2log (problem 3.3.6)

Percentage Accurate: 24.4% → 99.8%

Time: 13.8s

Alternatives: 7

Speedup: 68.3×

Specification

\[N > 1 \land N < 10^{+40}\]

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 24.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 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]

Derivation

1. Initial program 25.8%

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

   1. +-commutative 25.8%

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

   2. log1p-define 25.8%

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

3. Simplified 25.8%

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

   1. log1p-expm1-u 25.8%

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

   2. expm1-undefine 25.7%

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

   3. exp-diff 25.7%

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

   4. log1p-undefine 25.7%

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

   5. rem-exp-log 27.9%

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

   6. add-exp-log 28.7%

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

   7. +-commutative 28.7%

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

6. Applied egg-rr 28.7%

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

   1. flip3-- 28.7%

      \[\leadsto \mathsf{log1p}\left(\color{blue}{\frac{{\left(\frac{N + 1}{N}\right)}^{3} - {1}^{3}}{\frac{N + 1}{N} \cdot \frac{N + 1}{N} + \left(1 \cdot 1 + \frac{N + 1}{N} \cdot 1\right)}}\right) \]

   2. frac-2neg 28.7%

      \[\leadsto \mathsf{log1p}\left(\color{blue}{\frac{-\left({\left(\frac{N + 1}{N}\right)}^{3} - {1}^{3}\right)}{-\left(\frac{N + 1}{N} \cdot \frac{N + 1}{N} + \left(1 \cdot 1 + \frac{N + 1}{N} \cdot 1\right)\right)}}\right) \]

   3. pow3 28.7%

      \[\leadsto \mathsf{log1p}\left(\frac{-\left(\color{blue}{\left(\frac{N + 1}{N} \cdot \frac{N + 1}{N}\right) \cdot \frac{N + 1}{N}} - {1}^{3}\right)}{-\left(\frac{N + 1}{N} \cdot \frac{N + 1}{N} + \left(1 \cdot 1 + \frac{N + 1}{N} \cdot 1\right)\right)}\right) \]

   4. metadata-eval 28.7%

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

   5. sub-neg 28.7%

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

   6. pow3 28.7%

      \[\leadsto \mathsf{log1p}\left(\frac{-\left(\color{blue}{{\left(\frac{N + 1}{N}\right)}^{3}} + \left(-1\right)\right)}{-\left(\frac{N + 1}{N} \cdot \frac{N + 1}{N} + \left(1 \cdot 1 + \frac{N + 1}{N} \cdot 1\right)\right)}\right) \]

   7. metadata-eval 28.7%

      \[\leadsto \mathsf{log1p}\left(\frac{-\left({\left(\frac{N + 1}{N}\right)}^{3} + \color{blue}{-1}\right)}{-\left(\frac{N + 1}{N} \cdot \frac{N + 1}{N} + \left(1 \cdot 1 + \frac{N + 1}{N} \cdot 1\right)\right)}\right) \]

   8. pow2 28.7%

      \[\leadsto \mathsf{log1p}\left(\frac{-\left({\left(\frac{N + 1}{N}\right)}^{3} + -1\right)}{-\left(\color{blue}{{\left(\frac{N + 1}{N}\right)}^{2}} + \left(1 \cdot 1 + \frac{N + 1}{N} \cdot 1\right)\right)}\right) \]

   9. metadata-eval 28.7%

      \[\leadsto \mathsf{log1p}\left(\frac{-\left({\left(\frac{N + 1}{N}\right)}^{3} + -1\right)}{-\left({\left(\frac{N + 1}{N}\right)}^{2} + \left(\color{blue}{1} + \frac{N + 1}{N} \cdot 1\right)\right)}\right) \]

   10. *-rgt-identity 28.7%

       \[\leadsto \mathsf{log1p}\left(\frac{-\left({\left(\frac{N + 1}{N}\right)}^{3} + -1\right)}{-\left({\left(\frac{N + 1}{N}\right)}^{2} + \left(1 + \color{blue}{\frac{N + 1}{N}}\right)\right)}\right) \]

8. Applied egg-rr 28.7%

   \[\leadsto \mathsf{log1p}\left(\color{blue}{\frac{-\left({\left(\frac{N + 1}{N}\right)}^{3} + -1\right)}{-\left({\left(\frac{N + 1}{N}\right)}^{2} + \left(1 + \frac{N + 1}{N}\right)\right)}}\right) \]
9. Step-by-step derivation

   1. neg-sub0 28.7%

      \[\leadsto \mathsf{log1p}\left(\frac{\color{blue}{0 - \left({\left(\frac{N + 1}{N}\right)}^{3} + -1\right)}}{-\left({\left(\frac{N + 1}{N}\right)}^{2} + \left(1 + \frac{N + 1}{N}\right)\right)}\right) \]

   2. +-commutative 28.7%

      \[\leadsto \mathsf{log1p}\left(\frac{0 - \color{blue}{\left(-1 + {\left(\frac{N + 1}{N}\right)}^{3}\right)}}{-\left({\left(\frac{N + 1}{N}\right)}^{2} + \left(1 + \frac{N + 1}{N}\right)\right)}\right) \]

   3. associate--r+ 28.7%

      \[\leadsto \mathsf{log1p}\left(\frac{\color{blue}{\left(0 - -1\right) - {\left(\frac{N + 1}{N}\right)}^{3}}}{-\left({\left(\frac{N + 1}{N}\right)}^{2} + \left(1 + \frac{N + 1}{N}\right)\right)}\right) \]

   4. metadata-eval 28.7%

      \[\leadsto \mathsf{log1p}\left(\frac{\color{blue}{1} - {\left(\frac{N + 1}{N}\right)}^{3}}{-\left({\left(\frac{N + 1}{N}\right)}^{2} + \left(1 + \frac{N + 1}{N}\right)\right)}\right) \]

   5. neg-sub0 28.7%

      \[\leadsto \mathsf{log1p}\left(\frac{1 - {\left(\frac{N + 1}{N}\right)}^{3}}{\color{blue}{0 - \left({\left(\frac{N + 1}{N}\right)}^{2} + \left(1 + \frac{N + 1}{N}\right)\right)}}\right) \]

   6. +-commutative 28.7%

      \[\leadsto \mathsf{log1p}\left(\frac{1 - {\left(\frac{N + 1}{N}\right)}^{3}}{0 - \color{blue}{\left(\left(1 + \frac{N + 1}{N}\right) + {\left(\frac{N + 1}{N}\right)}^{2}\right)}}\right) \]

   7. +-commutative 28.7%

      \[\leadsto \mathsf{log1p}\left(\frac{1 - {\left(\frac{N + 1}{N}\right)}^{3}}{0 - \left(\color{blue}{\left(\frac{N + 1}{N} + 1\right)} + {\left(\frac{N + 1}{N}\right)}^{2}\right)}\right) \]

   8. associate--r+ 28.7%

      \[\leadsto \mathsf{log1p}\left(\frac{1 - {\left(\frac{N + 1}{N}\right)}^{3}}{\color{blue}{\left(0 - \left(\frac{N + 1}{N} + 1\right)\right) - {\left(\frac{N + 1}{N}\right)}^{2}}}\right) \]

10. Simplified 28.7%

    \[\leadsto \mathsf{log1p}\left(\color{blue}{\frac{1 - {\left(\frac{N + 1}{N}\right)}^{3}}{\left(-1 + \frac{-1 - N}{N}\right) - {\left(\frac{N + 1}{N}\right)}^{2}}}\right) \]

11. Taylor expanded in N around -inf 99.3%

    \[\leadsto \mathsf{log1p}\left(\frac{\color{blue}{-1 \cdot \frac{3 + \left(3 \cdot \frac{1}{N} + \frac{1}{{N}^{2}}\right)}{N}}}{\left(-1 + \frac{-1 - N}{N}\right) - {\left(\frac{N + 1}{N}\right)}^{2}}\right) \]
12. Step-by-step derivation

    1. mul-1-neg 99.3%

       \[\leadsto \mathsf{log1p}\left(\frac{\color{blue}{-\frac{3 + \left(3 \cdot \frac{1}{N} + \frac{1}{{N}^{2}}\right)}{N}}}{\left(-1 + \frac{-1 - N}{N}\right) - {\left(\frac{N + 1}{N}\right)}^{2}}\right) \]

    2. distribute-neg-frac2 99.3%

       \[\leadsto \mathsf{log1p}\left(\frac{\color{blue}{\frac{3 + \left(3 \cdot \frac{1}{N} + \frac{1}{{N}^{2}}\right)}{-N}}}{\left(-1 + \frac{-1 - N}{N}\right) - {\left(\frac{N + 1}{N}\right)}^{2}}\right) \]

    3. +-commutative 99.3%

       \[\leadsto \mathsf{log1p}\left(\frac{\frac{\color{blue}{\left(3 \cdot \frac{1}{N} + \frac{1}{{N}^{2}}\right) + 3}}{-N}}{\left(-1 + \frac{-1 - N}{N}\right) - {\left(\frac{N + 1}{N}\right)}^{2}}\right) \]

    4. associate-+l+ 99.4%

       \[\leadsto \mathsf{log1p}\left(\frac{\frac{\color{blue}{3 \cdot \frac{1}{N} + \left(\frac{1}{{N}^{2}} + 3\right)}}{-N}}{\left(-1 + \frac{-1 - N}{N}\right) - {\left(\frac{N + 1}{N}\right)}^{2}}\right) \]

    5. associate-*r/ 99.4%

       \[\leadsto \mathsf{log1p}\left(\frac{\frac{\color{blue}{\frac{3 \cdot 1}{N}} + \left(\frac{1}{{N}^{2}} + 3\right)}{-N}}{\left(-1 + \frac{-1 - N}{N}\right) - {\left(\frac{N + 1}{N}\right)}^{2}}\right) \]

    6. metadata-eval 99.4%

       \[\leadsto \mathsf{log1p}\left(\frac{\frac{\frac{\color{blue}{3}}{N} + \left(\frac{1}{{N}^{2}} + 3\right)}{-N}}{\left(-1 + \frac{-1 - N}{N}\right) - {\left(\frac{N + 1}{N}\right)}^{2}}\right) \]

13. Simplified 99.4%

    \[\leadsto \mathsf{log1p}\left(\frac{\color{blue}{\frac{\frac{3}{N} + \left(\frac{1}{{N}^{2}} + 3\right)}{-N}}}{\left(-1 + \frac{-1 - N}{N}\right) - {\left(\frac{N + 1}{N}\right)}^{2}}\right) \]

14. Taylor expanded in N around 0 99.8%

    \[\leadsto \mathsf{log1p}\left(\color{blue}{\frac{1}{N}}\right) \]
15. Add Preprocessing

Alternative 2: 96.1% accurate, 12.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 25.8%

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

   1. +-commutative 25.8%

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

   2. log1p-define 25.8%

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

3. Simplified 25.8%

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

   1. add-exp-log 25.8%

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

   2. *-un-lft-identity 25.8%

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

   3. exp-prod 25.8%

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

   4. exp-1-e 25.8%

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

6. Applied egg-rr 25.8%

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

7. Taylor expanded in N around -inf 91.2%

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

   1. associate-*r/ 91.2%

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

9. Simplified 91.2%

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

    1. e-exp-1 91.2%

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

    2. pow-exp 91.4%

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

    3. *-un-lft-identity 91.4%

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

    4. add-exp-log 95.9%

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

    5. clear-num 95.9%

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

11. Applied egg-rr 95.9%

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

Alternative 3: 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 25.8%

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

   1. +-commutative 25.8%

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

   2. log1p-define 25.8%

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

3. Simplified 25.8%

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

5. Taylor expanded in N around -inf 95.9%

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

   1. associate-*r/ 91.2%

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

7. Simplified 95.9%

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

Alternative 4: 94.8% 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 25.8%

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

   1. +-commutative 25.8%

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

   2. log1p-define 25.8%

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

3. Simplified 25.8%

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

   1. add-exp-log 25.8%

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

   2. *-un-lft-identity 25.8%

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

   3. exp-prod 25.8%

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

   4. exp-1-e 25.8%

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

6. Applied egg-rr 25.8%

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

7. Taylor expanded in N around -inf 91.2%

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

   1. associate-*r/ 91.2%

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

9. Simplified 91.2%

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

    1. e-exp-1 91.2%

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

    2. pow-exp 91.4%

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

    3. *-un-lft-identity 91.4%

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

    4. add-exp-log 95.9%

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

    5. clear-num 95.9%

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

11. Applied egg-rr 95.9%

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

12. Taylor expanded in N around inf 94.2%

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

    1. sub-neg 94.2%

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

    2. associate-*r/ 94.2%

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

    3. metadata-eval 94.2%

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

    4. metadata-eval 94.2%

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

14. Simplified 94.2%

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

15. Final simplification 94.2%

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

Alternative 5: 94.7% 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 25.8%

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

   1. +-commutative 25.8%

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

   2. log1p-define 25.8%

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

3. Simplified 25.8%

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

5. Taylor expanded in N around inf 94.1%

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

   1. sub-neg 94.1%

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

   2. associate-+l+ 94.2%

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

   3. sub-neg 94.2%

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

   4. unpow2 94.2%

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

   5. associate-/r* 94.2%

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

   6. metadata-eval 94.2%

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

   7. associate-*r/ 94.2%

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

   8. associate-*r/ 94.2%

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

   9. metadata-eval 94.2%

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

   10. div-sub 94.2%

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

   11. sub-neg 94.2%

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

   12. metadata-eval 94.2%

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

   13. +-commutative 94.2%

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

   14. associate-*r/ 94.2%

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

   15. metadata-eval 94.2%

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

7. Simplified 94.2%

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

Alternative 6: 92.0% 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 25.8%

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

   1. +-commutative 25.8%

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

   2. log1p-define 25.8%

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

3. Simplified 25.8%

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

5. Taylor expanded in N around inf 90.9%

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

   1. associate-*r/ 90.9%

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

   2. metadata-eval 90.9%

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

7. Simplified 90.9%

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

Alternative 7: 84.0% 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 25.8%

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

   1. +-commutative 25.8%

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

   2. log1p-define 25.8%

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

3. Simplified 25.8%

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

5. Taylor expanded in N around inf 82.9%

   \[\leadsto \color{blue}{\frac{1}{N}} \]
6. 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 2024106 
(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)))