2log (problem 3.3.6)

Percentage Accurate: 23.9% → 99.4%

Time: 10.5s

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: 23.9% 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: 99.4% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (N)
 :precision binary64
 (if (<= (- (log (+ N 1.0)) (log N)) 0.0005)
   (/
    1.0
    (/ N (+ 1.0 (/ (+ (/ (+ 0.3333333333333333 (/ -0.25 N)) N) -0.5) N))))
   (- (log (/ N (+ N 1.0))))))

C

double code(double N) {
	double tmp;
	if ((log((N + 1.0)) - log(N)) <= 0.0005) {
		tmp = 1.0 / (N / (1.0 + ((((0.3333333333333333 + (-0.25 / N)) / N) + -0.5) / N)));
	} else {
		tmp = -log((N / (N + 1.0)));
	}
	return tmp;
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((log((n + 1.0d0)) - log(n)) <= 0.0005d0) then
        tmp = 1.0d0 / (n / (1.0d0 + ((((0.3333333333333333d0 + ((-0.25d0) / n)) / n) + (-0.5d0)) / n)))
    else
        tmp = -log((n / (n + 1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double N) {
	double tmp;
	if ((Math.log((N + 1.0)) - Math.log(N)) <= 0.0005) {
		tmp = 1.0 / (N / (1.0 + ((((0.3333333333333333 + (-0.25 / N)) / N) + -0.5) / N)));
	} else {
		tmp = -Math.log((N / (N + 1.0)));
	}
	return tmp;
}

Python

def code(N):
	tmp = 0
	if (math.log((N + 1.0)) - math.log(N)) <= 0.0005:
		tmp = 1.0 / (N / (1.0 + ((((0.3333333333333333 + (-0.25 / N)) / N) + -0.5) / N)))
	else:
		tmp = -math.log((N / (N + 1.0)))
	return tmp

Julia

function code(N)
	tmp = 0.0
	if (Float64(log(Float64(N + 1.0)) - log(N)) <= 0.0005)
		tmp = Float64(1.0 / Float64(N / Float64(1.0 + Float64(Float64(Float64(Float64(0.3333333333333333 + Float64(-0.25 / N)) / N) + -0.5) / N))));
	else
		tmp = Float64(-log(Float64(N / Float64(N + 1.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(N)
	tmp = 0.0;
	if ((log((N + 1.0)) - log(N)) <= 0.0005)
		tmp = 1.0 / (N / (1.0 + ((((0.3333333333333333 + (-0.25 / N)) / N) + -0.5) / N)));
	else
		tmp = -log((N / (N + 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[N_] := If[LessEqual[N[(N[Log[N[(N + 1.0), $MachinePrecision]], $MachinePrecision] - N[Log[N], $MachinePrecision]), $MachinePrecision], 0.0005], N[(1.0 / N[(N / N[(1.0 + N[(N[(N[(N[(0.3333333333333333 + N[(-0.25 / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision] + -0.5), $MachinePrecision] / N), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-N[Log[N[(N / N[(N + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (log.f64 (+.f64 N #s(literal 1 binary64))) (log.f64 N)) < 5.0000000000000001e-4

   1. Initial program 17.8%

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

      1. +-commutative 17.8%

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

      2. log1p-define 17.8%

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

   3. Simplified 17.8%

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

      1. add-exp-log 17.8%

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

      2. *-un-lft-identity 17.8%

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

      3. exp-prod 17.8%

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

      4. exp-1-e 17.8%

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

   6. Applied egg-rr 17.8%

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

   7. Taylor expanded in N around -inf 94.3%

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

      1. associate-*r/ 94.3%

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

   9. Simplified 94.3%

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

       1. e-exp-1 94.3%

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

       2. pow-exp 94.9%

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

       3. *-un-lft-identity 94.9%

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

       4. add-exp-log 99.8%

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

       5. clear-num 99.9%

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

       6. +-commutative 99.9%

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

       7. +-commutative 99.9%

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

   11. Applied egg-rr 99.9%

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

   if 5.0000000000000001e-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 92.4%

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

   3. Simplified 92.4%

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

      1. add-log-exp 92.4%

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

      2. log1p-expm1-u 92.5%

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

      3. log1p-undefine 92.4%

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

      4. diff-log 92.1%

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

      5. log1p-undefine 92.1%

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

      6. rem-exp-log 93.4%

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

      7. +-commutative 93.4%

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

      8. add-exp-log 93.5%

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

      9. log1p-undefine 93.5%

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

      10. log1p-expm1-u 93.5%

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

      11. add-exp-log 94.5%

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

   6. Applied egg-rr 94.5%

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

      1. clear-num 94.4%

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

      2. log-div 95.5%

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

      3. metadata-eval 95.5%

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

   8. Applied egg-rr 95.5%

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

      1. neg-sub0 95.5%

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

   10. Simplified 95.5%

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

4. Final simplification 99.5%

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

Alternative 2: 99.4% accurate, 1.9× speedup

Math

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

FPCore

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

C

double code(double N) {
	double tmp;
	if (N <= 1400.0) {
		tmp = log(((N + 1.0) / N));
	} else {
		tmp = 1.0 / (N / (1.0 + ((((0.3333333333333333 + (-0.25 / N)) / N) + -0.5) / 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 / (n / (1.0d0 + ((((0.3333333333333333d0 + ((-0.25d0) / n)) / n) + (-0.5d0)) / 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 / (N / (1.0 + ((((0.3333333333333333 + (-0.25 / N)) / N) + -0.5) / N)));
	}
	return tmp;
}

Python

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

Julia

function code(N)
	tmp = 0.0
	if (N <= 1400.0)
		tmp = log(Float64(Float64(N + 1.0) / N));
	else
		tmp = Float64(1.0 / Float64(N / Float64(1.0 + Float64(Float64(Float64(Float64(0.3333333333333333 + Float64(-0.25 / N)) / N) + -0.5) / 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 / (N / (1.0 + ((((0.3333333333333333 + (-0.25 / N)) / N) + -0.5) / N)));
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if N < 1400

   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 92.4%

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

   3. Simplified 92.4%

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

      1. add-log-exp 92.4%

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

      2. log1p-expm1-u 92.5%

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

      3. log1p-undefine 92.4%

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

      4. diff-log 92.1%

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

      5. log1p-undefine 92.1%

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

      6. rem-exp-log 93.4%

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

      7. +-commutative 93.4%

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

      8. add-exp-log 93.5%

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

      9. log1p-undefine 93.5%

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

      10. log1p-expm1-u 93.5%

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

      11. add-exp-log 94.5%

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

   6. Applied egg-rr 94.5%

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

   if 1400 < N

   1. Initial program 17.8%

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

      1. +-commutative 17.8%

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

      2. log1p-define 17.8%

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

   3. Simplified 17.8%

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

      1. add-exp-log 17.8%

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

      2. *-un-lft-identity 17.8%

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

      3. exp-prod 17.8%

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

      4. exp-1-e 17.8%

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

   6. Applied egg-rr 17.8%

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

   7. Taylor expanded in N around -inf 94.3%

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

      1. associate-*r/ 94.3%

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

   9. Simplified 94.3%

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

       1. e-exp-1 94.3%

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

       2. pow-exp 94.9%

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

       3. *-un-lft-identity 94.9%

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

       4. add-exp-log 99.8%

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

       5. clear-num 99.9%

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

       6. +-commutative 99.9%

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

       7. +-commutative 99.9%

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

   11. Applied egg-rr 99.9%

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

4. Final simplification 99.5%

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

Alternative 3: 96.2% accurate, 12.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 23.4%

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

   1. +-commutative 23.4%

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

   2. log1p-define 23.4%

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

3. Simplified 23.4%

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

   1. add-exp-log 23.4%

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

   2. *-un-lft-identity 23.4%

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

   3. exp-prod 23.4%

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

   4. exp-1-e 23.4%

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

6. Applied egg-rr 23.4%

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

7. Taylor expanded in N around -inf 91.1%

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

   1. associate-*r/ 91.1%

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

9. Simplified 91.1%

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

    1. e-exp-1 91.1%

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

    2. pow-exp 91.7%

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

    3. *-un-lft-identity 91.7%

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

    4. add-exp-log 96.3%

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

    5. clear-num 96.3%

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

    6. +-commutative 96.3%

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

    7. +-commutative 96.3%

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

11. Applied egg-rr 96.3%

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

12. Final simplification 96.3%

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

Alternative 4: 96.2% accurate, 13.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 23.4%

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

   1. +-commutative 23.4%

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

   2. log1p-define 23.4%

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

3. Simplified 23.4%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(N\right) - \log N} \]
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. associate-*r/ 91.1%

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

7. Simplified 96.3%

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

8. Final simplification 96.3%

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

Alternative 5: 94.9% accurate, 15.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 23.4%

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

   1. +-commutative 23.4%

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

   2. log1p-define 23.4%

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

3. Simplified 23.4%

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

   1. add-exp-log 23.4%

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

   2. *-un-lft-identity 23.4%

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

   3. exp-prod 23.4%

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

   4. exp-1-e 23.4%

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

6. Applied egg-rr 23.4%

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

7. Taylor expanded in N around -inf 91.1%

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

   1. associate-*r/ 91.1%

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

9. Simplified 91.1%

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

    1. e-exp-1 91.1%

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

    2. pow-exp 91.7%

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

    3. *-un-lft-identity 91.7%

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

    4. add-exp-log 96.3%

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

    5. clear-num 96.3%

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

    6. +-commutative 96.3%

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

    7. +-commutative 96.3%

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

11. Applied egg-rr 96.3%

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

12. Taylor expanded in N around inf 95.1%

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

    1. sub-neg 95.1%

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

    2. associate-*r/ 95.1%

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

    3. metadata-eval 95.1%

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

    4. metadata-eval 95.1%

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

14. Simplified 95.1%

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

15. Final simplification 95.1%

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

Alternative 6: 94.9% accurate, 18.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 23.4%

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

   1. +-commutative 23.4%

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

   2. log1p-define 23.4%

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

3. Simplified 23.4%

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

5. Taylor expanded in N around inf 95.0%

   \[\leadsto \color{blue}{\frac{\left(1 + \frac{0.3333333333333333}{{N}^{2}}\right) - 0.5 \cdot \frac{1}{N}}{N}} \]
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. +-commutative 95.1%

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

   3. unpow2 95.1%

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

   4. associate-/r* 95.1%

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

   5. metadata-eval 95.1%

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

   6. associate-*r/ 95.1%

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

   7. associate-*r/ 95.1%

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

   8. metadata-eval 95.1%

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

   9. div-sub 95.1%

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

   10. remove-double-neg 95.1%

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

   11. mul-1-neg 95.1%

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

   12. metadata-eval 95.1%

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

   13. distribute-neg-in 95.1%

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

   14. metadata-eval 95.1%

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

   15. sub-neg 95.1%

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

7. Simplified 95.1%

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

8. Final simplification 95.1%

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

Alternative 7: 92.3% 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 23.4%

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

   1. +-commutative 23.4%

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

   2. log1p-define 23.4%

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

3. Simplified 23.4%

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

5. Taylor expanded in N around inf 92.5%

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

   1. associate-*r/ 92.5%

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

   2. metadata-eval 92.5%

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

7. Simplified 92.5%

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

   1. div-inv 92.5%

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

   2. sub-neg 92.5%

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

   3. distribute-neg-frac 92.5%

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

   4. metadata-eval 92.5%

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

9. Applied egg-rr 92.5%

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

10. Final simplification 92.5%

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

Alternative 8: 92.3% accurate, 22.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 23.4%

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

   1. +-commutative 23.4%

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

   2. log1p-define 23.4%

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

3. Simplified 23.4%

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

5. Taylor expanded in N around inf 92.5%

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

   1. associate-*r/ 92.5%

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

   2. metadata-eval 92.5%

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

7. Simplified 92.5%

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

   1. div-sub 92.5%

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

9. Applied egg-rr 92.5%

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

10. Final simplification 92.5%

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

Alternative 9: 92.4% accurate, 22.8× speedup

Math

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

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(1.0 / Float64(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[(1.0 / N[(N / N[(1.0 + N[(-0.5 / N), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 23.4%

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

   1. +-commutative 23.4%

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

   2. log1p-define 23.4%

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

3. Simplified 23.4%

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

   1. add-exp-log 23.4%

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

   2. *-un-lft-identity 23.4%

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

   3. exp-prod 23.4%

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

   4. exp-1-e 23.4%

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

6. Applied egg-rr 23.4%

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

7. Taylor expanded in N around -inf 91.1%

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

   1. associate-*r/ 91.1%

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

9. Simplified 91.1%

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

    1. e-exp-1 91.1%

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

    2. pow-exp 91.7%

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

    3. *-un-lft-identity 91.7%

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

    4. add-exp-log 96.3%

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

    5. clear-num 96.3%

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

    6. +-commutative 96.3%

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

    7. +-commutative 96.3%

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

11. Applied egg-rr 96.3%

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

12. Taylor expanded in N around inf 92.6%

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

13. Final simplification 92.6%

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

Alternative 10: 92.3% 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 23.4%

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

   1. +-commutative 23.4%

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

   2. log1p-define 23.4%

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

3. Simplified 23.4%

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

5. Taylor expanded in N around inf 92.5%

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

   1. associate-*r/ 92.5%

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

   2. metadata-eval 92.5%

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

7. Simplified 92.5%

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

8. Final simplification 92.5%

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

Alternative 11: 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 23.4%

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

   1. +-commutative 23.4%

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

   2. log1p-define 23.4%

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

3. Simplified 23.4%

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

5. Taylor expanded in N around inf 84.7%

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

6. Final simplification 84.7%

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

Alternative 12: 3.3% accurate, 205.0× speedup

Math

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

FPCore

(FPCore (N) :precision binary64 0.0)

C

double code(double N) {
	return 0.0;
}

Fortran

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

Java

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

Python

def code(N):
	return 0.0

Julia

function code(N)
	return 0.0
end

MATLAB

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

Wolfram

code[N_] := 0.0

Derivation

1. Initial program 23.4%

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

   1. +-commutative 23.4%

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

   2. log1p-define 23.4%

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

3. Simplified 23.4%

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

   1. sub-neg 23.4%

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

   2. +-commutative 23.4%

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

   3. add-sqr-sqrt 23.4%

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

   4. distribute-rgt-neg-in 23.4%

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

   5. fma-define 24.3%

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

6. Applied egg-rr 24.3%

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

7. Taylor expanded in N around inf 3.3%

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

   1. distribute-rgt1-in 3.3%

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

   2. metadata-eval 3.3%

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

   3. mul0-lft 3.3%

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

9. Simplified 3.3%

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

10. Final simplification 3.3%

    \[\leadsto 0 \]
11. 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 2024077 
(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)))