2log (problem 3.3.6)

Percentage Accurate: 53.9% → 99.9%

Time: 8.1s

Alternatives: 7

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: 53.9% 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, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 0.0002:\\ \;\;\;\;\frac{1}{N} + \left(\frac{0.3333333333333333}{{N}^{3}} + \left(\frac{-0.5}{{N}^{2}} - \frac{0.25}{{N}^{4}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= (- (log (+ N 1.0)) (log N)) 0.0002)
   (+
    (/ 1.0 N)
    (+
     (/ 0.3333333333333333 (pow N 3.0))
     (- (/ -0.5 (pow N 2.0)) (/ 0.25 (pow N 4.0)))))
   (log (/ (+ 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 / pow(N, 2.0)) - (0.25 / pow(N, 4.0))));
	} 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)) <= 0.0002d0) then
        tmp = (1.0d0 / n) + ((0.3333333333333333d0 / (n ** 3.0d0)) + (((-0.5d0) / (n ** 2.0d0)) - (0.25d0 / (n ** 4.0d0))))
    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)) <= 0.0002) {
		tmp = (1.0 / N) + ((0.3333333333333333 / Math.pow(N, 3.0)) + ((-0.5 / Math.pow(N, 2.0)) - (0.25 / Math.pow(N, 4.0))));
	} 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)) <= 0.0002:
		tmp = (1.0 / N) + ((0.3333333333333333 / math.pow(N, 3.0)) + ((-0.5 / math.pow(N, 2.0)) - (0.25 / math.pow(N, 4.0))))
	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)) <= 0.0002)
		tmp = Float64(Float64(1.0 / N) + Float64(Float64(0.3333333333333333 / (N ^ 3.0)) + Float64(Float64(-0.5 / (N ^ 2.0)) - Float64(0.25 / (N ^ 4.0)))));
	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)) <= 0.0002)
		tmp = (1.0 / N) + ((0.3333333333333333 / (N ^ 3.0)) + ((-0.5 / (N ^ 2.0)) - (0.25 / (N ^ 4.0))));
	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], 0.0002], N[(N[(1.0 / N), $MachinePrecision] + N[(N[(0.3333333333333333 / N[Power[N, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(-0.5 / N[Power[N, 2.0], $MachinePrecision]), $MachinePrecision] - N[(0.25 / N[Power[N, 4.0], $MachinePrecision]), $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)) < 2.0000000000000001e-4

   1. Initial program 8.2%

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

   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. sub-neg 100.0%

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

      2. +-commutative 100.0%

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

      3. associate-+l+ 100.0%

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

      4. associate-*r/ 100.0%

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

      5. metadata-eval 100.0%

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

      6. +-commutative 100.0%

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

      7. distribute-neg-in 100.0%

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

      8. unsub-neg 100.0%

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

      9. associate-*r/ 100.0%

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

      10. metadata-eval 100.0%

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

      11. distribute-neg-frac 100.0%

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

      12. metadata-eval 100.0%

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

      13. associate-*r/ 100.0%

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

      14. metadata-eval 100.0%

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

   4. Simplified 100.0%

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

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

   1. Initial program 99.9%

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

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 0.0002:\\ \;\;\;\;\frac{1}{N} + \left(\frac{0.3333333333333333}{{N}^{3}} + \left(\frac{-0.5}{{N}^{2}} - \frac{0.25}{{N}^{4}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \end{array} \]

Alternative 2: 99.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{N}, \frac{N + -0.5}{N}, 0.3333333333333333 \cdot {N}^{-3}\right)\\ \mathbf{else}:\\ \;\;\;\;-\log \left(\frac{N}{N + 1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= (- (log (+ N 1.0)) (log N)) 5e-6)
   (fma (/ 1.0 N) (/ (+ N -0.5) N) (* 0.3333333333333333 (pow N -3.0)))
   (- (log (/ N (+ N 1.0))))))

C

double code(double N) {
	double tmp;
	if ((log((N + 1.0)) - log(N)) <= 5e-6) {
		tmp = fma((1.0 / N), ((N + -0.5) / N), (0.3333333333333333 * pow(N, -3.0)));
	} else {
		tmp = -log((N / (N + 1.0)));
	}
	return tmp;
}

Julia

function code(N)
	tmp = 0.0
	if (Float64(log(Float64(N + 1.0)) - log(N)) <= 5e-6)
		tmp = fma(Float64(1.0 / N), Float64(Float64(N + -0.5) / N), Float64(0.3333333333333333 * (N ^ -3.0)));
	else
		tmp = Float64(-log(Float64(N / Float64(N + 1.0))));
	end
	return tmp
end

Wolfram

code[N_] := If[LessEqual[N[(N[Log[N[(N + 1.0), $MachinePrecision]], $MachinePrecision] - N[Log[N], $MachinePrecision]), $MachinePrecision], 5e-6], N[(N[(1.0 / N), $MachinePrecision] * N[(N[(N + -0.5), $MachinePrecision] / N), $MachinePrecision] + N[(0.3333333333333333 * N[Power[N, -3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-N[Log[N[(N / N[(N + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (log.f64 (+.f64 N 1)) (log.f64 N)) < 5.00000000000000041e-6

   1. Initial program 7.6%

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

   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. sub-neg 100.0%

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

      2. +-commutative 100.0%

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

      3. associate-+l+ 100.0%

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

      4. associate-*r/ 100.0%

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

      5. metadata-eval 100.0%

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

      6. +-commutative 100.0%

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

      7. distribute-neg-in 100.0%

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

      8. unsub-neg 100.0%

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

      9. associate-*r/ 100.0%

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

      10. metadata-eval 100.0%

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

      11. distribute-neg-frac 100.0%

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

      12. metadata-eval 100.0%

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

      13. associate-*r/ 100.0%

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

      14. metadata-eval 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in N around inf 100.0%

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

      1. associate-*r/ 100.0%

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

      2. metadata-eval 100.0%

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

      3. associate-*r/ 100.0%

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

      4. metadata-eval 100.0%

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

   7. Simplified 100.0%

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

   9. Applied egg-rr 100.0%

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

   if 5.00000000000000041e-6 < (-.f64 (log.f64 (+.f64 N 1)) (log.f64 N))

   1. Initial program 99.7%

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

      1. diff-log 99.9%

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

      2. clear-num 99.9%

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

      3. log-rec 99.9%

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

   3. Applied egg-rr 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{N}, \frac{N + -0.5}{N}, 0.3333333333333333 \cdot {N}^{-3}\right)\\ \mathbf{else}:\\ \;\;\;\;-\log \left(\frac{N}{N + 1}\right)\\ \end{array} \]

Alternative 3: 99.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 1.5 \cdot 10^{-6}:\\ \;\;\;\;\frac{\frac{N + -0.5}{N}}{N}\\ \mathbf{else}:\\ \;\;\;\;-\log \left(\frac{N}{N + 1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= (- (log (+ N 1.0)) (log N)) 1.5e-6)
   (/ (/ (+ N -0.5) N) N)
   (- (log (/ N (+ N 1.0))))))

C

double code(double N) {
	double tmp;
	if ((log((N + 1.0)) - log(N)) <= 1.5e-6) {
		tmp = ((N + -0.5) / N) / N;
	} else {
		tmp = -log((N / (N + 1.0)));
	}
	return tmp;
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((log((n + 1.0d0)) - log(n)) <= 1.5d-6) then
        tmp = ((n + (-0.5d0)) / n) / n
    else
        tmp = -log((n / (n + 1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double N) {
	double tmp;
	if ((Math.log((N + 1.0)) - Math.log(N)) <= 1.5e-6) {
		tmp = ((N + -0.5) / N) / N;
	} else {
		tmp = -Math.log((N / (N + 1.0)));
	}
	return tmp;
}

Python

def code(N):
	tmp = 0
	if (math.log((N + 1.0)) - math.log(N)) <= 1.5e-6:
		tmp = ((N + -0.5) / N) / N
	else:
		tmp = -math.log((N / (N + 1.0)))
	return tmp

Julia

function code(N)
	tmp = 0.0
	if (Float64(log(Float64(N + 1.0)) - log(N)) <= 1.5e-6)
		tmp = Float64(Float64(Float64(N + -0.5) / N) / N);
	else
		tmp = Float64(-log(Float64(N / Float64(N + 1.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(N)
	tmp = 0.0;
	if ((log((N + 1.0)) - log(N)) <= 1.5e-6)
		tmp = ((N + -0.5) / N) / N;
	else
		tmp = -log((N / (N + 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[N_] := If[LessEqual[N[(N[Log[N[(N + 1.0), $MachinePrecision]], $MachinePrecision] - N[Log[N], $MachinePrecision]), $MachinePrecision], 1.5e-6], N[(N[(N[(N + -0.5), $MachinePrecision] / N), $MachinePrecision] / N), $MachinePrecision], (-N[Log[N[(N / N[(N + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (log.f64 (+.f64 N 1)) (log.f64 N)) < 1.5e-6

   1. Initial program 6.6%

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

   2. Taylor expanded in N around inf 99.7%

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

      1. associate-*r/ 99.7%

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

      2. metadata-eval 99.7%

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

   4. Simplified 99.7%

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

      1. frac-sub 31.1%

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

      2. *-un-lft-identity 31.1%

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

      3. unpow2 31.1%

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

      4. distribute-lft-out-- 31.1%

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

      5. unpow2 31.1%

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

      6. cube-mult 31.0%

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

   6. Applied egg-rr 31.0%

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

      1. associate-/l* 34.5%

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

      2. associate-/r/ 34.6%

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

      3. sub-neg 34.6%

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

      4. metadata-eval 34.6%

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

   8. Simplified 34.6%

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

      1. clear-num 34.5%

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

      2. pow1 34.5%

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

      3. pow-div 50.9%

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

      4. metadata-eval 50.9%

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

      5. pow2 50.9%

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

      6. add-sqr-sqrt 50.9%

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

      7. sqrt-div 51.0%

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

      8. metadata-eval 51.0%

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

      9. sqrt-unprod 50.8%

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

      10. add-sqr-sqrt 51.0%

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

      11. sqrt-div 51.1%

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

      12. metadata-eval 51.1%

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

      13. sqrt-unprod 52.6%

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

      14. add-sqr-sqrt 52.8%

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

   10. Applied egg-rr 52.8%

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

       1. un-div-inv 52.9%

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

       2. associate-*l/ 99.7%

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

       3. associate-*l/ 99.7%

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

       4. *-un-lft-identity 99.7%

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

   12. Applied egg-rr 99.7%

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

   if 1.5e-6 < (-.f64 (log.f64 (+.f64 N 1)) (log.f64 N))

   1. Initial program 99.3%

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

      1. diff-log 99.5%

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

      2. clear-num 99.5%

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

      3. log-rec 99.6%

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

   3. Applied egg-rr 99.6%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 1.5 \cdot 10^{-6}:\\ \;\;\;\;\frac{\frac{N + -0.5}{N}}{N}\\ \mathbf{else}:\\ \;\;\;\;-\log \left(\frac{N}{N + 1}\right)\\ \end{array} \]

Alternative 4: 99.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;N \leq 300000:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{N + -0.5}{N}}{N}\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= N 300000.0) (log (/ (+ N 1.0) N)) (/ (/ (+ N -0.5) N) N)))

C

double code(double N) {
	double tmp;
	if (N <= 300000.0) {
		tmp = log(((N + 1.0) / N));
	} else {
		tmp = ((N + -0.5) / N) / N;
	}
	return tmp;
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= 300000.0d0) then
        tmp = log(((n + 1.0d0) / n))
    else
        tmp = ((n + (-0.5d0)) / n) / n
    end if
    code = tmp
end function

Java

public static double code(double N) {
	double tmp;
	if (N <= 300000.0) {
		tmp = Math.log(((N + 1.0) / N));
	} else {
		tmp = ((N + -0.5) / N) / N;
	}
	return tmp;
}

Python

def code(N):
	tmp = 0
	if N <= 300000.0:
		tmp = math.log(((N + 1.0) / N))
	else:
		tmp = ((N + -0.5) / N) / N
	return tmp

Julia

function code(N)
	tmp = 0.0
	if (N <= 300000.0)
		tmp = log(Float64(Float64(N + 1.0) / N));
	else
		tmp = Float64(Float64(Float64(N + -0.5) / N) / N);
	end
	return tmp
end

MATLAB

function tmp_2 = code(N)
	tmp = 0.0;
	if (N <= 300000.0)
		tmp = log(((N + 1.0) / N));
	else
		tmp = ((N + -0.5) / N) / N;
	end
	tmp_2 = tmp;
end

Wolfram

code[N_] := If[LessEqual[N, 300000.0], N[Log[N[(N[(N + 1.0), $MachinePrecision] / N), $MachinePrecision]], $MachinePrecision], N[(N[(N[(N + -0.5), $MachinePrecision] / N), $MachinePrecision] / N), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if N < 3e5

   1. Initial program 99.5%

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

      1. diff-log 99.7%

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

   3. Applied egg-rr 99.7%

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

   if 3e5 < N

   1. Initial program 7.1%

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

   2. Taylor expanded in N around inf 99.5%

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

      1. associate-*r/ 99.5%

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

      2. metadata-eval 99.5%

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

   4. Simplified 99.5%

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

      1. frac-sub 31.5%

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

      2. *-un-lft-identity 31.5%

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

      3. unpow2 31.5%

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

      4. distribute-lft-out-- 31.5%

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

      5. unpow2 31.5%

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

      6. cube-mult 31.4%

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

   6. Applied egg-rr 31.4%

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

      1. associate-/l* 34.8%

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

      2. associate-/r/ 34.9%

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

      3. sub-neg 34.9%

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

      4. metadata-eval 34.9%

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

   8. Simplified 34.9%

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

      1. clear-num 34.8%

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

      2. pow1 34.8%

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

      3. pow-div 51.1%

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

      4. metadata-eval 51.1%

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

      5. pow2 51.1%

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

      6. add-sqr-sqrt 51.1%

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

      7. sqrt-div 51.2%

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

      8. metadata-eval 51.2%

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

      9. sqrt-unprod 51.1%

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

      10. add-sqr-sqrt 51.2%

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

      11. sqrt-div 51.3%

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

      12. metadata-eval 51.3%

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

      13. sqrt-unprod 52.8%

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

      14. add-sqr-sqrt 53.0%

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

   10. Applied egg-rr 53.0%

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

       1. un-div-inv 53.1%

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

       2. associate-*l/ 99.5%

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

       3. associate-*l/ 99.6%

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

       4. *-un-lft-identity 99.6%

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

   12. Applied egg-rr 99.6%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;N \leq 300000:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{N + -0.5}{N}}{N}\\ \end{array} \]

Alternative 5: 99.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;N \leq 0.9:\\ \;\;\;\;N - \log N\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{N + -0.5}{N}}{N}\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= N 0.9) (- N (log N)) (/ (/ (+ N -0.5) N) N)))

C

double code(double N) {
	double tmp;
	if (N <= 0.9) {
		tmp = N - log(N);
	} else {
		tmp = ((N + -0.5) / N) / N;
	}
	return tmp;
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= 0.9d0) then
        tmp = n - log(n)
    else
        tmp = ((n + (-0.5d0)) / n) / n
    end if
    code = tmp
end function

Java

public static double code(double N) {
	double tmp;
	if (N <= 0.9) {
		tmp = N - Math.log(N);
	} else {
		tmp = ((N + -0.5) / N) / N;
	}
	return tmp;
}

Python

def code(N):
	tmp = 0
	if N <= 0.9:
		tmp = N - math.log(N)
	else:
		tmp = ((N + -0.5) / N) / N
	return tmp

Julia

function code(N)
	tmp = 0.0
	if (N <= 0.9)
		tmp = Float64(N - log(N));
	else
		tmp = Float64(Float64(Float64(N + -0.5) / N) / N);
	end
	return tmp
end

MATLAB

function tmp_2 = code(N)
	tmp = 0.0;
	if (N <= 0.9)
		tmp = N - log(N);
	else
		tmp = ((N + -0.5) / N) / N;
	end
	tmp_2 = tmp;
end

Wolfram

code[N_] := If[LessEqual[N, 0.9], N[(N - N[Log[N], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N + -0.5), $MachinePrecision] / N), $MachinePrecision] / N), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if N < 0.900000000000000022

   1. Initial program 100.0%

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

   2. Taylor expanded in N around 0 99.7%

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

      1. neg-mul-1 99.7%

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

      2. unsub-neg 99.7%

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

   4. Simplified 99.7%

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

   if 0.900000000000000022 < N

   1. Initial program 8.8%

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

   2. Taylor expanded in N around inf 98.6%

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

      1. associate-*r/ 98.6%

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

      2. metadata-eval 98.6%

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

   4. Simplified 98.6%

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

      1. frac-sub 32.0%

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

      2. *-un-lft-identity 32.0%

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

      3. unpow2 32.0%

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

      4. distribute-lft-out-- 32.0%

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

      5. unpow2 32.0%

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

      6. cube-mult 32.0%

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

   6. Applied egg-rr 32.0%

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

      1. associate-/l* 35.3%

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

      2. associate-/r/ 35.4%

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

      3. sub-neg 35.4%

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

      4. metadata-eval 35.4%

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

   8. Simplified 35.4%

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

      1. clear-num 35.3%

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

      2. pow1 35.3%

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

      3. pow-div 51.3%

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

      4. metadata-eval 51.3%

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

      5. pow2 51.3%

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

      6. add-sqr-sqrt 51.2%

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

      7. sqrt-div 51.3%

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

      8. metadata-eval 51.3%

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

      9. sqrt-unprod 51.2%

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

      10. add-sqr-sqrt 51.3%

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

      11. sqrt-div 51.4%

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

      12. metadata-eval 51.4%

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

      13. sqrt-unprod 52.9%

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

      14. add-sqr-sqrt 53.1%

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

   10. Applied egg-rr 53.1%

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

       1. un-div-inv 53.2%

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

       2. associate-*l/ 98.6%

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

       3. associate-*l/ 98.6%

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

       4. *-un-lft-identity 98.6%

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

   12. Applied egg-rr 98.6%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;N \leq 0.9:\\ \;\;\;\;N - \log N\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{N + -0.5}{N}}{N}\\ \end{array} \]

Alternative 6: 98.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;N \leq 0.67:\\ \;\;\;\;-\log N\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{N + -0.5}{N}}{N}\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= N 0.67) (- (log N)) (/ (/ (+ N -0.5) N) N)))

C

double code(double N) {
	double tmp;
	if (N <= 0.67) {
		tmp = -log(N);
	} else {
		tmp = ((N + -0.5) / N) / N;
	}
	return tmp;
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= 0.67d0) then
        tmp = -log(n)
    else
        tmp = ((n + (-0.5d0)) / n) / n
    end if
    code = tmp
end function

Java

public static double code(double N) {
	double tmp;
	if (N <= 0.67) {
		tmp = -Math.log(N);
	} else {
		tmp = ((N + -0.5) / N) / N;
	}
	return tmp;
}

Python

def code(N):
	tmp = 0
	if N <= 0.67:
		tmp = -math.log(N)
	else:
		tmp = ((N + -0.5) / N) / N
	return tmp

Julia

function code(N)
	tmp = 0.0
	if (N <= 0.67)
		tmp = Float64(-log(N));
	else
		tmp = Float64(Float64(Float64(N + -0.5) / N) / N);
	end
	return tmp
end

MATLAB

function tmp_2 = code(N)
	tmp = 0.0;
	if (N <= 0.67)
		tmp = -log(N);
	else
		tmp = ((N + -0.5) / N) / N;
	end
	tmp_2 = tmp;
end

Wolfram

code[N_] := If[LessEqual[N, 0.67], (-N[Log[N], $MachinePrecision]), N[(N[(N[(N + -0.5), $MachinePrecision] / N), $MachinePrecision] / N), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if N < 0.67000000000000004

   1. Initial program 100.0%

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

   2. Taylor expanded in N around 0 98.6%

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

      1. neg-mul-1 98.6%

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

   4. Simplified 98.6%

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

   if 0.67000000000000004 < N

   1. Initial program 8.8%

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

   2. Taylor expanded in N around inf 98.6%

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

      1. associate-*r/ 98.6%

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

      2. metadata-eval 98.6%

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

   4. Simplified 98.6%

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

      1. frac-sub 32.0%

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

      2. *-un-lft-identity 32.0%

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

      3. unpow2 32.0%

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

      4. distribute-lft-out-- 32.0%

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

      5. unpow2 32.0%

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

      6. cube-mult 32.0%

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

   6. Applied egg-rr 32.0%

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

      1. associate-/l* 35.3%

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

      2. associate-/r/ 35.4%

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

      3. sub-neg 35.4%

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

      4. metadata-eval 35.4%

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

   8. Simplified 35.4%

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

      1. clear-num 35.3%

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

      2. pow1 35.3%

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

      3. pow-div 51.3%

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

      4. metadata-eval 51.3%

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

      5. pow2 51.3%

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

      6. add-sqr-sqrt 51.2%

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

      7. sqrt-div 51.3%

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

      8. metadata-eval 51.3%

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

      9. sqrt-unprod 51.2%

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

      10. add-sqr-sqrt 51.3%

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

      11. sqrt-div 51.4%

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

      12. metadata-eval 51.4%

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

      13. sqrt-unprod 52.9%

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

      14. add-sqr-sqrt 53.1%

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

   10. Applied egg-rr 53.1%

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

       1. un-div-inv 53.2%

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

       2. associate-*l/ 98.6%

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

       3. associate-*l/ 98.6%

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

       4. *-un-lft-identity 98.6%

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

   12. Applied egg-rr 98.6%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;N \leq 0.67:\\ \;\;\;\;-\log N\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{N + -0.5}{N}}{N}\\ \end{array} \]

Alternative 7: 51.8% 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 53.3%

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

2. Taylor expanded in N around inf 52.5%

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

3. Final simplification 52.5%

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

Reproduce

herbie shell --seed 2023320 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1.0)) (log N)))