99.8%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in N around -inf 99.9%

      \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{N} - 0.3333333333333333}{N} - 0.5}{N} - 1}{N}} \]

   10. Simplified 99.9%

       \[\leadsto \color{blue}{\frac{1 - \frac{0.5 - \frac{0.3333333333333333 + \frac{-0.25}{N}}{N}}{N}}{N}} \]
   11. Step-by-step derivation

       1. div-sub 99.9%

          \[\leadsto \color{blue}{\frac{1}{N} - \frac{\frac{0.5 - \frac{0.3333333333333333 + \frac{-0.25}{N}}{N}}{N}}{N}} \]

   12. Applied egg-rr 99.9%

       \[\leadsto \color{blue}{\frac{1}{N} - \frac{\frac{0.5 - \frac{0.3333333333333333 + \frac{-0.25}{N}}{N}}{N}}{N}} \]
   13. Step-by-step derivation

       1. sub-div 99.9%

          \[\leadsto \color{blue}{\frac{1 - \frac{0.5 - \frac{0.3333333333333333 + \frac{-0.25}{N}}{N}}{N}}{N}} \]

       2. clear-num 99.9%

          \[\leadsto \color{blue}{\frac{1}{\frac{N}{1 - \frac{0.5 - \frac{0.3333333333333333 + \frac{-0.25}{N}}{N}}{N}}}} \]

       3. sub-neg 99.9%

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

       4. distribute-neg-frac 99.9%

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

       5. sub-neg 99.9%

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

       6. distribute-neg-in 99.9%

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

       7. metadata-eval 99.9%

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

       8. distribute-neg-frac2 99.9%

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

       9. distribute-frac-neg 99.9%

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

       10. frac-2neg 99.9%

           \[\leadsto \frac{1}{\frac{N}{1 + \frac{-0.5 + \color{blue}{\frac{0.3333333333333333 + \frac{-0.25}{N}}{N}}}{N}}} \]

   14. Applied egg-rr 99.9%

       \[\leadsto \color{blue}{\frac{1}{\frac{N}{1 + \frac{-0.5 + \frac{0.3333333333333333 + \frac{-0.25}{N}}{N}}{N}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;N \leq 1720:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{N}{-1 - \frac{-0.5 + \frac{0.3333333333333333 + \frac{-0.25}{N}}{N}}{N}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.2% accurate, 12.1× speedup

Math

\[\begin{array}{l} \\ \frac{-1}{\frac{N}{-1 - \frac{-0.5 + \frac{0.3333333333333333 + \frac{-0.25}{N}}{N}}{N}}} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (/
  -1.0
  (/ N (- -1.0 (/ (+ -0.5 (/ (+ 0.3333333333333333 (/ -0.25 N)) N)) N)))))

C

double code(double N) {
	return -1.0 / (N / (-1.0 - ((-0.5 + ((0.3333333333333333 + (-0.25 / N)) / N)) / N)));
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    code = (-1.0d0) / (n / ((-1.0d0) - (((-0.5d0) + ((0.3333333333333333d0 + ((-0.25d0) / n)) / n)) / n)))
end function

Java

public static double code(double N) {
	return -1.0 / (N / (-1.0 - ((-0.5 + ((0.3333333333333333 + (-0.25 / N)) / N)) / N)));
}

Python

def code(N):
	return -1.0 / (N / (-1.0 - ((-0.5 + ((0.3333333333333333 + (-0.25 / N)) / N)) / N)))

Julia

function code(N)
	return Float64(-1.0 / Float64(N / Float64(-1.0 - Float64(Float64(-0.5 + Float64(Float64(0.3333333333333333 + Float64(-0.25 / N)) / N)) / N))))
end

MATLAB

function tmp = code(N)
	tmp = -1.0 / (N / (-1.0 - ((-0.5 + ((0.3333333333333333 + (-0.25 / N)) / N)) / N)));
end

Wolfram

code[N_] := N[(-1.0 / N[(N / N[(-1.0 - N[(N[(-0.5 + N[(N[(0.3333333333333333 + N[(-0.25 / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 22.2%

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

   1. +-commutative 22.2%

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

   2. log1p-define 22.3%

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

3. Simplified 22.3%

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

5. Taylor expanded in N around inf 97.2%

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

7. Simplified 97.2%

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

8. Taylor expanded in N around -inf 97.2%

   \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{N} - 0.3333333333333333}{N} - 0.5}{N} - 1}{N}} \]

10. Simplified 97.2%

    \[\leadsto \color{blue}{\frac{1 - \frac{0.5 - \frac{0.3333333333333333 + \frac{-0.25}{N}}{N}}{N}}{N}} \]
11. Step-by-step derivation

    1. div-sub 97.2%

       \[\leadsto \color{blue}{\frac{1}{N} - \frac{\frac{0.5 - \frac{0.3333333333333333 + \frac{-0.25}{N}}{N}}{N}}{N}} \]

12. Applied egg-rr 97.2%

    \[\leadsto \color{blue}{\frac{1}{N} - \frac{\frac{0.5 - \frac{0.3333333333333333 + \frac{-0.25}{N}}{N}}{N}}{N}} \]
13. Step-by-step derivation

    1. sub-div 97.2%

       \[\leadsto \color{blue}{\frac{1 - \frac{0.5 - \frac{0.3333333333333333 + \frac{-0.25}{N}}{N}}{N}}{N}} \]

    2. clear-num 97.3%

       \[\leadsto \color{blue}{\frac{1}{\frac{N}{1 - \frac{0.5 - \frac{0.3333333333333333 + \frac{-0.25}{N}}{N}}{N}}}} \]

    3. sub-neg 97.3%

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

    4. distribute-neg-frac 97.3%

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

    5. sub-neg 97.3%

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

    6. distribute-neg-in 97.3%

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

    7. metadata-eval 97.3%

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

    8. distribute-neg-frac2 97.3%

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

    9. distribute-frac-neg 97.3%

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

    10. frac-2neg 97.3%

        \[\leadsto \frac{1}{\frac{N}{1 + \frac{-0.5 + \color{blue}{\frac{0.3333333333333333 + \frac{-0.25}{N}}{N}}}{N}}} \]

14. Applied egg-rr 97.3%

    \[\leadsto \color{blue}{\frac{1}{\frac{N}{1 + \frac{-0.5 + \frac{0.3333333333333333 + \frac{-0.25}{N}}{N}}{N}}}} \]

15. Final simplification 97.3%

    \[\leadsto \frac{-1}{\frac{N}{-1 - \frac{-0.5 + \frac{0.3333333333333333 + \frac{-0.25}{N}}{N}}{N}}} \]
16. Add Preprocessing

Alternative 4: 96.2% accurate, 13.7× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\frac{0.3333333333333333 + \frac{-0.25}{N}}{N} - 0.5}{N} + 1}{N} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (/ (+ (/ (- (/ (+ 0.3333333333333333 (/ -0.25 N)) N) 0.5) N) 1.0) N))

C

double code(double N) {
	return (((((0.3333333333333333 + (-0.25 / N)) / N) - 0.5) / N) + 1.0) / N;
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    code = (((((0.3333333333333333d0 + ((-0.25d0) / n)) / n) - 0.5d0) / n) + 1.0d0) / n
end function

Java

public static double code(double N) {
	return (((((0.3333333333333333 + (-0.25 / N)) / N) - 0.5) / N) + 1.0) / N;
}

Python

def code(N):
	return (((((0.3333333333333333 + (-0.25 / N)) / N) - 0.5) / N) + 1.0) / N

Julia

function code(N)
	return Float64(Float64(Float64(Float64(Float64(Float64(0.3333333333333333 + Float64(-0.25 / N)) / N) - 0.5) / N) + 1.0) / N)
end

MATLAB

function tmp = code(N)
	tmp = (((((0.3333333333333333 + (-0.25 / N)) / N) - 0.5) / N) + 1.0) / N;
end

Wolfram

code[N_] := N[(N[(N[(N[(N[(N[(0.3333333333333333 + N[(-0.25 / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision] - 0.5), $MachinePrecision] / N), $MachinePrecision] + 1.0), $MachinePrecision] / N), $MachinePrecision]

Derivation

1. Initial program 22.2%

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

   1. +-commutative 22.2%

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

   2. log1p-define 22.3%

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

3. Simplified 22.3%

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

5. Taylor expanded in N around inf 97.2%

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

7. Simplified 97.2%

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

8. Taylor expanded in N around -inf 97.2%

   \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{N} - 0.3333333333333333}{N} - 0.5}{N} - 1}{N}} \]

10. Simplified 97.2%

    \[\leadsto \color{blue}{\frac{1 - \frac{0.5 - \frac{0.3333333333333333 + \frac{-0.25}{N}}{N}}{N}}{N}} \]

11. Final simplification 97.2%

    \[\leadsto \frac{\frac{\frac{0.3333333333333333 + \frac{-0.25}{N}}{N} - 0.5}{N} + 1}{N} \]
12. Add Preprocessing

Alternative 5: 94.9% accurate, 15.8× speedup

Math

\[\begin{array}{l} \\ \frac{1}{N} - \frac{\frac{0.5 - \frac{0.3333333333333333}{N}}{N}}{N} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (- (/ 1.0 N) (/ (/ (- 0.5 (/ 0.3333333333333333 N)) N) N)))

C

double code(double N) {
	return (1.0 / N) - (((0.5 - (0.3333333333333333 / N)) / N) / N);
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    code = (1.0d0 / n) - (((0.5d0 - (0.3333333333333333d0 / n)) / n) / n)
end function

Java

public static double code(double N) {
	return (1.0 / N) - (((0.5 - (0.3333333333333333 / N)) / N) / N);
}

Python

def code(N):
	return (1.0 / N) - (((0.5 - (0.3333333333333333 / N)) / N) / N)

Julia

function code(N)
	return Float64(Float64(1.0 / N) - Float64(Float64(Float64(0.5 - Float64(0.3333333333333333 / N)) / N) / N))
end

MATLAB

function tmp = code(N)
	tmp = (1.0 / N) - (((0.5 - (0.3333333333333333 / N)) / N) / N);
end

Wolfram

code[N_] := N[(N[(1.0 / N), $MachinePrecision] - N[(N[(N[(0.5 - N[(0.3333333333333333 / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision] / N), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 22.2%

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

   1. +-commutative 22.2%

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

   2. log1p-define 22.3%

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

3. Simplified 22.3%

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

5. Taylor expanded in N around inf 97.2%

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

7. Simplified 97.2%

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

8. Taylor expanded in N around -inf 97.2%

   \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{0.25 \cdot \frac{1}{N} - 0.3333333333333333}{N} - 0.5}{N} - 1}{N}} \]

10. Simplified 97.2%

    \[\leadsto \color{blue}{\frac{1 - \frac{0.5 - \frac{0.3333333333333333 + \frac{-0.25}{N}}{N}}{N}}{N}} \]
11. Step-by-step derivation

    1. div-sub 97.2%

       \[\leadsto \color{blue}{\frac{1}{N} - \frac{\frac{0.5 - \frac{0.3333333333333333 + \frac{-0.25}{N}}{N}}{N}}{N}} \]

12. Applied egg-rr 97.2%

    \[\leadsto \color{blue}{\frac{1}{N} - \frac{\frac{0.5 - \frac{0.3333333333333333 + \frac{-0.25}{N}}{N}}{N}}{N}} \]

13. Taylor expanded in N around inf 96.1%

    \[\leadsto \frac{1}{N} - \frac{\color{blue}{\frac{0.5 - 0.3333333333333333 \cdot \frac{1}{N}}{N}}}{N} \]
14. Step-by-step derivation

    1. associate-*r/ 96.1%

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

    2. metadata-eval 96.1%

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

15. Simplified 96.1%

    \[\leadsto \frac{1}{N} - \frac{\color{blue}{\frac{0.5 - \frac{0.3333333333333333}{N}}{N}}}{N} \]
16. Add Preprocessing

Alternative 6: 94.9% accurate, 18.6× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{-0.5 + \frac{0.3333333333333333}{N}}{N} + 1}{N} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (/ (+ (/ (+ -0.5 (/ 0.3333333333333333 N)) N) 1.0) N))

C

double code(double N) {
	return (((-0.5 + (0.3333333333333333 / N)) / N) + 1.0) / N;
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    code = ((((-0.5d0) + (0.3333333333333333d0 / n)) / n) + 1.0d0) / n
end function

Java

public static double code(double N) {
	return (((-0.5 + (0.3333333333333333 / N)) / N) + 1.0) / N;
}

Python

def code(N):
	return (((-0.5 + (0.3333333333333333 / N)) / N) + 1.0) / N

Julia

function code(N)
	return Float64(Float64(Float64(Float64(-0.5 + Float64(0.3333333333333333 / N)) / N) + 1.0) / N)
end

MATLAB

function tmp = code(N)
	tmp = (((-0.5 + (0.3333333333333333 / N)) / N) + 1.0) / N;
end

Wolfram

code[N_] := N[(N[(N[(N[(-0.5 + N[(0.3333333333333333 / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision] + 1.0), $MachinePrecision] / N), $MachinePrecision]

Derivation

1. Initial program 22.2%

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

   1. +-commutative 22.2%

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

   2. log1p-define 22.3%

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

3. Simplified 22.3%

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

5. Taylor expanded in N around inf 96.0%

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

   1. associate--l+ 96.1%

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

   2. unpow2 96.1%

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

   3. associate-/r* 96.1%

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

   4. metadata-eval 96.1%

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

   5. associate-*r/ 96.1%

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

   6. associate-*r/ 96.1%

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

   7. metadata-eval 96.1%

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

   8. div-sub 96.1%

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

   9. sub-neg 96.1%

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

   10. metadata-eval 96.1%

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

   11. +-commutative 96.1%

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

   12. associate-*r/ 96.1%

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

   13. metadata-eval 96.1%

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

7. Simplified 96.1%

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

8. Final simplification 96.1%

   \[\leadsto \frac{\frac{-0.5 + \frac{0.3333333333333333}{N}}{N} + 1}{N} \]
9. Add Preprocessing

Alternative 7: 92.2% accurate, 29.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 22.2%

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

   1. +-commutative 22.2%

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

   2. log1p-define 22.3%

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

3. Simplified 22.3%

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

5. Taylor expanded in N around inf 93.5%

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

   1. associate-*r/ 93.5%

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

   2. metadata-eval 93.5%

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

7. Simplified 93.5%

   \[\leadsto \color{blue}{\frac{1 - \frac{0.5}{N}}{N}} \]
8. Add Preprocessing

Alternative 8: 84.1% 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 22.2%

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

   1. +-commutative 22.2%

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

   2. log1p-define 22.3%

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

3. Simplified 22.3%

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

5. Taylor expanded in N around inf 85.8%

   \[\leadsto \color{blue}{\frac{1}{N}} \]
6. Add Preprocessing

Alternative 9: 3.3% accurate, 205.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (N) :precision binary64 0.0)

C

double code(double N) {
	return 0.0;
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    code = 0.0d0
end function

Java

public static double code(double N) {
	return 0.0;
}

Python

def code(N):
	return 0.0

Julia

function code(N)
	return 0.0
end

MATLAB

function tmp = code(N)
	tmp = 0.0;
end

Wolfram

code[N_] := 0.0

Derivation

1. Initial program 22.2%

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

   1. +-commutative 22.2%

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

   2. log1p-define 22.3%

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

3. Simplified 22.3%

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

   1. sub-neg 22.3%

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

   2. +-commutative 22.3%

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

   3. add-sqr-sqrt 22.7%

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

   4. distribute-rgt-neg-in 22.7%

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

   5. fma-define 23.6%

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

6. Applied egg-rr 23.6%

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

7. Taylor expanded in N around inf 3.3%

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

   1. distribute-rgt1-in 3.3%

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

   2. metadata-eval 3.3%

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

   3. mul0-lft 3.3%

      \[\leadsto \color{blue}{0} \]

9. Simplified 3.3%

   \[\leadsto \color{blue}{0} \]
10. Add Preprocessing

Developer Target 1: 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 2024143 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  :pre (and (> N 1.0) (< N 1e+40))

  :alt
  (! :herbie-platform default (log1p (/ 1 N)))

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