2log (problem 3.3.6)

Percentage Accurate: 54.8% → 99.9%

Time: 6.0s

Alternatives: 6

Speedup: 2.0×

Specification

Math

\[\begin{array}{l} \\ \log \left(N + 1\right) - \log N \end{array} \]

FPCore

(FPCore (N) :precision binary64 (- (log (+ N 1.0)) (log N)))

C

double code(double N) {
	return log((N + 1.0)) - log(N);
}

Fortran

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

Java

public static double code(double N) {
	return Math.log((N + 1.0)) - Math.log(N);
}

Python

def code(N):
	return math.log((N + 1.0)) - math.log(N)

Julia

function code(N)
	return Float64(log(Float64(N + 1.0)) - log(N))
end

MATLAB

function tmp = code(N)
	tmp = log((N + 1.0)) - log(N);
end

Wolfram

code[N_] := N[(N[Log[N[(N + 1.0), $MachinePrecision]], $MachinePrecision] - N[Log[N], $MachinePrecision]), $MachinePrecision]

Initial Program: 54.8% 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

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

FPCore

(FPCore (N)
 :precision binary64
 (if (<= (- (log (+ N 1.0)) (log N)) 1e-5)
   (+ (/ 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)) <= 1e-5) {
		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)) <= 1d-5) 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)) <= 1e-5) {
		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)) <= 1e-5:
		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)) <= 1e-5)
		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)) <= 1e-5)
		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], 1e-5], 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)) < 1.00000000000000008e-5

   1. Initial program 7.3%

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

      1. +-commutative 7.3%

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

      2. log1p-def 7.3%

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

   3. Simplified 7.3%

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

      1. add-log-exp 7.3%

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

      2. log1p-expm1-u 7.3%

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

      3. log1p-udef 7.3%

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

      4. diff-log 7.2%

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

      5. log1p-udef 7.2%

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

      6. rem-exp-log 5.3%

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

      7. +-commutative 5.3%

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

      8. add-exp-log 5.3%

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

      9. log1p-udef 5.3%

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

      10. log1p-expm1-u 5.3%

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

      11. add-exp-log 7.7%

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

   5. Applied egg-rr 7.7%

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

   6. Taylor expanded in N around inf 100.0%

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

      1. associate--l+ 100.0%

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

      2. associate-*r/ 100.0%

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

      3. metadata-eval 100.0%

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

      4. *-lft-identity 100.0%

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

      5. *-lft-identity 100.0%

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

      6. *-inverses 46.2%

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

      7. associate-/r* 28.8%

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

      8. *-commutative 28.8%

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

      9. *-lft-identity 28.8%

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

      10. *-inverses 28.8%

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

      11. associate-*r/ 28.8%

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

      12. metadata-eval 28.8%

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

      13. times-frac 28.9%

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

      14. *-commutative 28.9%

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

      15. *-commutative 28.9%

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

      16. distribute-lft-neg-out 28.9%

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

      17. distribute-rgt-neg-in 28.9%

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

      18. metadata-eval 28.9%

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

      19. distribute-lft-neg-out 28.9%

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

      20. distribute-rgt-neg-out 28.9%

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

   8. Simplified 50.0%

      \[\leadsto \color{blue}{\frac{0.3333333333333333}{{N}^{3}} + \frac{N + -0.5}{{N}^{2}}} \]

   9. Taylor expanded in N around 0 100.0%

      \[\leadsto \color{blue}{\left(\frac{1}{N} + 0.3333333333333333 \cdot \frac{1}{{N}^{3}}\right) - 0.5 \cdot \frac{1}{{N}^{2}}} \]
   10. 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. *-rgt-identity 100.0%

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

       7. *-commutative 100.0%

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

       8. distribute-rgt-neg-in 100.0%

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

       9. unpow2 100.0%

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

       10. associate-/r* 100.0%

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

       11. *-rgt-identity 100.0%

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

       12. associate-*r/ 100.0%

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

       13. metadata-eval 100.0%

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

       14. associate-*r* 100.0%

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

       15. distribute-lft-in 100.0%

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

       16. lft-mult-inverse 99.7%

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

       17. distribute-lft-in 99.7%

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

       18. associate-*l/ 100.0%

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

   11. Simplified 100.0%

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

   if 1.00000000000000008e-5 < (-.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. +-commutative 99.9%

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

      2. log1p-def 99.9%

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

   3. Simplified 99.9%

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

      1. add-log-exp 99.9%

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

      2. log1p-expm1-u 7.9%

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

      3. log1p-udef 7.9%

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

      4. diff-log 7.9%

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

      5. log1p-udef 7.9%

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

      6. rem-exp-log 7.9%

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

      7. +-commutative 7.9%

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

      8. add-exp-log 7.9%

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

      9. log1p-udef 7.9%

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

      10. log1p-expm1-u 99.9%

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

      11. add-exp-log 99.9%

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

   5. 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 100.0%

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

Alternative 2: 99.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;N \leq 240000:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{0.5}{N}}{N}\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= N 240000.0) (log (/ (+ N 1.0) N)) (/ (- 1.0 (/ 0.5 N)) N)))

C

double code(double N) {
	double tmp;
	if (N <= 240000.0) {
		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 <= 240000.0d0) 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 <= 240000.0) {
		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 <= 240000.0:
		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 <= 240000.0)
		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 <= 240000.0)
		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, 240000.0], 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 < 2.4e5

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

      2. log1p-def 99.6%

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

   3. Simplified 99.6%

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

      1. add-log-exp 99.6%

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

      2. log1p-expm1-u 8.4%

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

      3. log1p-udef 8.4%

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

      4. diff-log 8.4%

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

      5. log1p-udef 8.4%

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

      6. rem-exp-log 8.4%

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

      7. +-commutative 8.4%

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

      8. add-exp-log 8.4%

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

      9. log1p-udef 8.4%

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

      10. log1p-expm1-u 99.6%

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

      11. add-exp-log 99.8%

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

   5. Applied egg-rr 99.8%

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

   if 2.4e5 < N

   1. Initial program 6.8%

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

      1. +-commutative 6.8%

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

      2. log1p-def 6.8%

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

   3. Simplified 6.8%

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

   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%

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

   6. Simplified 99.9%

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

      1. frac-sub 28.4%

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

      2. unpow2 28.4%

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

      3. cube-unmult 28.4%

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

      4. div-inv 28.4%

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

      5. *-un-lft-identity 28.4%

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

      6. unpow2 28.4%

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

      7. distribute-lft-out-- 28.4%

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

      8. metadata-eval 28.4%

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

      9. cube-div 28.9%

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

      10. inv-pow 28.9%

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

      11. pow-pow 29.0%

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

      12. metadata-eval 29.0%

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

   8. Applied egg-rr 29.0%

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

      1. *-commutative 29.0%

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

      2. associate-*r* 32.9%

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

      3. pow-plus 50.7%

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

      4. metadata-eval 50.7%

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

      5. sub-neg 50.7%

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

      6. metadata-eval 50.7%

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

   10. Simplified 50.7%

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

   11. Taylor expanded in N around 0 99.9%

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

       1. sub-neg 99.9%

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

       2. *-rgt-identity 99.9%

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

       3. *-commutative 99.9%

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

       4. distribute-rgt-neg-in 99.9%

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

       5. unpow2 99.9%

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

       6. associate-/r* 99.9%

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

       7. *-rgt-identity 99.9%

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

       8. associate-*r/ 99.9%

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

       9. metadata-eval 99.9%

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

       10. associate-*r* 99.9%

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

       11. distribute-lft-in 100.0%

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

       12. lft-mult-inverse 99.7%

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

       13. distribute-lft-in 99.7%

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

       14. associate-*l/ 100.0%

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

   13. Simplified 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;N \leq 240000:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{0.5}{N}}{N}\\ \end{array} \]

Alternative 3: 99.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;N \leq 0.9:\\ \;\;\;\;N - \log N\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{0.5}{N}}{N}\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= N 0.9) (- N (log N)) (/ (- 1.0 (/ 0.5 N)) N)))

C

double code(double N) {
	double tmp;
	if (N <= 0.9) {
		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 <= 0.9d0) 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 <= 0.9) {
		tmp = N - Math.log(N);
	} else {
		tmp = (1.0 - (0.5 / N)) / N;
	}
	return tmp;
}

Python

def code(N):
	tmp = 0
	if N <= 0.9:
		tmp = N - math.log(N)
	else:
		tmp = (1.0 - (0.5 / N)) / N
	return tmp

Julia

function code(N)
	tmp = 0.0
	if (N <= 0.9)
		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 <= 0.9)
		tmp = N - log(N);
	else
		tmp = (1.0 - (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[(1.0 - N[(0.5 / N), $MachinePrecision]), $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. 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 99.0%

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

      1. neg-mul-1 99.0%

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

      2. unsub-neg 99.0%

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

   6. Simplified 99.0%

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

   if 0.900000000000000022 < N

   1. Initial program 7.9%

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

      1. +-commutative 7.9%

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

      2. log1p-def 7.9%

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

   3. Simplified 7.9%

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

   4. Taylor expanded in N around inf 99.4%

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

      1. associate-*r/ 99.4%

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

      2. metadata-eval 99.4%

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

   6. Simplified 99.4%

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

      1. frac-sub 28.9%

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

      2. unpow2 28.9%

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

      3. cube-unmult 28.8%

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

      4. div-inv 28.8%

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

      5. *-un-lft-identity 28.8%

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

      6. unpow2 28.8%

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

      7. distribute-lft-out-- 28.8%

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

      8. metadata-eval 28.8%

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

      9. cube-div 29.4%

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

      10. inv-pow 29.4%

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

      11. pow-pow 29.5%

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

      12. metadata-eval 29.5%

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

   8. Applied egg-rr 29.5%

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

      1. *-commutative 29.5%

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

      2. associate-*r* 33.3%

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

      3. pow-plus 50.9%

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

      4. metadata-eval 50.9%

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

      5. sub-neg 50.9%

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

      6. metadata-eval 50.9%

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

   10. Simplified 50.9%

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

   11. Taylor expanded in N around 0 99.4%

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

       1. sub-neg 99.4%

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

       2. *-rgt-identity 99.4%

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

       3. *-commutative 99.4%

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

       4. distribute-rgt-neg-in 99.4%

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

       5. unpow2 99.4%

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

       6. associate-/r* 99.4%

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

       7. *-rgt-identity 99.4%

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

       8. associate-*r/ 99.4%

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

       9. metadata-eval 99.4%

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

       10. associate-*r* 99.4%

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

       11. distribute-lft-in 99.4%

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

       12. lft-mult-inverse 99.1%

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

       13. distribute-lft-in 99.1%

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

       14. associate-*l/ 99.4%

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

   13. Simplified 99.4%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;N \leq 0.9:\\ \;\;\;\;N - \log N\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{0.5}{N}}{N}\\ \end{array} \]

Alternative 4: 98.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if N < 0.680000000000000049

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. log1p-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in N around 0 97.7%

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

      1. neg-mul-1 97.7%

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

   6. Simplified 97.7%

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

   if 0.680000000000000049 < N

   1. Initial program 7.9%

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

      1. +-commutative 7.9%

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

      2. log1p-def 7.9%

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

   3. Simplified 7.9%

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

   4. Taylor expanded in N around inf 99.4%

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

      1. associate-*r/ 99.4%

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

      2. metadata-eval 99.4%

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

   6. Simplified 99.4%

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

      1. frac-sub 28.9%

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

      2. unpow2 28.9%

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

      3. cube-unmult 28.8%

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

      4. div-inv 28.8%

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

      5. *-un-lft-identity 28.8%

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

      6. unpow2 28.8%

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

      7. distribute-lft-out-- 28.8%

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

      8. metadata-eval 28.8%

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

      9. cube-div 29.4%

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

      10. inv-pow 29.4%

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

      11. pow-pow 29.5%

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

      12. metadata-eval 29.5%

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

   8. Applied egg-rr 29.5%

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

      1. *-commutative 29.5%

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

      2. associate-*r* 33.3%

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

      3. pow-plus 50.9%

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

      4. metadata-eval 50.9%

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

      5. sub-neg 50.9%

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

      6. metadata-eval 50.9%

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

   10. Simplified 50.9%

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

   11. Taylor expanded in N around 0 99.4%

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

       1. sub-neg 99.4%

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

       2. *-rgt-identity 99.4%

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

       3. *-commutative 99.4%

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

       4. distribute-rgt-neg-in 99.4%

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

       5. unpow2 99.4%

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

       6. associate-/r* 99.4%

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

       7. *-rgt-identity 99.4%

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

       8. associate-*r/ 99.4%

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

       9. metadata-eval 99.4%

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

       10. associate-*r* 99.4%

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

       11. distribute-lft-in 99.4%

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

       12. lft-mult-inverse 99.1%

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

       13. distribute-lft-in 99.1%

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

       14. associate-*l/ 99.4%

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

   13. Simplified 99.4%

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

4. Final simplification 98.6%

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

Alternative 5: 51.0% 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 51.4%

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

   1. +-commutative 51.4%

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

   2. log1p-def 51.4%

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

3. Simplified 51.4%

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

4. Taylor expanded in N around inf 54.5%

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

5. Final simplification 54.5%

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

Alternative 6: 4.6% 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 51.4%

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

   1. +-commutative 51.4%

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

   2. log1p-def 51.4%

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

3. Simplified 51.4%

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

4. Taylor expanded in N around 0 48.7%

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

   1. neg-mul-1 48.7%

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

   2. unsub-neg 48.7%

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

6. Simplified 48.7%

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

7. Taylor expanded in N around inf 4.4%

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

8. Final simplification 4.4%

   \[\leadsto N \]

Reproduce

herbie shell --seed 2023300 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1.0)) (log N)))