2log (problem 3.3.6)

Percentage Accurate: 55.0% → 100.0%

Time: 9.0s

Alternatives: 4

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: 55.0% 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 61.1%

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

   1. +-commutative 61.1%

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

   2. log1p-def 61.1%

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

3. Simplified 61.1%

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

   1. add-log-exp 61.1%

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

   2. log1p-expm1-u 7.7%

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

   3. log1p-udef 7.7%

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

   4. diff-log 7.8%

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

   5. log1p-udef 7.8%

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

   6. rem-exp-log 7.1%

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

   7. +-commutative 7.1%

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

   8. add-exp-log 7.1%

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

   9. log1p-udef 7.1%

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

   10. log1p-expm1-u 60.4%

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

   11. add-exp-log 61.4%

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

6. Applied egg-rr 61.4%

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

   1. add-cube-cbrt 61.2%

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

   2. unpow2 61.2%

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

   3. log-prod 60.9%

      \[\leadsto \color{blue}{\log \left({\left(\sqrt[3]{\frac{N + 1}{N}}\right)}^{2}\right) + \log \left(\sqrt[3]{\frac{N + 1}{N}}\right)} \]

   4. pow-to-exp 60.9%

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

   5. rem-log-exp 60.9%

      \[\leadsto \color{blue}{\log \left(\sqrt[3]{\frac{N + 1}{N}}\right) \cdot 2} + \log \left(\sqrt[3]{\frac{N + 1}{N}}\right) \]

   6. pow1/3 61.0%

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

   7. log-pow 61.2%

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

   8. log-div 61.0%

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

   9. +-commutative 61.0%

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

   10. log1p-udef 61.0%

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

   11. pow1/3 60.9%

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

   12. log-pow 60.9%

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

   13. log-div 60.9%

       \[\leadsto \left(0.3333333333333333 \cdot \left(\mathsf{log1p}\left(N\right) - \log N\right)\right) \cdot 2 + 0.3333333333333333 \cdot \color{blue}{\left(\log \left(N + 1\right) - \log N\right)} \]

   14. +-commutative 60.9%

       \[\leadsto \left(0.3333333333333333 \cdot \left(\mathsf{log1p}\left(N\right) - \log N\right)\right) \cdot 2 + 0.3333333333333333 \cdot \left(\log \color{blue}{\left(1 + N\right)} - \log N\right) \]

   15. log1p-udef 60.8%

       \[\leadsto \left(0.3333333333333333 \cdot \left(\mathsf{log1p}\left(N\right) - \log N\right)\right) \cdot 2 + 0.3333333333333333 \cdot \left(\color{blue}{\mathsf{log1p}\left(N\right)} - \log N\right) \]

8. Applied egg-rr 60.8%

   \[\leadsto \color{blue}{\left(0.3333333333333333 \cdot \left(\mathsf{log1p}\left(N\right) - \log N\right)\right) \cdot 2 + 0.3333333333333333 \cdot \left(\mathsf{log1p}\left(N\right) - \log N\right)} \]
9. Step-by-step derivation

   1. *-commutative 60.8%

      \[\leadsto \color{blue}{2 \cdot \left(0.3333333333333333 \cdot \left(\mathsf{log1p}\left(N\right) - \log N\right)\right)} + 0.3333333333333333 \cdot \left(\mathsf{log1p}\left(N\right) - \log N\right) \]

   2. associate-*r* 60.8%

      \[\leadsto \color{blue}{\left(2 \cdot 0.3333333333333333\right) \cdot \left(\mathsf{log1p}\left(N\right) - \log N\right)} + 0.3333333333333333 \cdot \left(\mathsf{log1p}\left(N\right) - \log N\right) \]

   3. metadata-eval 60.8%

      \[\leadsto \color{blue}{0.6666666666666666} \cdot \left(\mathsf{log1p}\left(N\right) - \log N\right) + 0.3333333333333333 \cdot \left(\mathsf{log1p}\left(N\right) - \log N\right) \]

   4. distribute-rgt-out 61.1%

      \[\leadsto \color{blue}{\left(\mathsf{log1p}\left(N\right) - \log N\right) \cdot \left(0.6666666666666666 + 0.3333333333333333\right)} \]

   5. metadata-eval 61.1%

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

   6. *-rgt-identity 61.1%

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

   7. log1p-def 61.1%

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

   8. +-commutative 61.1%

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

   9. log-div 61.4%

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

   10. *-lft-identity 61.4%

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

   11. associate-*r/ 61.4%

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

   12. associate-*l/ 61.2%

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

   13. distribute-lft-in 61.2%

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

   14. lft-mult-inverse 61.4%

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

   15. *-rgt-identity 61.4%

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

   16. log1p-def 100.0%

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

10. Simplified 100.0%

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

11. Final simplification 100.0%

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

Alternative 2: 98.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;N \leq 0.54:\\ \;\;\;\;-\log N\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N}\\ \end{array} \end{array} \]

FPCore

(FPCore (N) :precision binary64 (if (<= N 0.54) (- (log N)) (/ 1.0 N)))

C

double code(double N) {
	double tmp;
	if (N <= 0.54) {
		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 <= 0.54d0) 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 <= 0.54) {
		tmp = -Math.log(N);
	} else {
		tmp = 1.0 / N;
	}
	return tmp;
}

Python

def code(N):
	tmp = 0
	if N <= 0.54:
		tmp = -math.log(N)
	else:
		tmp = 1.0 / N
	return tmp

Julia

function code(N)
	tmp = 0.0
	if (N <= 0.54)
		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 <= 0.54)
		tmp = -log(N);
	else
		tmp = 1.0 / N;
	end
	tmp_2 = tmp;
end

Wolfram

code[N_] := If[LessEqual[N, 0.54], (-N[Log[N], $MachinePrecision]), N[(1.0 / N), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if N < 0.54000000000000004

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. log1p-def 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in N around 0 98.4%

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

      1. neg-mul-1 98.4%

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

   7. Simplified 98.4%

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

   if 0.54000000000000004 < N

   1. Initial program 9.5%

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

      1. +-commutative 9.5%

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

      2. log1p-def 9.5%

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

   3. Simplified 9.5%

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

   5. Taylor expanded in N around inf 97.0%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;N \leq 0.54:\\ \;\;\;\;-\log N\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 50.7% 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 61.1%

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

   1. +-commutative 61.1%

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

   2. log1p-def 61.1%

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

3. Simplified 61.1%

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

5. Taylor expanded in N around inf 44.8%

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

6. Final simplification 44.8%

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

Alternative 4: 4.6% accurate, 205.0× speedup

Math

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

FPCore

(FPCore (N) :precision binary64 N)

C

double code(double N) {
	return N;
}

Fortran

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

Java

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

Python

def code(N):
	return N

Julia

function code(N)
	return N
end

MATLAB

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

Wolfram

code[N_] := N

Derivation

1. Initial program 61.1%

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

   1. +-commutative 61.1%

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

   2. log1p-def 61.1%

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

3. Simplified 61.1%

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

5. Taylor expanded in N around 0 58.2%

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

   1. neg-mul-1 58.2%

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

   2. unsub-neg 58.2%

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

7. Simplified 58.2%

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

8. Taylor expanded in N around inf 4.7%

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

9. Final simplification 4.7%

   \[\leadsto N \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024017 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1.0)) (log N)))