2log (problem 3.3.6)

Percentage Accurate: 23.5% → 99.4%

Time: 8.9s

Alternatives: 12

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: 23.5% 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: 99.4% accurate, 0.4× speedup

Math

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

FPCore

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

C

double code(double N) {
	double tmp;
	if ((log((N + 1.0)) - log(N)) <= 0.0006) {
		tmp = -1.0 / (N / (-1.0 - ((-0.5 - (((0.25 / N) + -0.3333333333333333) / N)) / N)));
	} else {
		tmp = exp(log(log(((N + 1.0) / N))));
	}
	return tmp;
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((log((n + 1.0d0)) - log(n)) <= 0.0006d0) then
        tmp = (-1.0d0) / (n / ((-1.0d0) - (((-0.5d0) - (((0.25d0 / n) + (-0.3333333333333333d0)) / n)) / n)))
    else
        tmp = exp(log(log(((n + 1.0d0) / n))))
    end if
    code = tmp
end function

Java

public static double code(double N) {
	double tmp;
	if ((Math.log((N + 1.0)) - Math.log(N)) <= 0.0006) {
		tmp = -1.0 / (N / (-1.0 - ((-0.5 - (((0.25 / N) + -0.3333333333333333) / N)) / N)));
	} else {
		tmp = Math.exp(Math.log(Math.log(((N + 1.0) / N))));
	}
	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 - (((0.25 / N) + -0.3333333333333333) / N)) / N)))
	else:
		tmp = math.exp(math.log(math.log(((N + 1.0) / N))))
	return tmp

Julia

function code(N)
	tmp = 0.0
	if (Float64(log(Float64(N + 1.0)) - log(N)) <= 0.0006)
		tmp = Float64(-1.0 / Float64(N / Float64(-1.0 - Float64(Float64(-0.5 - Float64(Float64(Float64(0.25 / N) + -0.3333333333333333) / N)) / N))));
	else
		tmp = exp(log(log(Float64(Float64(N + 1.0) / N))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(N)
	tmp = 0.0;
	if ((log((N + 1.0)) - log(N)) <= 0.0006)
		tmp = -1.0 / (N / (-1.0 - ((-0.5 - (((0.25 / N) + -0.3333333333333333) / N)) / N)));
	else
		tmp = exp(log(log(((N + 1.0) / N))));
	end
	tmp_2 = tmp;
end

Wolfram

code[N_] := If[LessEqual[N[(N[Log[N[(N + 1.0), $MachinePrecision]], $MachinePrecision] - N[Log[N], $MachinePrecision]), $MachinePrecision], 0.0006], N[(-1.0 / N[(N / N[(-1.0 - N[(N[(-0.5 - N[(N[(N[(0.25 / N), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Exp[N[Log[N[Log[N[(N[(N + 1.0), $MachinePrecision] / N), $MachinePrecision]], $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 15.6%

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

      1. +-commutative 15.6%

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

      2. log1p-define 15.6%

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

   3. Simplified 15.6%

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

   5. Taylor expanded in N around -inf 99.8%

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

      1. mul-1-neg 99.8%

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

      2. distribute-neg-frac2 99.8%

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

   7. Simplified 99.8%

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

   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. clear-num 99.9%

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

       2. inv-pow 99.9%

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

   12. Applied egg-rr 99.9%

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

       1. unpow-1 99.9%

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

       2. *-lft-identity 99.9%

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

   14. Simplified 99.9%

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

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

   1. Initial program 92.4%

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

      1. +-commutative 92.4%

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

      2. log1p-define 92.7%

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

   3. Simplified 92.7%

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

      1. add-exp-log 92.8%

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

   6. Applied egg-rr 92.8%

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

      1. add-log-exp 92.8%

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

      2. log1p-expm1-u 92.8%

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

      3. log1p-undefine 92.8%

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

      4. diff-log 92.6%

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

      5. log1p-undefine 92.6%

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

      6. rem-exp-log 93.0%

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

      7. +-commutative 93.0%

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

      8. add-exp-log 92.9%

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

      9. log1p-undefine 92.9%

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

      10. log1p-expm1-u 92.9%

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

      11. add-exp-log 94.4%

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

   8. Applied egg-rr 94.4%

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

4. Final simplification 99.5%

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

Alternative 2: 99.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

   1. Initial program 15.6%

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

      1. +-commutative 15.6%

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

      2. log1p-define 15.6%

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

   3. Simplified 15.6%

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

   5. Taylor expanded in N around -inf 99.8%

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

      1. mul-1-neg 99.8%

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

      2. distribute-neg-frac2 99.8%

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

   7. Simplified 99.8%

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

   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. clear-num 99.9%

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

       2. inv-pow 99.9%

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

   12. Applied egg-rr 99.9%

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

       1. unpow-1 99.9%

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

       2. *-lft-identity 99.9%

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

   14. Simplified 99.9%

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

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

   1. Initial program 92.4%

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

      1. +-commutative 92.4%

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

      2. log1p-define 92.7%

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

   3. Simplified 92.7%

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

      1. add-log-exp 92.8%

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

      2. log1p-expm1-u 92.8%

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

      3. log1p-undefine 92.8%

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

      4. diff-log 92.6%

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

      5. log1p-undefine 92.6%

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

      6. rem-exp-log 93.0%

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

      7. +-commutative 93.0%

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

      8. add-exp-log 92.9%

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

      9. log1p-undefine 92.9%

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

      10. log1p-expm1-u 92.9%

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

      11. add-exp-log 94.4%

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

   6. Applied egg-rr 94.3%

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

4. Final simplification 99.5%

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

Alternative 3: 96.5% accurate, 12.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 21.3%

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

   1. +-commutative 21.3%

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

   2. log1p-define 21.3%

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

3. Simplified 21.3%

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

5. Taylor expanded in N around -inf 96.3%

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

   1. mul-1-neg 96.3%

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

   2. distribute-neg-frac2 96.3%

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

7. Simplified 96.3%

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

8. Taylor expanded in N around -inf 96.3%

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

10. Simplified 96.3%

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

    1. clear-num 96.4%

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

    2. inv-pow 96.4%

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

12. Applied egg-rr 96.4%

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

    1. unpow-1 96.4%

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

    2. *-lft-identity 96.4%

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

14. Simplified 96.4%

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

15. Final simplification 96.4%

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

Alternative 4: 96.4% accurate, 12.1× speedup

Math

\[\begin{array}{l} \\ \left(-1 - \frac{-0.5 - \frac{\frac{0.25}{N} + -0.3333333333333333}{N}}{N}\right) \cdot \frac{-1}{N} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 21.3%

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

   1. +-commutative 21.3%

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

   2. log1p-define 21.3%

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

3. Simplified 21.3%

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

5. Taylor expanded in N around -inf 96.3%

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

   1. mul-1-neg 96.3%

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

   2. distribute-neg-frac2 96.3%

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

7. Simplified 96.3%

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

   1. div-inv 96.3%

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

   2. add-sqr-sqrt 0.0%

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

   3. sqrt-unprod 1.7%

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

   4. sqr-neg 1.7%

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

   5. sqrt-prod 1.7%

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

   6. add-sqr-sqrt 1.7%

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

   7. frac-2neg 1.7%

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

   8. metadata-eval 1.7%

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

   9. add-sqr-sqrt 0.0%

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

   10. sqrt-unprod 96.3%

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

   11. sqr-neg 96.3%

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

   12. sqrt-prod 95.6%

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

   13. add-sqr-sqrt 96.3%

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

9. Applied egg-rr 96.3%

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

Alternative 5: 96.4% accurate, 13.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 21.3%

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

   1. +-commutative 21.3%

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

   2. log1p-define 21.3%

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

3. Simplified 21.3%

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

5. Taylor expanded in N around -inf 96.3%

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

   1. mul-1-neg 96.3%

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

   2. distribute-neg-frac2 96.3%

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

7. Simplified 96.3%

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

8. Taylor expanded in N around -inf 96.3%

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

10. Simplified 96.3%

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

11. Final simplification 96.3%

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

Alternative 6: 95.1% accurate, 15.8× speedup

Math

\[\begin{array}{l} \\ \frac{-1}{N} \cdot \left(-1 + \frac{\frac{-0.3333333333333333}{N} - -0.5}{N}\right) \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 21.3%

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

   1. +-commutative 21.3%

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

   2. log1p-define 21.3%

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

3. Simplified 21.3%

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

5. Taylor expanded in N around -inf 96.3%

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

   1. mul-1-neg 96.3%

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

   2. distribute-neg-frac2 96.3%

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

7. Simplified 96.3%

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

   1. div-inv 96.3%

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

   2. add-sqr-sqrt 0.0%

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

   3. sqrt-unprod 1.7%

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

   4. sqr-neg 1.7%

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

   5. sqrt-prod 1.7%

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

   6. add-sqr-sqrt 1.7%

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

   7. frac-2neg 1.7%

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

   8. metadata-eval 1.7%

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

   9. add-sqr-sqrt 0.0%

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

   10. sqrt-unprod 96.3%

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

   11. sqr-neg 96.3%

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

   12. sqrt-prod 95.6%

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

   13. add-sqr-sqrt 96.3%

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

9. Applied egg-rr 96.3%

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

10. Taylor expanded in N around inf 95.2%

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

11. Final simplification 95.2%

    \[\leadsto \frac{-1}{N} \cdot \left(-1 + \frac{\frac{-0.3333333333333333}{N} - -0.5}{N}\right) \]
12. Add Preprocessing

Alternative 7: 95.2% 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 21.3%

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

   1. +-commutative 21.3%

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

   2. log1p-define 21.3%

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

3. Simplified 21.3%

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

5. Taylor expanded in N around inf 95.1%

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

   1. associate--l+ 95.2%

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

   2. unpow2 95.2%

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

   3. associate-/r* 95.2%

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

   4. metadata-eval 95.2%

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

   5. associate-*r/ 95.2%

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

   6. associate-*r/ 95.2%

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

   7. metadata-eval 95.2%

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

   8. div-sub 95.2%

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

   9. sub-neg 95.2%

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

   10. metadata-eval 95.2%

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

   11. +-commutative 95.2%

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

   12. associate-*r/ 95.2%

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

   13. metadata-eval 95.2%

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

7. Simplified 95.2%

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

8. Final simplification 95.2%

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

Alternative 8: 92.6% accurate, 22.8× speedup

Math

\[\begin{array}{l} \\ \frac{-1}{N} \cdot \left(-1 - \frac{-0.5}{N}\right) \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 21.3%

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

   1. +-commutative 21.3%

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

   2. log1p-define 21.3%

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

3. Simplified 21.3%

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

5. Taylor expanded in N around -inf 96.3%

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

   1. mul-1-neg 96.3%

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

   2. distribute-neg-frac2 96.3%

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

7. Simplified 96.3%

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

   1. div-inv 96.3%

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

   2. add-sqr-sqrt 0.0%

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

   3. sqrt-unprod 1.7%

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

   4. sqr-neg 1.7%

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

   5. sqrt-prod 1.7%

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

   6. add-sqr-sqrt 1.7%

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

   7. frac-2neg 1.7%

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

   8. metadata-eval 1.7%

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

   9. add-sqr-sqrt 0.0%

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

   10. sqrt-unprod 96.3%

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

   11. sqr-neg 96.3%

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

   12. sqrt-prod 95.6%

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

   13. add-sqr-sqrt 96.3%

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

9. Applied egg-rr 96.3%

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

10. Taylor expanded in N around inf 93.0%

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

11. Final simplification 93.0%

    \[\leadsto \frac{-1}{N} \cdot \left(-1 - \frac{-0.5}{N}\right) \]
12. Add Preprocessing

Alternative 9: 92.6% 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 21.3%

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

   1. +-commutative 21.3%

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

   2. log1p-define 21.3%

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

3. Simplified 21.3%

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

5. Taylor expanded in N around inf 93.0%

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

   1. associate-*r/ 93.0%

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

   2. metadata-eval 93.0%

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

7. Simplified 93.0%

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

Alternative 10: 84.7% 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 21.3%

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

   1. +-commutative 21.3%

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

   2. log1p-define 21.3%

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

3. Simplified 21.3%

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

5. Taylor expanded in N around inf 86.2%

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

Alternative 11: 8.4% accurate, 68.3× speedup

Math

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

FPCore

(FPCore (N) :precision binary64 (+ N -0.5))

C

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

Fortran

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

Java

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

Python

def code(N):
	return N + -0.5

Julia

function code(N)
	return Float64(N + -0.5)
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 21.3%

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

   1. +-commutative 21.3%

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

   2. log1p-define 21.3%

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

3. Simplified 21.3%

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

5. Taylor expanded in N around inf 93.0%

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

   1. associate-*r/ 93.0%

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

   2. metadata-eval 93.0%

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

7. Simplified 93.0%

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

   1. div-sub 93.0%

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

   2. rem-exp-log 88.6%

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

   3. neg-log 88.6%

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

   4. add-sqr-sqrt 0.0%

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

   5. sqrt-unprod 8.3%

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

   6. sqr-neg 8.3%

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

   7. unpow2 8.3%

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

   8. sqrt-pow1 8.3%

      \[\leadsto e^{\color{blue}{{\log N}^{\left(\frac{2}{2}\right)}}} - \frac{\frac{0.5}{N}}{N} \]

   9. metadata-eval 8.3%

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

   10. pow1 8.3%

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

   11. add-exp-log 8.3%

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

9. Applied egg-rr 8.3%

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

    1. associate-/l* 8.3%

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

    2. *-inverses 8.3%

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

    3. cancel-sign-sub-inv 8.3%

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

    4. metadata-eval 8.3%

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

    5. metadata-eval 8.3%

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

11. Simplified 8.3%

    \[\leadsto \color{blue}{N + -0.5} \]
12. Add Preprocessing

Alternative 12: 8.4% accurate, 205.0× speedup

Math

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

FPCore

(FPCore (N) :precision binary64 N)

C

double code(double N) {
	return N;
}

Fortran

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

Java

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

Python

def code(N):
	return N

Julia

function code(N)
	return N
end

MATLAB

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

Wolfram

code[N_] := N

Derivation

1. Initial program 21.3%

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

   1. +-commutative 21.3%

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

   2. log1p-define 21.3%

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

3. Simplified 21.3%

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

5. Taylor expanded in N around inf 93.0%

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

   1. associate-*r/ 93.0%

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

   2. metadata-eval 93.0%

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

7. Simplified 93.0%

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

   1. div-sub 93.0%

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

   2. rem-exp-log 88.6%

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

   3. neg-log 88.6%

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

   4. add-sqr-sqrt 0.0%

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

   5. sqrt-unprod 8.3%

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

   6. sqr-neg 8.3%

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

   7. unpow2 8.3%

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

   8. sqrt-pow1 8.3%

      \[\leadsto e^{\color{blue}{{\log N}^{\left(\frac{2}{2}\right)}}} - \frac{\frac{0.5}{N}}{N} \]

   9. metadata-eval 8.3%

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

   10. pow1 8.3%

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

   11. add-exp-log 8.3%

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

9. Applied egg-rr 8.3%

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

    1. associate-/l* 8.3%

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

    2. *-inverses 8.3%

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

    3. cancel-sign-sub-inv 8.3%

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

    4. metadata-eval 8.3%

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

    5. metadata-eval 8.3%

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

11. Simplified 8.3%

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

12. Taylor expanded in N around inf 8.3%

    \[\leadsto \color{blue}{N} \]
13. 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]

Developer Target 2: 26.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Developer Target 3: 96.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \left(\left(\frac{1}{N} + \frac{-1}{2 \cdot {N}^{2}}\right) + \frac{1}{3 \cdot {N}^{3}}\right) + \frac{-1}{4 \cdot {N}^{4}} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (+
  (+ (+ (/ 1.0 N) (/ -1.0 (* 2.0 (pow N 2.0)))) (/ 1.0 (* 3.0 (pow N 3.0))))
  (/ -1.0 (* 4.0 (pow N 4.0)))))

C

double code(double N) {
	return (((1.0 / N) + (-1.0 / (2.0 * pow(N, 2.0)))) + (1.0 / (3.0 * pow(N, 3.0)))) + (-1.0 / (4.0 * pow(N, 4.0)));
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    code = (((1.0d0 / n) + ((-1.0d0) / (2.0d0 * (n ** 2.0d0)))) + (1.0d0 / (3.0d0 * (n ** 3.0d0)))) + ((-1.0d0) / (4.0d0 * (n ** 4.0d0)))
end function

Java

public static double code(double N) {
	return (((1.0 / N) + (-1.0 / (2.0 * Math.pow(N, 2.0)))) + (1.0 / (3.0 * Math.pow(N, 3.0)))) + (-1.0 / (4.0 * Math.pow(N, 4.0)));
}

Python

def code(N):
	return (((1.0 / N) + (-1.0 / (2.0 * math.pow(N, 2.0)))) + (1.0 / (3.0 * math.pow(N, 3.0)))) + (-1.0 / (4.0 * math.pow(N, 4.0)))

Julia

function code(N)
	return Float64(Float64(Float64(Float64(1.0 / N) + Float64(-1.0 / Float64(2.0 * (N ^ 2.0)))) + Float64(1.0 / Float64(3.0 * (N ^ 3.0)))) + Float64(-1.0 / Float64(4.0 * (N ^ 4.0))))
end

MATLAB

function tmp = code(N)
	tmp = (((1.0 / N) + (-1.0 / (2.0 * (N ^ 2.0)))) + (1.0 / (3.0 * (N ^ 3.0)))) + (-1.0 / (4.0 * (N ^ 4.0)));
end

Wolfram

code[N_] := N[(N[(N[(N[(1.0 / N), $MachinePrecision] + N[(-1.0 / N[(2.0 * N[Power[N, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(3.0 * N[Power[N, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[(4.0 * N[Power[N, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024181 
(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)))

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

  :alt
  (! :herbie-platform default (+ (/ 1 N) (/ -1 (* 2 (pow N 2))) (/ 1 (* 3 (pow N 3))) (/ -1 (* 4 (pow N 4)))))

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