2log (problem 3.3.6)

Percentage Accurate: 54.2% → 100.0%

Time: 5.2s

Alternatives: 5

Speedup: 41.0×

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: 54.2% 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 55.1%

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

   1. +-commutative 55.1%

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

   2. log1p-def 55.1%

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

3. Simplified 55.1%

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

   1. log1p-udef 55.1%

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

   2. diff-log 55.2%

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

   3. +-commutative 55.2%

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

5. Applied egg-rr 55.2%

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

6. Taylor expanded in N around 0 55.2%

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

   1. *-un-lft-identity 55.2%

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

   2. log-prod 55.2%

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

   3. metadata-eval 55.2%

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

   4. log1p-def 100.0%

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

8. Applied egg-rr 100.0%

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

   1. +-lft-identity 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%

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

Alternative 2: 98.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;N \leq 0.27:\\ \;\;\;\;-\log N\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N + 0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (N) :precision binary64 (if (<= N 0.27) (- (log N)) (/ 1.0 (+ N 0.5))))

C

double code(double N) {
	double tmp;
	if (N <= 0.27) {
		tmp = -log(N);
	} else {
		tmp = 1.0 / (N + 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= 0.27d0) then
        tmp = -log(n)
    else
        tmp = 1.0d0 / (n + 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double N) {
	double tmp;
	if (N <= 0.27) {
		tmp = -Math.log(N);
	} else {
		tmp = 1.0 / (N + 0.5);
	}
	return tmp;
}

Python

def code(N):
	tmp = 0
	if N <= 0.27:
		tmp = -math.log(N)
	else:
		tmp = 1.0 / (N + 0.5)
	return tmp

Julia

function code(N)
	tmp = 0.0
	if (N <= 0.27)
		tmp = Float64(-log(N));
	else
		tmp = Float64(1.0 / Float64(N + 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(N)
	tmp = 0.0;
	if (N <= 0.27)
		tmp = -log(N);
	else
		tmp = 1.0 / (N + 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[N_] := If[LessEqual[N, 0.27], (-N[Log[N], $MachinePrecision]), N[(1.0 / N[(N + 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if N < 0.27000000000000002

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. log1p-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in N around 0 97.0%

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

      1. neg-mul-1 97.0%

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

   6. Simplified 97.0%

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

   if 0.27000000000000002 < N

   1. Initial program 8.0%

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

      1. +-commutative 8.0%

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

      2. log1p-def 8.0%

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

   3. Simplified 8.0%

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

   4. Taylor expanded in N around inf 99.4%

      \[\leadsto \color{blue}{\frac{1}{N} - 0.5 \cdot \frac{1}{{N}^{2}}} \]
   5. Step-by-step derivation

      1. associate-*r/ 99.4%

         \[\leadsto \frac{1}{N} - \color{blue}{\frac{0.5 \cdot 1}{{N}^{2}}} \]

      2. metadata-eval 99.4%

         \[\leadsto \frac{1}{N} - \frac{\color{blue}{0.5}}{{N}^{2}} \]

      3. unpow2 99.4%

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

      4. associate-/r* 99.4%

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

   6. Simplified 99.4%

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

      1. sub-div 99.4%

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

      2. clear-num 99.4%

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

      3. sub-neg 99.4%

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

      4. distribute-neg-frac 99.4%

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

      5. metadata-eval 99.4%

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

   8. Applied egg-rr 99.4%

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

   9. Taylor expanded in N around inf 99.5%

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

       1. +-commutative 99.5%

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

   11. Simplified 99.5%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;N \leq 0.27:\\ \;\;\;\;-\log N\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N + 0.5}\\ \end{array} \]

Alternative 3: 56.1% accurate, 40.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;N \leq 0.5:\\ \;\;\;\;2\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N}\\ \end{array} \end{array} \]

FPCore

(FPCore (N) :precision binary64 (if (<= N 0.5) 2.0 (/ 1.0 N)))

C

double code(double N) {
	double tmp;
	if (N <= 0.5) {
		tmp = 2.0;
	} else {
		tmp = 1.0 / N;
	}
	return tmp;
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= 0.5d0) then
        tmp = 2.0d0
    else
        tmp = 1.0d0 / n
    end if
    code = tmp
end function

Java

public static double code(double N) {
	double tmp;
	if (N <= 0.5) {
		tmp = 2.0;
	} else {
		tmp = 1.0 / N;
	}
	return tmp;
}

Python

def code(N):
	tmp = 0
	if N <= 0.5:
		tmp = 2.0
	else:
		tmp = 1.0 / N
	return tmp

Julia

function code(N)
	tmp = 0.0
	if (N <= 0.5)
		tmp = 2.0;
	else
		tmp = Float64(1.0 / N);
	end
	return tmp
end

MATLAB

function tmp_2 = code(N)
	tmp = 0.0;
	if (N <= 0.5)
		tmp = 2.0;
	else
		tmp = 1.0 / N;
	end
	tmp_2 = tmp;
end

Wolfram

code[N_] := If[LessEqual[N, 0.5], 2.0, N[(1.0 / N), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if N < 0.5

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. log1p-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in N around inf 0.8%

      \[\leadsto \color{blue}{\frac{1}{N} - 0.5 \cdot \frac{1}{{N}^{2}}} \]
   5. Step-by-step derivation

      1. associate-*r/ 0.8%

         \[\leadsto \frac{1}{N} - \color{blue}{\frac{0.5 \cdot 1}{{N}^{2}}} \]

      2. metadata-eval 0.8%

         \[\leadsto \frac{1}{N} - \frac{\color{blue}{0.5}}{{N}^{2}} \]

      3. unpow2 0.8%

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

      4. associate-/r* 0.8%

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

   6. Simplified 0.8%

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

      1. sub-div 0.8%

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

      2. clear-num 0.8%

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

      3. sub-neg 0.8%

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

      4. distribute-neg-frac 0.8%

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

      5. metadata-eval 0.8%

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

   8. Applied egg-rr 0.8%

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

   9. Taylor expanded in N around inf 14.5%

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

       1. +-commutative 14.5%

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

   11. Simplified 14.5%

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

   12. Taylor expanded in N around 0 14.6%

       \[\leadsto \color{blue}{2} \]

   if 0.5 < N

   1. Initial program 8.0%

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

      1. +-commutative 8.0%

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

      2. log1p-def 8.0%

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

   3. Simplified 8.0%

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

   4. Taylor expanded in N around inf 98.3%

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

4. Final simplification 55.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;N \leq 0.5:\\ \;\;\;\;2\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N}\\ \end{array} \]

Alternative 4: 56.6% accurate, 41.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.1%

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

   1. +-commutative 55.1%

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

   2. log1p-def 55.1%

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

3. Simplified 55.1%

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

4. Taylor expanded in N around inf 49.0%

   \[\leadsto \color{blue}{\frac{1}{N} - 0.5 \cdot \frac{1}{{N}^{2}}} \]
5. Step-by-step derivation

   1. associate-*r/ 49.0%

      \[\leadsto \frac{1}{N} - \color{blue}{\frac{0.5 \cdot 1}{{N}^{2}}} \]

   2. metadata-eval 49.0%

      \[\leadsto \frac{1}{N} - \frac{\color{blue}{0.5}}{{N}^{2}} \]

   3. unpow2 49.0%

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

   4. associate-/r* 49.0%

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

6. Simplified 49.0%

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

   1. sub-div 49.0%

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

   2. clear-num 49.0%

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

   3. sub-neg 49.0%

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

   4. distribute-neg-frac 49.0%

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

   5. metadata-eval 49.0%

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

8. Applied egg-rr 49.0%

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

9. Taylor expanded in N around inf 56.0%

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

    1. +-commutative 56.0%

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

11. Simplified 56.0%

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

12. Final simplification 56.0%

    \[\leadsto \frac{1}{N + 0.5} \]

Alternative 5: 9.9% accurate, 205.0× speedup

Math

\[\begin{array}{l} \\ 2 \end{array} \]

FPCore

(FPCore (N) :precision binary64 2.0)

C

double code(double N) {
	return 2.0;
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    code = 2.0d0
end function

Java

public static double code(double N) {
	return 2.0;
}

Python

def code(N):
	return 2.0

Julia

function code(N)
	return 2.0
end

MATLAB

function tmp = code(N)
	tmp = 2.0;
end

Wolfram

code[N_] := 2.0

Derivation

1. Initial program 55.1%

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

   1. +-commutative 55.1%

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

   2. log1p-def 55.1%

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

3. Simplified 55.1%

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

4. Taylor expanded in N around inf 49.0%

   \[\leadsto \color{blue}{\frac{1}{N} - 0.5 \cdot \frac{1}{{N}^{2}}} \]
5. Step-by-step derivation

   1. associate-*r/ 49.0%

      \[\leadsto \frac{1}{N} - \color{blue}{\frac{0.5 \cdot 1}{{N}^{2}}} \]

   2. metadata-eval 49.0%

      \[\leadsto \frac{1}{N} - \frac{\color{blue}{0.5}}{{N}^{2}} \]

   3. unpow2 49.0%

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

   4. associate-/r* 49.0%

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

6. Simplified 49.0%

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

   1. sub-div 49.0%

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

   2. clear-num 49.0%

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

   3. sub-neg 49.0%

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

   4. distribute-neg-frac 49.0%

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

   5. metadata-eval 49.0%

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

8. Applied egg-rr 49.0%

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

9. Taylor expanded in N around inf 56.0%

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

    1. +-commutative 56.0%

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

11. Simplified 56.0%

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

12. Taylor expanded in N around 0 10.0%

    \[\leadsto \color{blue}{2} \]

13. Final simplification 10.0%

    \[\leadsto 2 \]

Reproduce

herbie shell --seed 2023293 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1.0)) (log N)))