2log (problem 3.3.6)

Percentage Accurate: 54.2% → 100.0%

Time: 5.5s

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: 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 61.2%

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

   1. +-commutative 61.2%

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

   2. log1p-def 61.2%

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

3. Simplified 61.2%

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

   1. log1p-udef 61.2%

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

   2. diff-log 61.4%

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

   3. +-commutative 61.4%

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

5. Applied egg-rr 61.4%

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

   1. log-div 61.2%

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

   2. +-commutative 61.2%

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

   3. log1p-udef 61.2%

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

   4. sub-neg 61.2%

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

7. Applied egg-rr 61.2%

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

   1. +-commutative 61.2%

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

   2. log-rec 61.2%

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

   3. log1p-def 61.2%

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

   4. +-commutative 61.2%

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

   5. log-prod 61.3%

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

   6. distribute-rgt-in 61.2%

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

   7. rgt-mult-inverse 61.4%

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

   8. *-lft-identity 61.4%

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

   9. log1p-def 100.0%

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

9. Simplified 100.0%

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

10. Final simplification 100.0%

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

Alternative 2: 98.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;N \leq 0.68:\\ \;\;\;\;-\log N\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{-0.5}{N}}{N}\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= N 0.68) (- (log N)) (/ (+ 1.0 (/ -0.5 N)) N)))

C

double code(double N) {
	double tmp;
	if (N <= 0.68) {
		tmp = -log(N);
	} else {
		tmp = (1.0 + (-0.5 / N)) / N;
	}
	return tmp;
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= 0.68d0) then
        tmp = -log(n)
    else
        tmp = (1.0d0 + ((-0.5d0) / n)) / n
    end if
    code = tmp
end function

Java

public static double code(double N) {
	double tmp;
	if (N <= 0.68) {
		tmp = -Math.log(N);
	} else {
		tmp = (1.0 + (-0.5 / N)) / N;
	}
	return tmp;
}

Python

def code(N):
	tmp = 0
	if N <= 0.68:
		tmp = -math.log(N)
	else:
		tmp = (1.0 + (-0.5 / N)) / N
	return tmp

Julia

function code(N)
	tmp = 0.0
	if (N <= 0.68)
		tmp = Float64(-log(N));
	else
		tmp = Float64(Float64(1.0 + Float64(-0.5 / N)) / N);
	end
	return tmp
end

MATLAB

function tmp_2 = code(N)
	tmp = 0.0;
	if (N <= 0.68)
		tmp = -log(N);
	else
		tmp = (1.0 + (-0.5 / N)) / N;
	end
	tmp_2 = tmp;
end

Wolfram

code[N_] := If[LessEqual[N, 0.68], (-N[Log[N], $MachinePrecision]), N[(N[(1.0 + N[(-0.5 / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if N < 0.680000000000000049

   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 98.0%

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

      1. neg-mul-1 98.0%

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

   6. Simplified 98.0%

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

   if 0.680000000000000049 < N

   1. Initial program 8.1%

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

      1. +-commutative 8.1%

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

      2. log1p-def 8.1%

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

   3. Simplified 8.1%

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

      1. expm1-log1p-u 8.1%

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

   5. Applied egg-rr 8.1%

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

   6. Taylor expanded in N around inf 99.2%

      \[\leadsto \mathsf{expm1}\left(\color{blue}{\frac{1}{N} - \frac{1}{{N}^{2}}}\right) \]
   7. Step-by-step derivation

      1. unpow2 99.2%

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

   8. Simplified 99.2%

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

   9. Taylor expanded in N around inf 99.2%

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

   11. Simplified 99.2%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;N \leq 0.68:\\ \;\;\;\;-\log N\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{-0.5}{N}}{N}\\ \end{array} \]

Alternative 3: 51.6% 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 61.2%

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

   1. +-commutative 61.2%

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

   2. log1p-def 61.2%

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

3. Simplified 61.2%

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

4. Taylor expanded in N around inf 44.5%

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

5. Final simplification 44.5%

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

Reproduce

herbie shell --seed 2023201 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1.0)) (log N)))