2log (problem 3.3.6)

Percentage Accurate: 23.9% → 99.7%

Time: 10.2s

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: 23.9% 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: 99.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \mathsf{log1p}\left(\frac{\frac{1}{N} + 2}{N} \cdot \frac{-1}{-1 + \frac{-1 - N}{N}}\right) \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (log1p (* (/ (+ (/ 1.0 N) 2.0) N) (/ -1.0 (+ -1.0 (/ (- -1.0 N) N))))))

C

double code(double N) {
	return log1p(((((1.0 / N) + 2.0) / N) * (-1.0 / (-1.0 + ((-1.0 - N) / N)))));
}

Java

public static double code(double N) {
	return Math.log1p(((((1.0 / N) + 2.0) / N) * (-1.0 / (-1.0 + ((-1.0 - N) / N)))));
}

Python

def code(N):
	return math.log1p(((((1.0 / N) + 2.0) / N) * (-1.0 / (-1.0 + ((-1.0 - N) / N)))))

Julia

function code(N)
	return log1p(Float64(Float64(Float64(Float64(1.0 / N) + 2.0) / N) * Float64(-1.0 / Float64(-1.0 + Float64(Float64(-1.0 - N) / N)))))
end

Wolfram

code[N_] := N[Log[1 + N[(N[(N[(N[(1.0 / N), $MachinePrecision] + 2.0), $MachinePrecision] / N), $MachinePrecision] * N[(-1.0 / N[(-1.0 + N[(N[(-1.0 - N), $MachinePrecision] / N), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 22.8%

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

   1. +-commutative 22.8%

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

   2. log1p-define 22.8%

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

3. Simplified 22.8%

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

   1. log1p-expm1-u 22.8%

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

   2. expm1-undefine 22.9%

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

   3. exp-diff 22.8%

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

   4. log1p-undefine 22.8%

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

   5. rem-exp-log 25.2%

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

   6. add-exp-log 25.5%

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

   7. +-commutative 25.5%

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

6. Applied egg-rr 25.5%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(\frac{N + 1}{N} - 1\right)} \]
7. Step-by-step derivation

   1. flip-- 25.4%

      \[\leadsto \mathsf{log1p}\left(\color{blue}{\frac{\frac{N + 1}{N} \cdot \frac{N + 1}{N} - 1 \cdot 1}{\frac{N + 1}{N} + 1}}\right) \]

   2. div-inv 25.4%

      \[\leadsto \mathsf{log1p}\left(\color{blue}{\left(\frac{N + 1}{N} \cdot \frac{N + 1}{N} - 1 \cdot 1\right) \cdot \frac{1}{\frac{N + 1}{N} + 1}}\right) \]

   3. metadata-eval 25.4%

      \[\leadsto \mathsf{log1p}\left(\left(\frac{N + 1}{N} \cdot \frac{N + 1}{N} - \color{blue}{1}\right) \cdot \frac{1}{\frac{N + 1}{N} + 1}\right) \]

   4. sub-neg 25.4%

      \[\leadsto \mathsf{log1p}\left(\color{blue}{\left(\frac{N + 1}{N} \cdot \frac{N + 1}{N} + \left(-1\right)\right)} \cdot \frac{1}{\frac{N + 1}{N} + 1}\right) \]

   5. pow2 25.4%

      \[\leadsto \mathsf{log1p}\left(\left(\color{blue}{{\left(\frac{N + 1}{N}\right)}^{2}} + \left(-1\right)\right) \cdot \frac{1}{\frac{N + 1}{N} + 1}\right) \]

   6. metadata-eval 25.4%

      \[\leadsto \mathsf{log1p}\left(\left({\left(\frac{N + 1}{N}\right)}^{2} + \color{blue}{-1}\right) \cdot \frac{1}{\frac{N + 1}{N} + 1}\right) \]

8. Applied egg-rr 25.4%

   \[\leadsto \mathsf{log1p}\left(\color{blue}{\left({\left(\frac{N + 1}{N}\right)}^{2} + -1\right) \cdot \frac{1}{\frac{N + 1}{N} + 1}}\right) \]

9. Taylor expanded in N around inf 99.7%

   \[\leadsto \mathsf{log1p}\left(\color{blue}{\frac{2 + \frac{1}{N}}{N}} \cdot \frac{1}{\frac{N + 1}{N} + 1}\right) \]
10. Step-by-step derivation

    1. +-commutative 99.7%

       \[\leadsto \mathsf{log1p}\left(\frac{\color{blue}{\frac{1}{N} + 2}}{N} \cdot \frac{1}{\frac{N + 1}{N} + 1}\right) \]

11. Simplified 99.7%

    \[\leadsto \mathsf{log1p}\left(\color{blue}{\frac{\frac{1}{N} + 2}{N}} \cdot \frac{1}{\frac{N + 1}{N} + 1}\right) \]

12. Final simplification 99.7%

    \[\leadsto \mathsf{log1p}\left(\frac{\frac{1}{N} + 2}{N} \cdot \frac{-1}{-1 + \frac{-1 - N}{N}}\right) \]
13. Add Preprocessing

Alternative 2: 99.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;N \leq 1400:\\ \;\;\;\;-\log \left(\frac{N}{1 + N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{-0.5 + \frac{0.3333333333333333 + \frac{-0.25}{N}}{N}}{N}}{N}\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= N 1400.0)
   (- (log (/ N (+ 1.0 N))))
   (/ (+ 1.0 (/ (+ -0.5 (/ (+ 0.3333333333333333 (/ -0.25 N)) N)) N)) N)))

C

double code(double N) {
	double tmp;
	if (N <= 1400.0) {
		tmp = -log((N / (1.0 + N)));
	} else {
		tmp = (1.0 + ((-0.5 + ((0.3333333333333333 + (-0.25 / N)) / N)) / N)) / N;
	}
	return tmp;
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= 1400.0d0) then
        tmp = -log((n / (1.0d0 + n)))
    else
        tmp = (1.0d0 + (((-0.5d0) + ((0.3333333333333333d0 + ((-0.25d0) / n)) / n)) / n)) / n
    end if
    code = tmp
end function

Java

public static double code(double N) {
	double tmp;
	if (N <= 1400.0) {
		tmp = -Math.log((N / (1.0 + N)));
	} else {
		tmp = (1.0 + ((-0.5 + ((0.3333333333333333 + (-0.25 / N)) / N)) / N)) / N;
	}
	return tmp;
}

Python

def code(N):
	tmp = 0
	if N <= 1400.0:
		tmp = -math.log((N / (1.0 + N)))
	else:
		tmp = (1.0 + ((-0.5 + ((0.3333333333333333 + (-0.25 / N)) / N)) / N)) / N
	return tmp

Julia

function code(N)
	tmp = 0.0
	if (N <= 1400.0)
		tmp = Float64(-log(Float64(N / Float64(1.0 + N))));
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(-0.5 + Float64(Float64(0.3333333333333333 + Float64(-0.25 / N)) / N)) / N)) / N);
	end
	return tmp
end

MATLAB

function tmp_2 = code(N)
	tmp = 0.0;
	if (N <= 1400.0)
		tmp = -log((N / (1.0 + N)));
	else
		tmp = (1.0 + ((-0.5 + ((0.3333333333333333 + (-0.25 / N)) / N)) / N)) / N;
	end
	tmp_2 = tmp;
end

Wolfram

code[N_] := If[LessEqual[N, 1400.0], (-N[Log[N[(N / N[(1.0 + N), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), N[(N[(1.0 + N[(N[(-0.5 + N[(N[(0.3333333333333333 + N[(-0.25 / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if N < 1400

   1. Initial program 91.2%

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

      1. +-commutative 91.2%

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

      2. log1p-define 91.2%

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

   3. Simplified 91.2%

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

      1. log1p-expm1-u 91.2%

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

      2. expm1-undefine 91.6%

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

      3. exp-diff 91.3%

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

      4. log1p-undefine 91.3%

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

      5. rem-exp-log 91.0%

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

      6. add-exp-log 93.8%

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

      7. +-commutative 93.8%

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

   6. Applied egg-rr 93.8%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\frac{N + 1}{N} - 1\right)} \]
   7. Step-by-step derivation

      1. add-exp-log 93.8%

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

      2. expm1-define 93.8%

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

      3. log1p-expm1-u 93.8%

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

      4. clear-num 93.6%

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

      5. log-div 94.4%

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

      6. metadata-eval 94.4%

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

   8. Applied egg-rr 94.4%

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

      1. neg-sub0 94.4%

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

   10. Simplified 94.4%

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

   if 1400 < N

   1. Initial program 16.7%

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

      1. +-commutative 16.7%

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

      2. log1p-define 16.7%

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

   3. Simplified 16.7%

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

   5. Taylor expanded in N around -inf 99.9%

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

      1. mul-1-neg 99.9%

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

      2. distribute-neg-frac2 99.9%

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

   7. Simplified 99.9%

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

   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%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;N \leq 1400:\\ \;\;\;\;-\log \left(\frac{N}{1 + N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{-0.5 + \frac{0.3333333333333333 + \frac{-0.25}{N}}{N}}{N}}{N}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;N \leq 1150:\\ \;\;\;\;\log \left(\frac{1}{N} + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-0.5 + \frac{0.3333333333333333 + \frac{-0.25}{N}}{N}}{N} + 1}{N}\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= N 1150.0)
   (log (+ (/ 1.0 N) 1.0))
   (/ (+ (/ (+ -0.5 (/ (+ 0.3333333333333333 (/ -0.25 N)) N)) N) 1.0) N)))

C

double code(double N) {
	double tmp;
	if (N <= 1150.0) {
		tmp = log(((1.0 / N) + 1.0));
	} else {
		tmp = (((-0.5 + ((0.3333333333333333 + (-0.25 / N)) / N)) / N) + 1.0) / N;
	}
	return tmp;
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= 1150.0d0) then
        tmp = log(((1.0d0 / n) + 1.0d0))
    else
        tmp = ((((-0.5d0) + ((0.3333333333333333d0 + ((-0.25d0) / n)) / n)) / n) + 1.0d0) / n
    end if
    code = tmp
end function

Java

public static double code(double N) {
	double tmp;
	if (N <= 1150.0) {
		tmp = Math.log(((1.0 / N) + 1.0));
	} else {
		tmp = (((-0.5 + ((0.3333333333333333 + (-0.25 / N)) / N)) / N) + 1.0) / N;
	}
	return tmp;
}

Python

def code(N):
	tmp = 0
	if N <= 1150.0:
		tmp = math.log(((1.0 / N) + 1.0))
	else:
		tmp = (((-0.5 + ((0.3333333333333333 + (-0.25 / N)) / N)) / N) + 1.0) / N
	return tmp

Julia

function code(N)
	tmp = 0.0
	if (N <= 1150.0)
		tmp = log(Float64(Float64(1.0 / N) + 1.0));
	else
		tmp = Float64(Float64(Float64(Float64(-0.5 + Float64(Float64(0.3333333333333333 + Float64(-0.25 / N)) / N)) / N) + 1.0) / N);
	end
	return tmp
end

MATLAB

function tmp_2 = code(N)
	tmp = 0.0;
	if (N <= 1150.0)
		tmp = log(((1.0 / N) + 1.0));
	else
		tmp = (((-0.5 + ((0.3333333333333333 + (-0.25 / N)) / N)) / N) + 1.0) / N;
	end
	tmp_2 = tmp;
end

Wolfram

code[N_] := If[LessEqual[N, 1150.0], N[Log[N[(N[(1.0 / N), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision], 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. Split input into 2 regimes

2. if N < 1150

   1. Initial program 91.2%

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

      1. +-commutative 91.2%

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

      2. log1p-define 91.2%

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

   3. Simplified 91.2%

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

      1. add-log-exp 91.2%

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

      2. log1p-expm1-u 91.2%

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

      3. log1p-undefine 91.2%

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

      4. diff-log 91.3%

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

      5. log1p-undefine 91.3%

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

      6. rem-exp-log 91.0%

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

      7. +-commutative 91.0%

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

      8. add-exp-log 91.0%

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

      9. log1p-undefine 91.0%

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

      10. log1p-expm1-u 91.0%

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

      11. add-exp-log 93.8%

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

   6. Applied egg-rr 93.8%

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

   7. Taylor expanded in N around inf 94.0%

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

   if 1150 < N

   1. Initial program 16.7%

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

      1. +-commutative 16.7%

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

      2. log1p-define 16.7%

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

   3. Simplified 16.7%

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

   5. Taylor expanded in N around -inf 99.9%

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

      1. mul-1-neg 99.9%

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

      2. distribute-neg-frac2 99.9%

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

   7. Simplified 99.9%

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

   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%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;N \leq 1150:\\ \;\;\;\;\log \left(\frac{1}{N} + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-0.5 + \frac{0.3333333333333333 + \frac{-0.25}{N}}{N}}{N} + 1}{N}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.2% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{-0.5 + \frac{0.3333333333333333 + \frac{-0.25 + \frac{0.375 + \frac{-0.28125}{N}}{N}}{N}}{N}}{N} + 1}{N} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (/
  (+
   (/
    (+
     -0.5
     (/
      (+ 0.3333333333333333 (/ (+ -0.25 (/ (+ 0.375 (/ -0.28125 N)) N)) N))
      N))
    N)
   1.0)
  N))

C

double code(double N) {
	return (((-0.5 + ((0.3333333333333333 + ((-0.25 + ((0.375 + (-0.28125 / N)) / N)) / N)) / N)) / N) + 1.0) / N;
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    code = ((((-0.5d0) + ((0.3333333333333333d0 + (((-0.25d0) + ((0.375d0 + ((-0.28125d0) / n)) / n)) / n)) / n)) / n) + 1.0d0) / n
end function

Java

public static double code(double N) {
	return (((-0.5 + ((0.3333333333333333 + ((-0.25 + ((0.375 + (-0.28125 / N)) / N)) / N)) / N)) / N) + 1.0) / N;
}

Python

def code(N):
	return (((-0.5 + ((0.3333333333333333 + ((-0.25 + ((0.375 + (-0.28125 / N)) / N)) / N)) / N)) / N) + 1.0) / N

Julia

function code(N)
	return Float64(Float64(Float64(Float64(-0.5 + Float64(Float64(0.3333333333333333 + Float64(Float64(-0.25 + Float64(Float64(0.375 + Float64(-0.28125 / N)) / N)) / N)) / N)) / N) + 1.0) / N)
end

MATLAB

function tmp = code(N)
	tmp = (((-0.5 + ((0.3333333333333333 + ((-0.25 + ((0.375 + (-0.28125 / N)) / N)) / N)) / N)) / N) + 1.0) / N;
end

Wolfram

code[N_] := N[(N[(N[(N[(-0.5 + N[(N[(0.3333333333333333 + N[(N[(-0.25 + N[(N[(0.375 + N[(-0.28125 / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision] + 1.0), $MachinePrecision] / N), $MachinePrecision]

Derivation

1. Initial program 22.8%

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

   1. +-commutative 22.8%

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

   2. log1p-define 22.8%

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

3. Simplified 22.8%

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

5. Taylor expanded in N around -inf 96.4%

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

   1. mul-1-neg 96.4%

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

   2. distribute-neg-frac2 96.4%

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

7. Simplified 96.4%

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

   1. flip-+ 96.4%

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

   2. frac-2neg 96.4%

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

   3. cancel-sign-sub-inv 96.4%

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

   4. metadata-eval 96.4%

      \[\leadsto \frac{-1 - \frac{-0.5 + \frac{\frac{-\left(\color{blue}{0.1111111111111111} + \left(-\frac{-0.25}{N}\right) \cdot \frac{-0.25}{N}\right)}{-\left(0.3333333333333333 - \frac{-0.25}{N}\right)}}{N}}{N}}{-N} \]

   5. frac-2neg 96.4%

      \[\leadsto \frac{-1 - \frac{-0.5 + \frac{\frac{-\left(0.1111111111111111 + \left(-\color{blue}{\frac{--0.25}{-N}}\right) \cdot \frac{-0.25}{N}\right)}{-\left(0.3333333333333333 - \frac{-0.25}{N}\right)}}{N}}{N}}{-N} \]

   6. add-sqr-sqrt 0.0%

      \[\leadsto \frac{-1 - \frac{-0.5 + \frac{\frac{-\left(0.1111111111111111 + \left(-\frac{--0.25}{\color{blue}{\sqrt{-N} \cdot \sqrt{-N}}}\right) \cdot \frac{-0.25}{N}\right)}{-\left(0.3333333333333333 - \frac{-0.25}{N}\right)}}{N}}{N}}{-N} \]

   7. sqrt-unprod 96.4%

      \[\leadsto \frac{-1 - \frac{-0.5 + \frac{\frac{-\left(0.1111111111111111 + \left(-\frac{--0.25}{\color{blue}{\sqrt{\left(-N\right) \cdot \left(-N\right)}}}\right) \cdot \frac{-0.25}{N}\right)}{-\left(0.3333333333333333 - \frac{-0.25}{N}\right)}}{N}}{N}}{-N} \]

   8. sqr-neg 96.4%

      \[\leadsto \frac{-1 - \frac{-0.5 + \frac{\frac{-\left(0.1111111111111111 + \left(-\frac{--0.25}{\sqrt{\color{blue}{N \cdot N}}}\right) \cdot \frac{-0.25}{N}\right)}{-\left(0.3333333333333333 - \frac{-0.25}{N}\right)}}{N}}{N}}{-N} \]

   9. sqrt-unprod 96.4%

      \[\leadsto \frac{-1 - \frac{-0.5 + \frac{\frac{-\left(0.1111111111111111 + \left(-\frac{--0.25}{\color{blue}{\sqrt{N} \cdot \sqrt{N}}}\right) \cdot \frac{-0.25}{N}\right)}{-\left(0.3333333333333333 - \frac{-0.25}{N}\right)}}{N}}{N}}{-N} \]

   10. add-sqr-sqrt 96.4%

       \[\leadsto \frac{-1 - \frac{-0.5 + \frac{\frac{-\left(0.1111111111111111 + \left(-\frac{--0.25}{\color{blue}{N}}\right) \cdot \frac{-0.25}{N}\right)}{-\left(0.3333333333333333 - \frac{-0.25}{N}\right)}}{N}}{N}}{-N} \]

   11. distribute-frac-neg2 96.4%

       \[\leadsto \frac{-1 - \frac{-0.5 + \frac{\frac{-\left(0.1111111111111111 + \color{blue}{\frac{--0.25}{-N}} \cdot \frac{-0.25}{N}\right)}{-\left(0.3333333333333333 - \frac{-0.25}{N}\right)}}{N}}{N}}{-N} \]

   12. frac-2neg 96.4%

       \[\leadsto \frac{-1 - \frac{-0.5 + \frac{\frac{-\left(0.1111111111111111 + \color{blue}{\frac{-0.25}{N}} \cdot \frac{-0.25}{N}\right)}{-\left(0.3333333333333333 - \frac{-0.25}{N}\right)}}{N}}{N}}{-N} \]

   13. frac-times 96.4%

       \[\leadsto \frac{-1 - \frac{-0.5 + \frac{\frac{-\left(0.1111111111111111 + \color{blue}{\frac{-0.25 \cdot -0.25}{N \cdot N}}\right)}{-\left(0.3333333333333333 - \frac{-0.25}{N}\right)}}{N}}{N}}{-N} \]

   14. metadata-eval 96.4%

       \[\leadsto \frac{-1 - \frac{-0.5 + \frac{\frac{-\left(0.1111111111111111 + \frac{\color{blue}{0.0625}}{N \cdot N}\right)}{-\left(0.3333333333333333 - \frac{-0.25}{N}\right)}}{N}}{N}}{-N} \]

   15. pow2 96.4%

       \[\leadsto \frac{-1 - \frac{-0.5 + \frac{\frac{-\left(0.1111111111111111 + \frac{0.0625}{\color{blue}{{N}^{2}}}\right)}{-\left(0.3333333333333333 - \frac{-0.25}{N}\right)}}{N}}{N}}{-N} \]

   16. sub-neg 96.4%

       \[\leadsto \frac{-1 - \frac{-0.5 + \frac{\frac{-\left(0.1111111111111111 + \frac{0.0625}{{N}^{2}}\right)}{-\color{blue}{\left(0.3333333333333333 + \left(-\frac{-0.25}{N}\right)\right)}}}{N}}{N}}{-N} \]

   17. frac-2neg 96.4%

       \[\leadsto \frac{-1 - \frac{-0.5 + \frac{\frac{-\left(0.1111111111111111 + \frac{0.0625}{{N}^{2}}\right)}{-\left(0.3333333333333333 + \left(-\color{blue}{\frac{--0.25}{-N}}\right)\right)}}{N}}{N}}{-N} \]

   18. add-sqr-sqrt 0.0%

       \[\leadsto \frac{-1 - \frac{-0.5 + \frac{\frac{-\left(0.1111111111111111 + \frac{0.0625}{{N}^{2}}\right)}{-\left(0.3333333333333333 + \left(-\frac{--0.25}{\color{blue}{\sqrt{-N} \cdot \sqrt{-N}}}\right)\right)}}{N}}{N}}{-N} \]

9. Applied egg-rr 96.4%

   \[\leadsto \frac{-1 - \frac{-0.5 + \frac{\color{blue}{\frac{-\left(0.1111111111111111 + \frac{0.0625}{{N}^{2}}\right)}{\frac{-0.25}{N} + -0.3333333333333333}}}{N}}{N}}{-N} \]
10. Step-by-step derivation

    1. distribute-neg-in 96.4%

       \[\leadsto \frac{-1 - \frac{-0.5 + \frac{\frac{\color{blue}{\left(-0.1111111111111111\right) + \left(-\frac{0.0625}{{N}^{2}}\right)}}{\frac{-0.25}{N} + -0.3333333333333333}}{N}}{N}}{-N} \]

    2. metadata-eval 96.4%

       \[\leadsto \frac{-1 - \frac{-0.5 + \frac{\frac{\color{blue}{-0.1111111111111111} + \left(-\frac{0.0625}{{N}^{2}}\right)}{\frac{-0.25}{N} + -0.3333333333333333}}{N}}{N}}{-N} \]

    3. distribute-neg-frac 96.4%

       \[\leadsto \frac{-1 - \frac{-0.5 + \frac{\frac{-0.1111111111111111 + \color{blue}{\frac{-0.0625}{{N}^{2}}}}{\frac{-0.25}{N} + -0.3333333333333333}}{N}}{N}}{-N} \]

    4. metadata-eval 96.4%

       \[\leadsto \frac{-1 - \frac{-0.5 + \frac{\frac{-0.1111111111111111 + \frac{\color{blue}{-0.0625}}{{N}^{2}}}{\frac{-0.25}{N} + -0.3333333333333333}}{N}}{N}}{-N} \]

11. Simplified 96.4%

    \[\leadsto \frac{-1 - \frac{-0.5 + \frac{\color{blue}{\frac{-0.1111111111111111 + \frac{-0.0625}{{N}^{2}}}{\frac{-0.25}{N} + -0.3333333333333333}}}{N}}{N}}{-N} \]

12. Taylor expanded in N around inf 96.4%

    \[\leadsto \frac{-1 - \frac{-0.5 + \color{blue}{\frac{\left(0.3333333333333333 + \frac{0.375}{{N}^{2}}\right) - \left(0.25 \cdot \frac{1}{N} + 0.28125 \cdot \frac{1}{{N}^{3}}\right)}{N}}}{N}}{-N} \]

14. Simplified 96.4%

    \[\leadsto \frac{-1 - \frac{-0.5 + \color{blue}{\frac{0.3333333333333333 + \frac{\frac{0.375 + \frac{-0.28125}{N}}{N} + -0.25}{N}}{N}}}{N}}{-N} \]

15. Final simplification 96.4%

    \[\leadsto \frac{\frac{-0.5 + \frac{0.3333333333333333 + \frac{-0.25 + \frac{0.375 + \frac{-0.28125}{N}}{N}}{N}}{N}}{N} + 1}{N} \]
16. Add Preprocessing

Alternative 5: 96.1% accurate, 13.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[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] / N), $MachinePrecision]

Derivation

1. Initial program 22.8%

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

   1. +-commutative 22.8%

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

   2. log1p-define 22.8%

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

3. Simplified 22.8%

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

5. Taylor expanded in N around -inf 96.4%

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

   1. mul-1-neg 96.4%

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

   2. distribute-neg-frac2 96.4%

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

7. Simplified 96.4%

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

8. Taylor expanded in N around -inf 96.4%

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

10. Simplified 96.4%

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

11. Final simplification 96.4%

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

Alternative 6: 94.8% 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 22.8%

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

   1. +-commutative 22.8%

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

   2. log1p-define 22.8%

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

3. Simplified 22.8%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(N\right) - \log N} \]
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+ 95.1%

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

   2. unpow2 95.1%

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

   3. associate-/r* 95.1%

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

   4. metadata-eval 95.1%

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

   5. associate-*r/ 95.1%

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

   6. associate-*r/ 95.1%

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

   7. metadata-eval 95.1%

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

   8. div-sub 95.1%

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

   9. sub-neg 95.1%

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

   10. metadata-eval 95.1%

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

   11. +-commutative 95.1%

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

   12. associate-*r/ 95.1%

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

   13. metadata-eval 95.1%

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

7. Simplified 95.1%

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

8. Final simplification 95.1%

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

Alternative 7: 84.7% accurate, 29.3× speedup

Math

\[\begin{array}{l} \\ \frac{1 - \frac{0.16666666666666666}{N}}{N} \end{array} \]

FPCore

(FPCore (N) :precision binary64 (/ (- 1.0 (/ 0.16666666666666666 N)) N))

C

double code(double N) {
	return (1.0 - (0.16666666666666666 / N)) / N;
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    code = (1.0d0 - (0.16666666666666666d0 / n)) / n
end function

Java

public static double code(double N) {
	return (1.0 - (0.16666666666666666 / N)) / N;
}

Python

def code(N):
	return (1.0 - (0.16666666666666666 / N)) / N

Julia

function code(N)
	return Float64(Float64(1.0 - Float64(0.16666666666666666 / N)) / N)
end

MATLAB

function tmp = code(N)
	tmp = (1.0 - (0.16666666666666666 / N)) / N;
end

Wolfram

code[N_] := N[(N[(1.0 - N[(0.16666666666666666 / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision]

Derivation

1. Initial program 22.8%

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

   1. +-commutative 22.8%

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

   2. log1p-define 22.8%

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

3. Simplified 22.8%

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

5. Taylor expanded in N around -inf 96.4%

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

   1. mul-1-neg 96.4%

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

   2. distribute-neg-frac2 96.4%

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

7. Simplified 96.4%

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

9. Applied egg-rr 84.3%

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

10. Taylor expanded in N around inf 85.4%

    \[\leadsto \color{blue}{\frac{1 - 0.16666666666666666 \cdot \frac{1}{N}}{N}} \]
11. Step-by-step derivation

    1. associate-*r/ 85.4%

       \[\leadsto \frac{1 - \color{blue}{\frac{0.16666666666666666 \cdot 1}{N}}}{N} \]

    2. metadata-eval 85.4%

       \[\leadsto \frac{1 - \frac{\color{blue}{0.16666666666666666}}{N}}{N} \]

12. Simplified 85.4%

    \[\leadsto \color{blue}{\frac{1 - \frac{0.16666666666666666}{N}}{N}} \]

13. Final simplification 85.4%

    \[\leadsto \frac{1 - \frac{0.16666666666666666}{N}}{N} \]
14. Add Preprocessing

Alternative 8: 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 22.8%

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

   1. +-commutative 22.8%

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

   2. log1p-define 22.8%

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

3. Simplified 22.8%

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

5. Taylor expanded in N around inf 92.3%

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

   1. associate-*r/ 92.3%

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

   2. metadata-eval 92.3%

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

7. Simplified 92.3%

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

8. Final simplification 92.3%

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

Alternative 9: 84.4% 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 22.8%

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

   1. +-commutative 22.8%

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

   2. log1p-define 22.8%

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

3. Simplified 22.8%

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

5. Taylor expanded in N around inf 85.1%

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

6. Final simplification 85.1%

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

Developer target: 99.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Reproduce

herbie shell --seed 2024057 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  :pre (and (> N 1.0) (< N 1e+40))

  :alt
  (log1p (/ 1.0 N))

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