2log (problem 3.3.6)

Percentage Accurate: 54.3% → 99.9%

Time: 5.2s

Alternatives: 7

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: 54.3% 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: 99.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 4 \cdot 10^{-7}:\\ \;\;\;\;\frac{0.3333333333333333}{{N}^{3}} + \frac{1 + \frac{-0.5}{N}}{N}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= (- (log (+ N 1.0)) (log N)) 4e-7)
   (+ (/ 0.3333333333333333 (pow N 3.0)) (/ (+ 1.0 (/ -0.5 N)) N))
   (log (/ (+ N 1.0) N))))

C

double code(double N) {
	double tmp;
	if ((log((N + 1.0)) - log(N)) <= 4e-7) {
		tmp = (0.3333333333333333 / pow(N, 3.0)) + ((1.0 + (-0.5 / N)) / N);
	} else {
		tmp = log(((N + 1.0) / N));
	}
	return tmp;
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((log((n + 1.0d0)) - log(n)) <= 4d-7) then
        tmp = (0.3333333333333333d0 / (n ** 3.0d0)) + ((1.0d0 + ((-0.5d0) / n)) / n)
    else
        tmp = log(((n + 1.0d0) / n))
    end if
    code = tmp
end function

Java

public static double code(double N) {
	double tmp;
	if ((Math.log((N + 1.0)) - Math.log(N)) <= 4e-7) {
		tmp = (0.3333333333333333 / Math.pow(N, 3.0)) + ((1.0 + (-0.5 / N)) / N);
	} else {
		tmp = Math.log(((N + 1.0) / N));
	}
	return tmp;
}

Python

def code(N):
	tmp = 0
	if (math.log((N + 1.0)) - math.log(N)) <= 4e-7:
		tmp = (0.3333333333333333 / math.pow(N, 3.0)) + ((1.0 + (-0.5 / N)) / N)
	else:
		tmp = math.log(((N + 1.0) / N))
	return tmp

Julia

function code(N)
	tmp = 0.0
	if (Float64(log(Float64(N + 1.0)) - log(N)) <= 4e-7)
		tmp = Float64(Float64(0.3333333333333333 / (N ^ 3.0)) + Float64(Float64(1.0 + Float64(-0.5 / N)) / N));
	else
		tmp = log(Float64(Float64(N + 1.0) / N));
	end
	return tmp
end

MATLAB

function tmp_2 = code(N)
	tmp = 0.0;
	if ((log((N + 1.0)) - log(N)) <= 4e-7)
		tmp = (0.3333333333333333 / (N ^ 3.0)) + ((1.0 + (-0.5 / N)) / N);
	else
		tmp = log(((N + 1.0) / N));
	end
	tmp_2 = tmp;
end

Wolfram

code[N_] := If[LessEqual[N[(N[Log[N[(N + 1.0), $MachinePrecision]], $MachinePrecision] - N[Log[N], $MachinePrecision]), $MachinePrecision], 4e-7], N[(N[(0.3333333333333333 / N[Power[N, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 + N[(-0.5 / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision]), $MachinePrecision], N[Log[N[(N[(N + 1.0), $MachinePrecision] / N), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (log.f64 (+.f64 N 1)) (log.f64 N)) < 3.9999999999999998e-7

   1. Initial program 6.7%

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

      1. +-commutative 6.7%

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

      2. log1p-def 6.7%

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

   3. Simplified 6.7%

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

      1. log1p-udef 6.7%

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

      2. diff-log 6.8%

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

      3. +-commutative 6.8%

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

   5. Applied egg-rr 6.8%

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

   6. Taylor expanded in N around inf 100.0%

      \[\leadsto \color{blue}{\left(\frac{1}{N} + 0.3333333333333333 \cdot \frac{1}{{N}^{3}}\right) - 0.5 \cdot \frac{1}{{N}^{2}}} \]
   7. Step-by-step derivation

      1. +-commutative 100.0%

         \[\leadsto \color{blue}{\left(0.3333333333333333 \cdot \frac{1}{{N}^{3}} + \frac{1}{N}\right)} - 0.5 \cdot \frac{1}{{N}^{2}} \]

      2. associate--l+ 100.0%

         \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{1}{{N}^{3}} + \left(\frac{1}{N} - 0.5 \cdot \frac{1}{{N}^{2}}\right)} \]

      3. associate-*r/ 100.0%

         \[\leadsto \color{blue}{\frac{0.3333333333333333 \cdot 1}{{N}^{3}}} + \left(\frac{1}{N} - 0.5 \cdot \frac{1}{{N}^{2}}\right) \]

      4. metadata-eval 100.0%

         \[\leadsto \frac{\color{blue}{0.3333333333333333}}{{N}^{3}} + \left(\frac{1}{N} - 0.5 \cdot \frac{1}{{N}^{2}}\right) \]

      5. sub-neg 100.0%

         \[\leadsto \frac{0.3333333333333333}{{N}^{3}} + \color{blue}{\left(\frac{1}{N} + \left(-0.5 \cdot \frac{1}{{N}^{2}}\right)\right)} \]

      6. *-lft-identity 100.0%

         \[\leadsto \frac{0.3333333333333333}{{N}^{3}} + \left(\color{blue}{1 \cdot \frac{1}{N}} + \left(-0.5 \cdot \frac{1}{{N}^{2}}\right)\right) \]

      7. distribute-lft-neg-in 100.0%

         \[\leadsto \frac{0.3333333333333333}{{N}^{3}} + \left(1 \cdot \frac{1}{N} + \color{blue}{\left(-0.5\right) \cdot \frac{1}{{N}^{2}}}\right) \]

      8. metadata-eval 100.0%

         \[\leadsto \frac{0.3333333333333333}{{N}^{3}} + \left(1 \cdot \frac{1}{N} + \color{blue}{-0.5} \cdot \frac{1}{{N}^{2}}\right) \]

      9. unpow2 100.0%

         \[\leadsto \frac{0.3333333333333333}{{N}^{3}} + \left(1 \cdot \frac{1}{N} + -0.5 \cdot \frac{1}{\color{blue}{N \cdot N}}\right) \]

      10. associate-*r/ 100.0%

          \[\leadsto \frac{0.3333333333333333}{{N}^{3}} + \left(1 \cdot \frac{1}{N} + \color{blue}{\frac{-0.5 \cdot 1}{N \cdot N}}\right) \]

      11. associate-*l/ 100.0%

          \[\leadsto \frac{0.3333333333333333}{{N}^{3}} + \left(1 \cdot \frac{1}{N} + \color{blue}{\frac{-0.5}{N \cdot N} \cdot 1}\right) \]

      12. associate-/l/ 100.0%

          \[\leadsto \frac{0.3333333333333333}{{N}^{3}} + \left(1 \cdot \frac{1}{N} + \color{blue}{\frac{\frac{-0.5}{N}}{N}} \cdot 1\right) \]

      13. associate-*l/ 100.0%

          \[\leadsto \frac{0.3333333333333333}{{N}^{3}} + \left(1 \cdot \frac{1}{N} + \color{blue}{\frac{\frac{-0.5}{N} \cdot 1}{N}}\right) \]

      14. associate-*r/ 100.0%

          \[\leadsto \frac{0.3333333333333333}{{N}^{3}} + \left(1 \cdot \frac{1}{N} + \color{blue}{\frac{-0.5}{N} \cdot \frac{1}{N}}\right) \]

      15. distribute-rgt-in 100.0%

          \[\leadsto \frac{0.3333333333333333}{{N}^{3}} + \color{blue}{\frac{1}{N} \cdot \left(1 + \frac{-0.5}{N}\right)} \]

      16. associate-*l/ 100.0%

          \[\leadsto \frac{0.3333333333333333}{{N}^{3}} + \color{blue}{\frac{1 \cdot \left(1 + \frac{-0.5}{N}\right)}{N}} \]

      17. *-lft-identity 100.0%

          \[\leadsto \frac{0.3333333333333333}{{N}^{3}} + \frac{\color{blue}{1 + \frac{-0.5}{N}}}{N} \]

   8. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{0.3333333333333333}{{N}^{3}} + \frac{1 + \frac{-0.5}{N}}{N}} \]

   if 3.9999999999999998e-7 < (-.f64 (log.f64 (+.f64 N 1)) (log.f64 N))

   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. Step-by-step derivation

      1. log1p-udef 100.0%

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

      2. diff-log 100.0%

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

      3. +-commutative 100.0%

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

   5. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\log \left(\frac{N + 1}{N}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 4 \cdot 10^{-7}:\\ \;\;\;\;\frac{0.3333333333333333}{{N}^{3}} + \frac{1 + \frac{-0.5}{N}}{N}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \end{array} \]

Alternative 2: 99.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;N \leq 106000:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N + 0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= N 106000.0) (log (/ (+ N 1.0) N)) (/ 1.0 (+ N 0.5))))

C

double code(double N) {
	double tmp;
	if (N <= 106000.0) {
		tmp = log(((N + 1.0) / 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 <= 106000.0d0) then
        tmp = log(((n + 1.0d0) / 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 <= 106000.0) {
		tmp = Math.log(((N + 1.0) / N));
	} else {
		tmp = 1.0 / (N + 0.5);
	}
	return tmp;
}

Python

def code(N):
	tmp = 0
	if N <= 106000.0:
		tmp = math.log(((N + 1.0) / N))
	else:
		tmp = 1.0 / (N + 0.5)
	return tmp

Julia

function code(N)
	tmp = 0.0
	if (N <= 106000.0)
		tmp = log(Float64(Float64(N + 1.0) / 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 <= 106000.0)
		tmp = log(((N + 1.0) / N));
	else
		tmp = 1.0 / (N + 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[N_] := If[LessEqual[N, 106000.0], N[Log[N[(N[(N + 1.0), $MachinePrecision] / N), $MachinePrecision]], $MachinePrecision], N[(1.0 / N[(N + 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if N < 106000

   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. Step-by-step derivation

      1. log1p-udef 100.0%

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

      2. diff-log 100.0%

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

      3. +-commutative 100.0%

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

   5. Applied egg-rr 100.0%

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

   if 106000 < N

   1. Initial program 6.7%

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

      1. +-commutative 6.7%

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

      2. log1p-def 6.7%

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

   3. Simplified 6.7%

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

   4. Taylor expanded in N around inf 99.9%

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

      1. associate-*r/ 99.9%

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

      2. metadata-eval 99.9%

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

      3. unpow2 99.9%

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

      4. associate-/r* 99.9%

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

   6. Simplified 99.9%

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

      1. frac-sub 58.9%

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

      2. clear-num 59.0%

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

      3. *-un-lft-identity 59.0%

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

      4. clear-num 59.0%

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

      5. un-div-inv 59.0%

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

      6. div-inv 59.0%

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

      7. metadata-eval 59.0%

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

   8. Applied egg-rr 59.0%

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

   9. Taylor expanded in N around inf 99.9%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;N \leq 106000:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N + 0.5}\\ \end{array} \]

Alternative 3: 99.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;N \leq 0.6:\\ \;\;\;\;N - \log N\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N + 0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= N 0.6) (- N (log N)) (/ 1.0 (+ N 0.5))))

C

double code(double N) {
	double tmp;
	if (N <= 0.6) {
		tmp = N - 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 <= 0.6d0) then
        tmp = n - 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 <= 0.6) {
		tmp = N - Math.log(N);
	} else {
		tmp = 1.0 / (N + 0.5);
	}
	return tmp;
}

Python

def code(N):
	tmp = 0
	if N <= 0.6:
		tmp = N - math.log(N)
	else:
		tmp = 1.0 / (N + 0.5)
	return tmp

Julia

function code(N)
	tmp = 0.0
	if (N <= 0.6)
		tmp = Float64(N - 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 <= 0.6)
		tmp = N - log(N);
	else
		tmp = 1.0 / (N + 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[N_] := If[LessEqual[N, 0.6], N[(N - N[Log[N], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N + 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if N < 0.599999999999999978

   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 98.8%

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

      1. neg-mul-1 98.8%

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

      2. unsub-neg 98.8%

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

   6. Simplified 98.8%

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

   if 0.599999999999999978 < N

   1. Initial program 6.7%

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

      1. +-commutative 6.7%

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

      2. log1p-def 6.7%

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

   3. Simplified 6.7%

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

   4. Taylor expanded in N around inf 99.9%

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

      1. associate-*r/ 99.9%

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

      2. metadata-eval 99.9%

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

      3. unpow2 99.9%

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

      4. associate-/r* 99.9%

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

   6. Simplified 99.9%

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

      1. frac-sub 58.9%

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

      2. clear-num 59.0%

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

      3. *-un-lft-identity 59.0%

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

      4. clear-num 59.0%

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

      5. un-div-inv 59.0%

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

      6. div-inv 59.0%

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

      7. metadata-eval 59.0%

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

   8. Applied egg-rr 59.0%

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

   9. Taylor expanded in N around inf 99.9%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;N \leq 0.6:\\ \;\;\;\;N - \log N\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N + 0.5}\\ \end{array} \]

Alternative 4: 98.7% 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 99.1%

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

      1. neg-mul-1 99.1%

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

   6. Simplified 99.1%

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

   if 0.27000000000000002 < N

   1. Initial program 7.5%

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

      1. +-commutative 7.5%

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

      2. log1p-def 7.5%

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

   3. Simplified 7.5%

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

   4. Taylor expanded in N around inf 99.1%

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

      1. associate-*r/ 99.1%

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

      2. metadata-eval 99.1%

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

      3. unpow2 99.1%

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

      4. associate-/r* 99.1%

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

   6. Simplified 99.1%

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

      1. frac-sub 58.4%

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

      2. clear-num 58.5%

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

      3. *-un-lft-identity 58.5%

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

      4. clear-num 58.5%

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

      5. un-div-inv 58.5%

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

      6. div-inv 58.5%

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

      7. metadata-eval 58.5%

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

   8. Applied egg-rr 58.5%

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

   9. Taylor expanded in N around inf 99.3%

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

4. Final simplification 99.2%

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

Alternative 5: 55.9% 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. frac-sub 0.8%

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

      2. clear-num 0.8%

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

      3. *-un-lft-identity 0.8%

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

      4. clear-num 0.8%

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

      5. un-div-inv 0.8%

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

      6. div-inv 0.8%

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

      7. metadata-eval 0.8%

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

   8. Applied egg-rr 0.8%

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

   9. Taylor expanded in N around inf 14.4%

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

   10. Taylor expanded in N around 0 14.4%

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

   if 0.5 < N

   1. Initial program 6.7%

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

      1. +-commutative 6.7%

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

      2. log1p-def 6.7%

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

   3. Simplified 6.7%

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

   4. Taylor expanded in N around inf 99.1%

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

4. Final simplification 53.4%

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

Alternative 6: 56.5% 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 57.0%

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

   1. +-commutative 57.0%

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

   2. log1p-def 57.0%

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

3. Simplified 57.0%

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

4. Taylor expanded in N around inf 46.5%

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

   1. associate-*r/ 46.5%

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

   2. metadata-eval 46.5%

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

   3. unpow2 46.5%

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

   4. associate-/r* 46.5%

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

6. Simplified 46.5%

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

   1. frac-sub 27.6%

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

   2. clear-num 27.6%

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

   3. *-un-lft-identity 27.6%

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

   4. clear-num 27.6%

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

   5. un-div-inv 27.6%

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

   6. div-inv 27.6%

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

   7. metadata-eval 27.6%

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

8. Applied egg-rr 27.6%

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

9. Taylor expanded in N around inf 53.8%

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

10. Final simplification 53.8%

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

Alternative 7: 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 57.0%

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

   1. +-commutative 57.0%

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

   2. log1p-def 57.0%

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

3. Simplified 57.0%

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

4. Taylor expanded in N around inf 46.5%

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

   1. associate-*r/ 46.5%

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

   2. metadata-eval 46.5%

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

   3. unpow2 46.5%

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

   4. associate-/r* 46.5%

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

6. Simplified 46.5%

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

   1. frac-sub 27.6%

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

   2. clear-num 27.6%

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

   3. *-un-lft-identity 27.6%

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

   4. clear-num 27.6%

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

   5. un-div-inv 27.6%

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

   6. div-inv 27.6%

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

   7. metadata-eval 27.6%

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

8. Applied egg-rr 27.6%

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

9. Taylor expanded in N around inf 53.8%

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

10. Taylor expanded in N around 0 10.2%

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

11. Final simplification 10.2%

    \[\leadsto 2 \]

Reproduce

herbie shell --seed 2023195 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1.0)) (log N)))