2log (problem 3.3.6)

Percentage Accurate: 53.1% → 99.9%

Time: 8.4s

Alternatives: 8

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: 53.1% 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, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 0.0002:\\ \;\;\;\;\left(\frac{1}{N} + \frac{0.3333333333333333}{{N}^{3}}\right) - \left(\frac{0.5}{N \cdot N} + \frac{0.25}{{N}^{4}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{1 - N \cdot N}{N \cdot \left(1 - N\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= (- (log (+ N 1.0)) (log N)) 0.0002)
   (-
    (+ (/ 1.0 N) (/ 0.3333333333333333 (pow N 3.0)))
    (+ (/ 0.5 (* N N)) (/ 0.25 (pow N 4.0))))
   (log (/ (- 1.0 (* N N)) (* N (- 1.0 N))))))

C

double code(double N) {
	double tmp;
	if ((log((N + 1.0)) - log(N)) <= 0.0002) {
		tmp = ((1.0 / N) + (0.3333333333333333 / pow(N, 3.0))) - ((0.5 / (N * N)) + (0.25 / pow(N, 4.0)));
	} else {
		tmp = log(((1.0 - (N * N)) / (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)) <= 0.0002d0) then
        tmp = ((1.0d0 / n) + (0.3333333333333333d0 / (n ** 3.0d0))) - ((0.5d0 / (n * n)) + (0.25d0 / (n ** 4.0d0)))
    else
        tmp = log(((1.0d0 - (n * n)) / (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)) <= 0.0002) {
		tmp = ((1.0 / N) + (0.3333333333333333 / Math.pow(N, 3.0))) - ((0.5 / (N * N)) + (0.25 / Math.pow(N, 4.0)));
	} else {
		tmp = Math.log(((1.0 - (N * N)) / (N * (1.0 - N))));
	}
	return tmp;
}

Python

def code(N):
	tmp = 0
	if (math.log((N + 1.0)) - math.log(N)) <= 0.0002:
		tmp = ((1.0 / N) + (0.3333333333333333 / math.pow(N, 3.0))) - ((0.5 / (N * N)) + (0.25 / math.pow(N, 4.0)))
	else:
		tmp = math.log(((1.0 - (N * N)) / (N * (1.0 - N))))
	return tmp

Julia

function code(N)
	tmp = 0.0
	if (Float64(log(Float64(N + 1.0)) - log(N)) <= 0.0002)
		tmp = Float64(Float64(Float64(1.0 / N) + Float64(0.3333333333333333 / (N ^ 3.0))) - Float64(Float64(0.5 / Float64(N * N)) + Float64(0.25 / (N ^ 4.0))));
	else
		tmp = log(Float64(Float64(1.0 - Float64(N * N)) / Float64(N * Float64(1.0 - N))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(N)
	tmp = 0.0;
	if ((log((N + 1.0)) - log(N)) <= 0.0002)
		tmp = ((1.0 / N) + (0.3333333333333333 / (N ^ 3.0))) - ((0.5 / (N * N)) + (0.25 / (N ^ 4.0)));
	else
		tmp = log(((1.0 - (N * N)) / (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], 0.0002], N[(N[(N[(1.0 / N), $MachinePrecision] + N[(0.3333333333333333 / N[Power[N, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(0.5 / N[(N * N), $MachinePrecision]), $MachinePrecision] + N[(0.25 / N[Power[N, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[N[(N[(1.0 - N[(N * N), $MachinePrecision]), $MachinePrecision] / N[(N * N[(1.0 - N), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (log.f64 (+.f64 N 1)) (log.f64 N)) < 2.0000000000000001e-4

   1. Initial program 6.6%

      \[\log \left(N + 1\right) - \log N \]

   2. Taylor expanded in N around inf 100.0%

      \[\leadsto \color{blue}{\left(0.3333333333333333 \cdot \frac{1}{{N}^{3}} + \frac{1}{N}\right) - \left(0.25 \cdot \frac{1}{{N}^{4}} + 0.5 \cdot \frac{1}{{N}^{2}}\right)} \]
   3. Step-by-step derivation

      1. +-commutative 100.0%

         \[\leadsto \color{blue}{\left(\frac{1}{N} + 0.3333333333333333 \cdot \frac{1}{{N}^{3}}\right)} - \left(0.25 \cdot \frac{1}{{N}^{4}} + 0.5 \cdot \frac{1}{{N}^{2}}\right) \]

      2. associate-*r/ 100.0%

         \[\leadsto \left(\frac{1}{N} + \color{blue}{\frac{0.3333333333333333 \cdot 1}{{N}^{3}}}\right) - \left(0.25 \cdot \frac{1}{{N}^{4}} + 0.5 \cdot \frac{1}{{N}^{2}}\right) \]

      3. metadata-eval 100.0%

         \[\leadsto \left(\frac{1}{N} + \frac{\color{blue}{0.3333333333333333}}{{N}^{3}}\right) - \left(0.25 \cdot \frac{1}{{N}^{4}} + 0.5 \cdot \frac{1}{{N}^{2}}\right) \]

      4. +-commutative 100.0%

         \[\leadsto \left(\frac{1}{N} + \frac{0.3333333333333333}{{N}^{3}}\right) - \color{blue}{\left(0.5 \cdot \frac{1}{{N}^{2}} + 0.25 \cdot \frac{1}{{N}^{4}}\right)} \]

      5. associate-*r/ 100.0%

         \[\leadsto \left(\frac{1}{N} + \frac{0.3333333333333333}{{N}^{3}}\right) - \left(\color{blue}{\frac{0.5 \cdot 1}{{N}^{2}}} + 0.25 \cdot \frac{1}{{N}^{4}}\right) \]

      6. metadata-eval 100.0%

         \[\leadsto \left(\frac{1}{N} + \frac{0.3333333333333333}{{N}^{3}}\right) - \left(\frac{\color{blue}{0.5}}{{N}^{2}} + 0.25 \cdot \frac{1}{{N}^{4}}\right) \]

      7. unpow2 100.0%

         \[\leadsto \left(\frac{1}{N} + \frac{0.3333333333333333}{{N}^{3}}\right) - \left(\frac{0.5}{\color{blue}{N \cdot N}} + 0.25 \cdot \frac{1}{{N}^{4}}\right) \]

      8. associate-*r/ 100.0%

         \[\leadsto \left(\frac{1}{N} + \frac{0.3333333333333333}{{N}^{3}}\right) - \left(\frac{0.5}{N \cdot N} + \color{blue}{\frac{0.25 \cdot 1}{{N}^{4}}}\right) \]

      9. metadata-eval 100.0%

         \[\leadsto \left(\frac{1}{N} + \frac{0.3333333333333333}{{N}^{3}}\right) - \left(\frac{0.5}{N \cdot N} + \frac{\color{blue}{0.25}}{{N}^{4}}\right) \]

   4. Simplified 100.0%

      \[\leadsto \color{blue}{\left(\frac{1}{N} + \frac{0.3333333333333333}{{N}^{3}}\right) - \left(\frac{0.5}{N \cdot N} + \frac{0.25}{{N}^{4}}\right)} \]

   if 2.0000000000000001e-4 < (-.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. diff-log 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. +-commutative 100.0%

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

      2. expm1-log1p-u 100.0%

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

   5. Applied egg-rr 100.0%

      \[\leadsto \log \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1 + N}{N}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. expm1-log1p-u 100.0%

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

      2. div-inv 100.0%

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

      3. flip-+ 100.0%

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

      4. frac-times 100.0%

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

      5. metadata-eval 100.0%

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

   7. Applied egg-rr 100.0%

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

      1. *-rgt-identity 100.0%

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

      2. *-commutative 100.0%

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

   9. Simplified 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 0.0002:\\ \;\;\;\;\left(\frac{1}{N} + \frac{0.3333333333333333}{{N}^{3}}\right) - \left(\frac{0.5}{N \cdot N} + \frac{0.25}{{N}^{4}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{1 - N \cdot N}{N \cdot \left(1 - N\right)}\right)\\ \end{array} \]

Alternative 2: 99.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 10^{-6}:\\ \;\;\;\;\frac{1}{N} + \left(\frac{0.3333333333333333}{{N}^{3}} + \frac{-0.5}{N \cdot N}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= (- (log (+ N 1.0)) (log N)) 1e-6)
   (+ (/ 1.0 N) (+ (/ 0.3333333333333333 (pow N 3.0)) (/ -0.5 (* N N))))
   (log (/ (+ N 1.0) N))))

C

double code(double N) {
	double tmp;
	if ((log((N + 1.0)) - log(N)) <= 1e-6) {
		tmp = (1.0 / N) + ((0.3333333333333333 / pow(N, 3.0)) + (-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)) <= 1d-6) then
        tmp = (1.0d0 / n) + ((0.3333333333333333d0 / (n ** 3.0d0)) + ((-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)) <= 1e-6) {
		tmp = (1.0 / N) + ((0.3333333333333333 / Math.pow(N, 3.0)) + (-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)) <= 1e-6:
		tmp = (1.0 / N) + ((0.3333333333333333 / math.pow(N, 3.0)) + (-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)) <= 1e-6)
		tmp = Float64(Float64(1.0 / N) + Float64(Float64(0.3333333333333333 / (N ^ 3.0)) + Float64(-0.5 / Float64(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)) <= 1e-6)
		tmp = (1.0 / N) + ((0.3333333333333333 / (N ^ 3.0)) + (-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], 1e-6], N[(N[(1.0 / N), $MachinePrecision] + N[(N[(0.3333333333333333 / N[Power[N, 3.0], $MachinePrecision]), $MachinePrecision] + N[(-0.5 / N[(N * N), $MachinePrecision]), $MachinePrecision]), $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)) < 9.99999999999999955e-7

   1. Initial program 6.1%

      \[\log \left(N + 1\right) - \log N \]

   2. Taylor expanded in N around inf 100.0%

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

      1. sub-neg 100.0%

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

      2. +-commutative 100.0%

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

      3. associate-+l+ 100.0%

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

      4. associate-*r/ 100.0%

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

      5. metadata-eval 100.0%

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

      6. associate-*r/ 100.0%

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

      7. metadata-eval 100.0%

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

      8. distribute-neg-frac 100.0%

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

      9. metadata-eval 100.0%

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

      10. unpow2 100.0%

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

   4. Simplified 100.0%

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

   if 9.99999999999999955e-7 < (-.f64 (log.f64 (+.f64 N 1)) (log.f64 N))

   1. Initial program 99.8%

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

      1. diff-log 99.9%

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

   3. Applied egg-rr 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 10^{-6}:\\ \;\;\;\;\frac{1}{N} + \left(\frac{0.3333333333333333}{{N}^{3}} + \frac{-0.5}{N \cdot N}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \end{array} \]

Alternative 3: 99.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;N \leq 9000:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{0.5}{N}}{N} + 0.3333333333333333 \cdot {N}^{-3}\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= N 9000.0)
   (log (/ (+ N 1.0) N))
   (+ (/ (- 1.0 (/ 0.5 N)) N) (* 0.3333333333333333 (pow N -3.0)))))

C

double code(double N) {
	double tmp;
	if (N <= 9000.0) {
		tmp = log(((N + 1.0) / N));
	} else {
		tmp = ((1.0 - (0.5 / N)) / N) + (0.3333333333333333 * pow(N, -3.0));
	}
	return tmp;
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= 9000.0d0) then
        tmp = log(((n + 1.0d0) / n))
    else
        tmp = ((1.0d0 - (0.5d0 / n)) / n) + (0.3333333333333333d0 * (n ** (-3.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double N) {
	double tmp;
	if (N <= 9000.0) {
		tmp = Math.log(((N + 1.0) / N));
	} else {
		tmp = ((1.0 - (0.5 / N)) / N) + (0.3333333333333333 * Math.pow(N, -3.0));
	}
	return tmp;
}

Python

def code(N):
	tmp = 0
	if N <= 9000.0:
		tmp = math.log(((N + 1.0) / N))
	else:
		tmp = ((1.0 - (0.5 / N)) / N) + (0.3333333333333333 * math.pow(N, -3.0))
	return tmp

Julia

function code(N)
	tmp = 0.0
	if (N <= 9000.0)
		tmp = log(Float64(Float64(N + 1.0) / N));
	else
		tmp = Float64(Float64(Float64(1.0 - Float64(0.5 / N)) / N) + Float64(0.3333333333333333 * (N ^ -3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(N)
	tmp = 0.0;
	if (N <= 9000.0)
		tmp = log(((N + 1.0) / N));
	else
		tmp = ((1.0 - (0.5 / N)) / N) + (0.3333333333333333 * (N ^ -3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[N_] := If[LessEqual[N, 9000.0], N[Log[N[(N[(N + 1.0), $MachinePrecision] / N), $MachinePrecision]], $MachinePrecision], N[(N[(N[(1.0 - N[(0.5 / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision] + N[(0.3333333333333333 * N[Power[N, -3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if N < 9e3

   1. Initial program 99.8%

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

      1. diff-log 99.9%

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

   3. Applied egg-rr 99.9%

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

   if 9e3 < N

   1. Initial program 6.1%

      \[\log \left(N + 1\right) - \log N \]

   2. Taylor expanded in N around inf 100.0%

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

      1. sub-neg 100.0%

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

      2. +-commutative 100.0%

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

      3. associate-+l+ 100.0%

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

      4. associate-*r/ 100.0%

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

      5. metadata-eval 100.0%

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

      6. associate-*r/ 100.0%

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

      7. metadata-eval 100.0%

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

      8. distribute-neg-frac 100.0%

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

      9. metadata-eval 100.0%

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

      10. unpow2 100.0%

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

   4. Simplified 100.0%

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

      1. expm1-log1p-u 100.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{N} + \left(\frac{0.3333333333333333}{{N}^{3}} + \frac{-0.5}{N \cdot N}\right)\right)\right)} \]

      2. expm1-udef 6.2%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{N} + \left(\frac{0.3333333333333333}{{N}^{3}} + \frac{-0.5}{N \cdot N}\right)\right)} - 1} \]

   6. Applied egg-rr 6.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1 - \frac{0.5}{N}}{N} + 0.3333333333333333 \cdot {N}^{-3}\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 100.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1 - \frac{0.5}{N}}{N} + 0.3333333333333333 \cdot {N}^{-3}\right)\right)} \]

      2. expm1-log1p 100.0%

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

   8. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{1 - \frac{0.5}{N}}{N} + 0.3333333333333333 \cdot {N}^{-3}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;N \leq 9000:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{0.5}{N}}{N} + 0.3333333333333333 \cdot {N}^{-3}\\ \end{array} \]

Alternative 4: 99.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;N \leq 350000:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{0.5}{N}}{N}\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= N 350000.0) (log (/ (+ N 1.0) N)) (/ (- 1.0 (/ 0.5 N)) N)))

C

double code(double N) {
	double tmp;
	if (N <= 350000.0) {
		tmp = log(((N + 1.0) / 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 <= 350000.0d0) then
        tmp = log(((n + 1.0d0) / 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 <= 350000.0) {
		tmp = Math.log(((N + 1.0) / N));
	} else {
		tmp = (1.0 - (0.5 / N)) / N;
	}
	return tmp;
}

Python

def code(N):
	tmp = 0
	if N <= 350000.0:
		tmp = math.log(((N + 1.0) / N))
	else:
		tmp = (1.0 - (0.5 / N)) / N
	return tmp

Julia

function code(N)
	tmp = 0.0
	if (N <= 350000.0)
		tmp = log(Float64(Float64(N + 1.0) / 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 <= 350000.0)
		tmp = log(((N + 1.0) / N));
	else
		tmp = (1.0 - (0.5 / N)) / N;
	end
	tmp_2 = tmp;
end

Wolfram

code[N_] := If[LessEqual[N, 350000.0], N[Log[N[(N[(N + 1.0), $MachinePrecision] / N), $MachinePrecision]], $MachinePrecision], N[(N[(1.0 - N[(0.5 / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if N < 3.5e5

   1. Initial program 99.8%

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

      1. diff-log 99.9%

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

   3. Applied egg-rr 99.9%

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

   if 3.5e5 < N

   1. Initial program 6.1%

      \[\log \left(N + 1\right) - \log N \]

   2. Taylor expanded in N around inf 99.9%

      \[\leadsto \color{blue}{\frac{1}{N} - 0.5 \cdot \frac{1}{{N}^{2}}} \]
   3. 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. Simplified 99.9%

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

      1. *-un-lft-identity 99.9%

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

      2. associate-/r* 99.9%

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

      3. sub-div 99.9%

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

   6. Applied egg-rr 99.9%

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

      1. *-lft-identity 99.9%

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

   8. Simplified 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;N \leq 350000:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{0.5}{N}}{N}\\ \end{array} \]

Alternative 5: 99.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;N \leq 0.92:\\ \;\;\;\;N - \log N\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{0.5}{N}}{N}\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= N 0.92) (- N (log N)) (/ (- 1.0 (/ 0.5 N)) N)))

C

double code(double N) {
	double tmp;
	if (N <= 0.92) {
		tmp = N - 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 <= 0.92d0) then
        tmp = n - 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 <= 0.92) {
		tmp = N - Math.log(N);
	} else {
		tmp = (1.0 - (0.5 / N)) / N;
	}
	return tmp;
}

Python

def code(N):
	tmp = 0
	if N <= 0.92:
		tmp = N - math.log(N)
	else:
		tmp = (1.0 - (0.5 / N)) / N
	return tmp

Julia

function code(N)
	tmp = 0.0
	if (N <= 0.92)
		tmp = Float64(N - 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 <= 0.92)
		tmp = N - log(N);
	else
		tmp = (1.0 - (0.5 / N)) / N;
	end
	tmp_2 = tmp;
end

Wolfram

code[N_] := If[LessEqual[N, 0.92], N[(N - N[Log[N], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(0.5 / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if N < 0.92000000000000004

   1. Initial program 100.0%

      \[\log \left(N + 1\right) - \log N \]

   2. Taylor expanded in N around 0 98.8%

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

      1. neg-mul-1 98.8%

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

      2. unsub-neg 98.8%

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

   4. Simplified 98.8%

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

   if 0.92000000000000004 < N

   1. Initial program 6.6%

      \[\log \left(N + 1\right) - \log N \]

   2. Taylor expanded in N around inf 99.6%

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

      1. associate-*r/ 99.6%

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

      2. metadata-eval 99.6%

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

      3. unpow2 99.6%

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

   4. Simplified 99.6%

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

      1. *-un-lft-identity 99.6%

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

      2. associate-/r* 99.6%

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

      3. sub-div 99.6%

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

   6. Applied egg-rr 99.6%

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

      1. *-lft-identity 99.6%

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

   8. Simplified 99.6%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;N \leq 0.92:\\ \;\;\;\;N - \log N\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{0.5}{N}}{N}\\ \end{array} \]

Alternative 6: 98.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;N \leq 0.66:\\ \;\;\;\;-\log N\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{0.5}{N}}{N}\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= N 0.66) (- (log N)) (/ (- 1.0 (/ 0.5 N)) N)))

C

double code(double N) {
	double tmp;
	if (N <= 0.66) {
		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 <= 0.66d0) 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 <= 0.66) {
		tmp = -Math.log(N);
	} else {
		tmp = (1.0 - (0.5 / N)) / N;
	}
	return tmp;
}

Python

def code(N):
	tmp = 0
	if N <= 0.66:
		tmp = -math.log(N)
	else:
		tmp = (1.0 - (0.5 / N)) / N
	return tmp

Julia

function code(N)
	tmp = 0.0
	if (N <= 0.66)
		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 <= 0.66)
		tmp = -log(N);
	else
		tmp = (1.0 - (0.5 / N)) / N;
	end
	tmp_2 = tmp;
end

Wolfram

code[N_] := If[LessEqual[N, 0.66], (-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 < 0.660000000000000031

   1. Initial program 100.0%

      \[\log \left(N + 1\right) - \log N \]

   2. Taylor expanded in N around 0 97.9%

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

      1. neg-mul-1 97.9%

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

   4. Simplified 97.9%

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

   if 0.660000000000000031 < N

   1. Initial program 6.6%

      \[\log \left(N + 1\right) - \log N \]

   2. Taylor expanded in N around inf 99.6%

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

      1. associate-*r/ 99.6%

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

      2. metadata-eval 99.6%

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

      3. unpow2 99.6%

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

   4. Simplified 99.6%

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

      1. *-un-lft-identity 99.6%

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

      2. associate-/r* 99.6%

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

      3. sub-div 99.6%

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

   6. Applied egg-rr 99.6%

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

      1. *-lft-identity 99.6%

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

   8. Simplified 99.6%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;N \leq 0.66:\\ \;\;\;\;-\log N\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{0.5}{N}}{N}\\ \end{array} \]

Alternative 7: 52.6% 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 51.8%

   \[\log \left(N + 1\right) - \log N \]

2. Taylor expanded in N around inf 53.6%

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

3. Final simplification 53.6%

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

Alternative 8: 4.5% 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 51.8%

   \[\log \left(N + 1\right) - \log N \]

2. Taylor expanded in N around 0 49.8%

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

   1. neg-mul-1 49.8%

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

   2. unsub-neg 49.8%

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

4. Simplified 49.8%

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

5. Taylor expanded in N around inf 4.5%

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

6. Final simplification 4.5%

   \[\leadsto N \]

Reproduce

herbie shell --seed 2023275 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1.0)) (log N)))