2log (problem 3.3.6)

Percentage Accurate: 53.0% → 100.0%

Time: 10.0s

Alternatives: 3

Speedup: 2.0×

Specification

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: 53.0% 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: 100.0% 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 48.0%

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

   1. add-cbrt-cube 47.6%

      \[\leadsto \color{blue}{\sqrt[3]{\left(\left(\log \left(N + 1\right) - \log N\right) \cdot \left(\log \left(N + 1\right) - \log N\right)\right) \cdot \left(\log \left(N + 1\right) - \log N\right)}} \]

   2. pow1/3 46.9%

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

   3. add-sqr-sqrt 46.9%

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

   4. associate-*r* 46.9%

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

   5. unpow-prod-down 46.9%

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

4. Applied egg-rr 46.9%

   \[\leadsto \color{blue}{{\left({\left(\mathsf{log1p}\left(N\right) - \log N\right)}^{2.5}\right)}^{0.3333333333333333} \cdot {\left(\sqrt{\mathsf{log1p}\left(N\right) - \log N}\right)}^{0.3333333333333333}} \]
5. Step-by-step derivation

   1. unpow1/3 47.2%

      \[\leadsto {\left({\left(\mathsf{log1p}\left(N\right) - \log N\right)}^{2.5}\right)}^{0.3333333333333333} \cdot \color{blue}{\sqrt[3]{\sqrt{\mathsf{log1p}\left(N\right) - \log N}}} \]

   2. *-commutative 47.2%

      \[\leadsto \color{blue}{\sqrt[3]{\sqrt{\mathsf{log1p}\left(N\right) - \log N}} \cdot {\left({\left(\mathsf{log1p}\left(N\right) - \log N\right)}^{2.5}\right)}^{0.3333333333333333}} \]

   3. log1p-def 47.2%

      \[\leadsto \sqrt[3]{\sqrt{\color{blue}{\log \left(1 + N\right)} - \log N}} \cdot {\left({\left(\mathsf{log1p}\left(N\right) - \log N\right)}^{2.5}\right)}^{0.3333333333333333} \]

   4. +-commutative 47.2%

      \[\leadsto \sqrt[3]{\sqrt{\log \color{blue}{\left(N + 1\right)} - \log N}} \cdot {\left({\left(\mathsf{log1p}\left(N\right) - \log N\right)}^{2.5}\right)}^{0.3333333333333333} \]

   5. log-div 47.2%

      \[\leadsto \sqrt[3]{\sqrt{\color{blue}{\log \left(\frac{N + 1}{N}\right)}}} \cdot {\left({\left(\mathsf{log1p}\left(N\right) - \log N\right)}^{2.5}\right)}^{0.3333333333333333} \]

   6. *-rgt-identity 47.2%

      \[\leadsto \sqrt[3]{\sqrt{\log \left(\frac{\color{blue}{\left(N + 1\right) \cdot 1}}{N}\right)}} \cdot {\left({\left(\mathsf{log1p}\left(N\right) - \log N\right)}^{2.5}\right)}^{0.3333333333333333} \]

   7. associate-*r/ 46.9%

      \[\leadsto \sqrt[3]{\sqrt{\log \color{blue}{\left(\left(N + 1\right) \cdot \frac{1}{N}\right)}}} \cdot {\left({\left(\mathsf{log1p}\left(N\right) - \log N\right)}^{2.5}\right)}^{0.3333333333333333} \]

   8. *-commutative 46.9%

      \[\leadsto \sqrt[3]{\sqrt{\log \color{blue}{\left(\frac{1}{N} \cdot \left(N + 1\right)\right)}}} \cdot {\left({\left(\mathsf{log1p}\left(N\right) - \log N\right)}^{2.5}\right)}^{0.3333333333333333} \]

   9. distribute-lft-in 46.9%

      \[\leadsto \sqrt[3]{\sqrt{\log \color{blue}{\left(\frac{1}{N} \cdot N + \frac{1}{N} \cdot 1\right)}}} \cdot {\left({\left(\mathsf{log1p}\left(N\right) - \log N\right)}^{2.5}\right)}^{0.3333333333333333} \]

   10. lft-mult-inverse 47.2%

       \[\leadsto \sqrt[3]{\sqrt{\log \left(\color{blue}{1} + \frac{1}{N} \cdot 1\right)}} \cdot {\left({\left(\mathsf{log1p}\left(N\right) - \log N\right)}^{2.5}\right)}^{0.3333333333333333} \]

   11. *-rgt-identity 47.2%

       \[\leadsto \sqrt[3]{\sqrt{\log \left(1 + \color{blue}{\frac{1}{N}}\right)}} \cdot {\left({\left(\mathsf{log1p}\left(N\right) - \log N\right)}^{2.5}\right)}^{0.3333333333333333} \]

   12. unpow1/3 47.5%

       \[\leadsto \sqrt[3]{\sqrt{\log \left(1 + \frac{1}{N}\right)}} \cdot \color{blue}{\sqrt[3]{{\left(\mathsf{log1p}\left(N\right) - \log N\right)}^{2.5}}} \]

6. Simplified 47.6%

   \[\leadsto \color{blue}{\sqrt[3]{\sqrt{\log \left(1 + \frac{1}{N}\right)}} \cdot \sqrt[3]{{\log \left(1 + \frac{1}{N}\right)}^{2.5}}} \]
7. Step-by-step derivation

   1. expm1-log1p-u 47.5%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{\sqrt{\log \left(1 + \frac{1}{N}\right)}}\right)\right)} \cdot \sqrt[3]{{\log \left(1 + \frac{1}{N}\right)}^{2.5}} \]

   2. expm1-udef 47.5%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt[3]{\sqrt{\log \left(1 + \frac{1}{N}\right)}}\right)} - 1\right)} \cdot \sqrt[3]{{\log \left(1 + \frac{1}{N}\right)}^{2.5}} \]

   3. pow1/3 47.5%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{{\left(\sqrt{\log \left(1 + \frac{1}{N}\right)}\right)}^{0.3333333333333333}}\right)} - 1\right) \cdot \sqrt[3]{{\log \left(1 + \frac{1}{N}\right)}^{2.5}} \]

   4. sqrt-pow2 47.5%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{{\log \left(1 + \frac{1}{N}\right)}^{\left(\frac{0.3333333333333333}{2}\right)}}\right)} - 1\right) \cdot \sqrt[3]{{\log \left(1 + \frac{1}{N}\right)}^{2.5}} \]

   5. log1p-def 47.5%

      \[\leadsto \left(e^{\mathsf{log1p}\left({\color{blue}{\left(\mathsf{log1p}\left(\frac{1}{N}\right)\right)}}^{\left(\frac{0.3333333333333333}{2}\right)}\right)} - 1\right) \cdot \sqrt[3]{{\log \left(1 + \frac{1}{N}\right)}^{2.5}} \]

   6. metadata-eval 47.5%

      \[\leadsto \left(e^{\mathsf{log1p}\left({\left(\mathsf{log1p}\left(\frac{1}{N}\right)\right)}^{\color{blue}{0.16666666666666666}}\right)} - 1\right) \cdot \sqrt[3]{{\log \left(1 + \frac{1}{N}\right)}^{2.5}} \]

8. Applied egg-rr 47.5%

   \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left({\left(\mathsf{log1p}\left(\frac{1}{N}\right)\right)}^{0.16666666666666666}\right)} - 1\right)} \cdot \sqrt[3]{{\log \left(1 + \frac{1}{N}\right)}^{2.5}} \]
9. Step-by-step derivation

   1. expm1-def 47.5%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\mathsf{log1p}\left(\frac{1}{N}\right)\right)}^{0.16666666666666666}\right)\right)} \cdot \sqrt[3]{{\log \left(1 + \frac{1}{N}\right)}^{2.5}} \]

   2. expm1-log1p 47.7%

      \[\leadsto \color{blue}{{\left(\mathsf{log1p}\left(\frac{1}{N}\right)\right)}^{0.16666666666666666}} \cdot \sqrt[3]{{\log \left(1 + \frac{1}{N}\right)}^{2.5}} \]

10. Simplified 47.7%

    \[\leadsto \color{blue}{{\left(\mathsf{log1p}\left(\frac{1}{N}\right)\right)}^{0.16666666666666666}} \cdot \sqrt[3]{{\log \left(1 + \frac{1}{N}\right)}^{2.5}} \]
11. Step-by-step derivation

    1. *-un-lft-identity 47.7%

       \[\leadsto {\left(\mathsf{log1p}\left(\frac{1}{N}\right)\right)}^{0.16666666666666666} \cdot \sqrt[3]{{\log \color{blue}{\left(1 \cdot \left(1 + \frac{1}{N}\right)\right)}}^{2.5}} \]

    2. log-prod 47.7%

       \[\leadsto {\left(\mathsf{log1p}\left(\frac{1}{N}\right)\right)}^{0.16666666666666666} \cdot \sqrt[3]{{\color{blue}{\left(\log 1 + \log \left(1 + \frac{1}{N}\right)\right)}}^{2.5}} \]

    3. metadata-eval 47.7%

       \[\leadsto {\left(\mathsf{log1p}\left(\frac{1}{N}\right)\right)}^{0.16666666666666666} \cdot \sqrt[3]{{\left(\color{blue}{0} + \log \left(1 + \frac{1}{N}\right)\right)}^{2.5}} \]

    4. log1p-def 67.3%

       \[\leadsto {\left(\mathsf{log1p}\left(\frac{1}{N}\right)\right)}^{0.16666666666666666} \cdot \sqrt[3]{{\left(0 + \color{blue}{\mathsf{log1p}\left(\frac{1}{N}\right)}\right)}^{2.5}} \]

12. Applied egg-rr 67.3%

    \[\leadsto {\left(\mathsf{log1p}\left(\frac{1}{N}\right)\right)}^{0.16666666666666666} \cdot \sqrt[3]{{\color{blue}{\left(0 + \mathsf{log1p}\left(\frac{1}{N}\right)\right)}}^{2.5}} \]
13. Step-by-step derivation

    1. +-lft-identity 67.3%

       \[\leadsto {\left(\mathsf{log1p}\left(\frac{1}{N}\right)\right)}^{0.16666666666666666} \cdot \sqrt[3]{{\color{blue}{\left(\mathsf{log1p}\left(\frac{1}{N}\right)\right)}}^{2.5}} \]

14. Simplified 67.3%

    \[\leadsto {\left(\mathsf{log1p}\left(\frac{1}{N}\right)\right)}^{0.16666666666666666} \cdot \sqrt[3]{{\color{blue}{\left(\mathsf{log1p}\left(\frac{1}{N}\right)\right)}}^{2.5}} \]
15. Step-by-step derivation

    1. pow1/3 65.9%

       \[\leadsto {\left(\mathsf{log1p}\left(\frac{1}{N}\right)\right)}^{0.16666666666666666} \cdot \color{blue}{{\left({\left(\mathsf{log1p}\left(\frac{1}{N}\right)\right)}^{2.5}\right)}^{0.3333333333333333}} \]

    2. pow-pow 92.4%

       \[\leadsto {\left(\mathsf{log1p}\left(\frac{1}{N}\right)\right)}^{0.16666666666666666} \cdot \color{blue}{{\left(\mathsf{log1p}\left(\frac{1}{N}\right)\right)}^{\left(2.5 \cdot 0.3333333333333333\right)}} \]

    3. metadata-eval 94.5%

       \[\leadsto {\left(\mathsf{log1p}\left(\frac{1}{N}\right)\right)}^{0.16666666666666666} \cdot {\left(\mathsf{log1p}\left(\frac{1}{N}\right)\right)}^{\color{blue}{0.8333333333333334}} \]

    4. pow-prod-up 100.0%

       \[\leadsto \color{blue}{{\left(\mathsf{log1p}\left(\frac{1}{N}\right)\right)}^{\left(0.16666666666666666 + 0.8333333333333334\right)}} \]

    5. metadata-eval 100.0%

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

    6. add-log-exp 48.1%

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

    7. pow1 48.1%

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

    8. *-un-lft-identity 48.1%

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

    9. log-prod 48.1%

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

    10. metadata-eval 48.1%

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

    11. add-log-exp 100.0%

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

16. Applied egg-rr 100.0%

    \[\leadsto \color{blue}{0 + \mathsf{log1p}\left(\frac{1}{N}\right)} \]
17. Step-by-step derivation

    1. +-lft-identity 100.0%

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

18. Simplified 100.0%

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

19. Final simplification 100.0%

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

Alternative 2: 98.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;N \leq 0.52:\\ \;\;\;\;-\log N\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N}\\ \end{array} \end{array} \]

FPCore

(FPCore (N) :precision binary64 (if (<= N 0.52) (- (log N)) (/ 1.0 N)))

C

double code(double N) {
	double tmp;
	if (N <= 0.52) {
		tmp = -log(N);
	} else {
		tmp = 1.0 / N;
	}
	return tmp;
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= 0.52d0) then
        tmp = -log(n)
    else
        tmp = 1.0d0 / n
    end if
    code = tmp
end function

Java

public static double code(double N) {
	double tmp;
	if (N <= 0.52) {
		tmp = -Math.log(N);
	} else {
		tmp = 1.0 / N;
	}
	return tmp;
}

Python

def code(N):
	tmp = 0
	if N <= 0.52:
		tmp = -math.log(N)
	else:
		tmp = 1.0 / N
	return tmp

Julia

function code(N)
	tmp = 0.0
	if (N <= 0.52)
		tmp = Float64(-log(N));
	else
		tmp = Float64(1.0 / N);
	end
	return tmp
end

MATLAB

function tmp_2 = code(N)
	tmp = 0.0;
	if (N <= 0.52)
		tmp = -log(N);
	else
		tmp = 1.0 / N;
	end
	tmp_2 = tmp;
end

Wolfram

code[N_] := If[LessEqual[N, 0.52], (-N[Log[N], $MachinePrecision]), N[(1.0 / N), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if N < 0.52000000000000002

   1. Initial program 100.0%

      \[\log \left(N + 1\right) - \log N \]
   2. Add Preprocessing

   3. Taylor expanded in N around 0 97.7%

      \[\leadsto \color{blue}{-1 \cdot \log N} \]
   4. Step-by-step derivation

      1. neg-mul-1 97.7%

         \[\leadsto \color{blue}{-\log N} \]

   5. Simplified 97.7%

      \[\leadsto \color{blue}{-\log N} \]

   if 0.52000000000000002 < N

   1. Initial program 6.9%

      \[\log \left(N + 1\right) - \log N \]
   2. Add Preprocessing

   3. Taylor expanded in N around inf 98.8%

      \[\leadsto \color{blue}{\frac{1}{N}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;N \leq 0.52:\\ \;\;\;\;-\log N\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 52.7% accurate, 68.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(N):
	return 1.0 / N

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 48.0%

   \[\log \left(N + 1\right) - \log N \]
2. Add Preprocessing

3. Taylor expanded in N around inf 57.6%

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

4. Final simplification 57.6%

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

Reproduce

herbie shell --seed 2024019 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1.0)) (log N)))