2log (problem 3.3.6)

Percentage Accurate: 54.7% → 99.9%

Time: 11.2s

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: 54.7% 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

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 0.0005:\\ \;\;\;\;\frac{1 - \frac{0.5}{N}}{N} + \left(\langle \left( 0.3333333333333333 \cdot {N}^{-3} \right)_{\text{binary32}} \rangle_{\text{binary64}} + \frac{\frac{-0.25}{N \cdot N}}{N \cdot N}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(N\right) - \log N\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= (- (log (+ N 1.0)) (log N)) 0.0005)
   (+
    (/ (- 1.0 (/ 0.5 N)) N)
    (+
     (cast (! :precision binary32 (* 0.3333333333333333 (pow N -3.0))))
     (/ (/ -0.25 (* N N)) (* N N))))
   (- (log1p N) (log N))))

C

double code(double N) {
	double tmp_1;
	if ((log((N + 1.0)) - log(N)) <= 0.0005) {
		float tmp_2 = 0.3333333333333333f * powf(N, -3.0f);
		tmp_1 = ((1.0 - (0.5 / N)) / N) + (((double) ((double) tmp_2)) + ((-0.25 / (N * N)) / (N * N)));
	} else {
		tmp_1 = log1p(N) - log(N);
	}
	return tmp_1;
}

Julia

function code(N)
	tmp_1 = 0.0
	if (Float64(log(Float64(N + 1.0)) - log(N)) <= 0.0005)
		tmp_2 = Float32(Float32(0.3333333333333333) * (N ^ Float32(-3.0)))
		tmp_1 = Float64(Float64(Float64(1.0 - Float64(0.5 / N)) / N) + Float64(Float64(Float64(tmp_2)) + Float64(Float64(-0.25 / Float64(N * N)) / Float64(N * N))));
	else
		tmp_1 = Float64(log1p(N) - log(N));
	end
	return tmp_1
end

Derivation

1. Split input into 2 regimes

2. if (-.f64 (log.f64 (+.f64 N 1)) (log.f64 N)) < 5.0000000000000001e-4

   1. Initial program 8.8%

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

   2. Taylor expanded in N around inf 99.9%

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

      1. +-commutative 99.9%

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

      2. associate-*r/ 99.9%

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

      3. metadata-eval 99.9%

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

      4. +-commutative 99.9%

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

      5. unpow2 99.9%

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

      6. associate-*r/ 99.9%

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

      7. metadata-eval 99.9%

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

      8. associate-*r/ 99.9%

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

      9. metadata-eval 99.9%

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

   4. Simplified 99.9%

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

      1. metadata-eval 99.9%

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

      2. metadata-eval 99.9%

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

      3. pow-prod-up 99.9%

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

      4. pow-prod-down 99.9%

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

      5. pow2 99.9%

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

      6. frac-times 99.9%

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

   6. Applied egg-rr 99.9%

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

      1. associate--r+ 99.9%

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

      2. cancel-sign-sub-inv 99.9%

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

      3. +-commutative 99.9%

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

      4. associate--l+ 99.9%

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

      5. +-commutative 99.9%

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

      6. associate-/r* 99.9%

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

      7. sub-div 99.9%

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

      8. div-inv 99.9%

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

      9. pow-flip 99.9%

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

      10. metadata-eval 99.9%

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

      11. *-commutative 99.9%

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

      12. distribute-neg-frac 99.9%

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

      13. metadata-eval 99.9%

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

   8. Applied egg-rr 99.9%

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

      1. associate-+l+ 99.9%

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

      2. associate-*r/ 99.9%

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

      3. associate-*l/ 99.9%

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

      4. metadata-eval 99.9%

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

   10. Simplified 99.9%

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

       1. rewrite-binary64/binary32 100.0%

          \[\leadsto \color{blue}{\frac{1 - \frac{0.5}{N}}{N} + \left(\langle \left( 0.3333333333333333 \cdot {N}^{-3} \right)_{\text{binary32}} \rangle_{\text{binary64}} + \frac{\frac{-0.25}{N \cdot N}}{N \cdot N}\right)} \]

   12. Applied rewrite-once 100.0%

       \[\leadsto \frac{1 - \frac{0.5}{N}}{N} + \left(\color{blue}{\langle \color{blue}{\left( \color{blue}{0.3333333333333333 \cdot {N}^{-3}} \right)_{\text{binary32}}} \rangle_{\text{binary64}}} + \frac{\frac{-0.25}{N \cdot N}}{N \cdot N}\right) \]

   if 5.0000000000000001e-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. +-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%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(N\right) - \log N} \]
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 0.0005:\\ \;\;\;\;\frac{1 - \frac{0.5}{N}}{N} + \left(\langle \left( 0.3333333333333333 \cdot {N}^{-3} \right)_{\text{binary32}} \rangle_{\text{binary64}} + \frac{\frac{-0.25}{N \cdot N}}{N \cdot N}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(N\right) - \log N\\ \end{array} \]

Alternative 2: 99.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 0.0005:\\ \;\;\;\;\frac{1 - \frac{0.5}{N}}{N} + \left(\frac{\frac{-0.25}{N \cdot N}}{N \cdot N} + \frac{\frac{0.3333333333333333}{N \cdot N}}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(N\right) - \log N\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= (- (log (+ N 1.0)) (log N)) 0.0005)
   (+
    (/ (- 1.0 (/ 0.5 N)) N)
    (+ (/ (/ -0.25 (* N N)) (* N N)) (/ (/ 0.3333333333333333 (* N N)) N)))
   (- (log1p N) (log N))))

C

double code(double N) {
	double tmp;
	if ((log((N + 1.0)) - log(N)) <= 0.0005) {
		tmp = ((1.0 - (0.5 / N)) / N) + (((-0.25 / (N * N)) / (N * N)) + ((0.3333333333333333 / (N * N)) / N));
	} else {
		tmp = log1p(N) - log(N);
	}
	return tmp;
}

Java

public static double code(double N) {
	double tmp;
	if ((Math.log((N + 1.0)) - Math.log(N)) <= 0.0005) {
		tmp = ((1.0 - (0.5 / N)) / N) + (((-0.25 / (N * N)) / (N * N)) + ((0.3333333333333333 / (N * N)) / N));
	} else {
		tmp = Math.log1p(N) - Math.log(N);
	}
	return tmp;
}

Python

def code(N):
	tmp = 0
	if (math.log((N + 1.0)) - math.log(N)) <= 0.0005:
		tmp = ((1.0 - (0.5 / N)) / N) + (((-0.25 / (N * N)) / (N * N)) + ((0.3333333333333333 / (N * N)) / N))
	else:
		tmp = math.log1p(N) - math.log(N)
	return tmp

Julia

function code(N)
	tmp = 0.0
	if (Float64(log(Float64(N + 1.0)) - log(N)) <= 0.0005)
		tmp = Float64(Float64(Float64(1.0 - Float64(0.5 / N)) / N) + Float64(Float64(Float64(-0.25 / Float64(N * N)) / Float64(N * N)) + Float64(Float64(0.3333333333333333 / Float64(N * N)) / N)));
	else
		tmp = Float64(log1p(N) - log(N));
	end
	return tmp
end

Wolfram

code[N_] := If[LessEqual[N[(N[Log[N[(N + 1.0), $MachinePrecision]], $MachinePrecision] - N[Log[N], $MachinePrecision]), $MachinePrecision], 0.0005], N[(N[(N[(1.0 - N[(0.5 / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision] + N[(N[(N[(-0.25 / N[(N * N), $MachinePrecision]), $MachinePrecision] / N[(N * N), $MachinePrecision]), $MachinePrecision] + N[(N[(0.3333333333333333 / N[(N * N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Log[1 + N], $MachinePrecision] - N[Log[N], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (log.f64 (+.f64 N 1)) (log.f64 N)) < 5.0000000000000001e-4

   1. Initial program 8.8%

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

   2. Taylor expanded in N around inf 99.9%

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

      1. +-commutative 99.9%

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

      2. associate-*r/ 99.9%

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

      3. metadata-eval 99.9%

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

      4. +-commutative 99.9%

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

      5. unpow2 99.9%

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

      6. associate-*r/ 99.9%

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

      7. metadata-eval 99.9%

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

      8. associate-*r/ 99.9%

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

      9. metadata-eval 99.9%

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

   4. Simplified 99.9%

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

      1. metadata-eval 99.9%

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

      2. metadata-eval 99.9%

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

      3. pow-prod-up 99.9%

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

      4. pow-prod-down 99.9%

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

      5. pow2 99.9%

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

      6. frac-times 99.9%

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

   6. Applied egg-rr 99.9%

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

      1. associate--r+ 99.9%

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

      2. cancel-sign-sub-inv 99.9%

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

      3. +-commutative 99.9%

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

      4. associate--l+ 99.9%

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

      5. +-commutative 99.9%

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

      6. associate-/r* 99.9%

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

      7. sub-div 99.9%

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

      8. div-inv 99.9%

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

      9. pow-flip 99.9%

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

      10. metadata-eval 99.9%

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

      11. *-commutative 99.9%

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

      12. distribute-neg-frac 99.9%

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

      13. metadata-eval 99.9%

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

   8. Applied egg-rr 99.9%

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

      1. associate-+l+ 99.9%

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

      2. associate-*r/ 99.9%

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

      3. associate-*l/ 99.9%

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

      4. metadata-eval 99.9%

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

   10. Simplified 99.9%

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

       1. metadata-eval 99.9%

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

       2. pow-flip 99.9%

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

       3. div-inv 99.9%

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

       4. unpow3 99.9%

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

       5. associate-/r* 99.9%

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

   12. Applied egg-rr 99.9%

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

   if 5.0000000000000001e-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. +-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%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(N\right) - \log N} \]
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 0.0005:\\ \;\;\;\;\frac{1 - \frac{0.5}{N}}{N} + \left(\frac{\frac{-0.25}{N \cdot N}}{N \cdot N} + \frac{\frac{0.3333333333333333}{N \cdot N}}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(N\right) - \log N\\ \end{array} \]

Alternative 3: 99.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;N \leq 1400:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{0.5}{N}}{N} + \left(\frac{\frac{-0.25}{N \cdot N}}{N \cdot N} + \frac{\frac{0.3333333333333333}{N \cdot N}}{N}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= N 1400.0)
   (log (/ (+ N 1.0) N))
   (+
    (/ (- 1.0 (/ 0.5 N)) N)
    (+ (/ (/ -0.25 (* N N)) (* N N)) (/ (/ 0.3333333333333333 (* N N)) N)))))

C

double code(double N) {
	double tmp;
	if (N <= 1400.0) {
		tmp = log(((N + 1.0) / N));
	} else {
		tmp = ((1.0 - (0.5 / N)) / N) + (((-0.25 / (N * N)) / (N * N)) + ((0.3333333333333333 / (N * N)) / N));
	}
	return tmp;
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= 1400.0d0) then
        tmp = log(((n + 1.0d0) / n))
    else
        tmp = ((1.0d0 - (0.5d0 / n)) / n) + ((((-0.25d0) / (n * n)) / (n * n)) + ((0.3333333333333333d0 / (n * n)) / n))
    end if
    code = tmp
end function

Java

public static double code(double N) {
	double tmp;
	if (N <= 1400.0) {
		tmp = Math.log(((N + 1.0) / N));
	} else {
		tmp = ((1.0 - (0.5 / N)) / N) + (((-0.25 / (N * N)) / (N * N)) + ((0.3333333333333333 / (N * N)) / N));
	}
	return tmp;
}

Python

def code(N):
	tmp = 0
	if N <= 1400.0:
		tmp = math.log(((N + 1.0) / N))
	else:
		tmp = ((1.0 - (0.5 / N)) / N) + (((-0.25 / (N * N)) / (N * N)) + ((0.3333333333333333 / (N * N)) / N))
	return tmp

Julia

function code(N)
	tmp = 0.0
	if (N <= 1400.0)
		tmp = log(Float64(Float64(N + 1.0) / N));
	else
		tmp = Float64(Float64(Float64(1.0 - Float64(0.5 / N)) / N) + Float64(Float64(Float64(-0.25 / Float64(N * N)) / Float64(N * N)) + Float64(Float64(0.3333333333333333 / Float64(N * N)) / N)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(N)
	tmp = 0.0;
	if (N <= 1400.0)
		tmp = log(((N + 1.0) / N));
	else
		tmp = ((1.0 - (0.5 / N)) / N) + (((-0.25 / (N * N)) / (N * N)) + ((0.3333333333333333 / (N * N)) / N));
	end
	tmp_2 = tmp;
end

Wolfram

code[N_] := If[LessEqual[N, 1400.0], 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[(N[(N[(-0.25 / N[(N * N), $MachinePrecision]), $MachinePrecision] / N[(N * N), $MachinePrecision]), $MachinePrecision] + N[(N[(0.3333333333333333 / N[(N * N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if N < 1400

   1. Initial program 100.0%

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

      1. diff-log 99.2%

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

   3. Applied egg-rr 99.2%

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

   if 1400 < N

   1. Initial program 8.8%

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

   2. Taylor expanded in N around inf 99.9%

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

      1. +-commutative 99.9%

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

      2. associate-*r/ 99.9%

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

      3. metadata-eval 99.9%

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

      4. +-commutative 99.9%

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

      5. unpow2 99.9%

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

      6. associate-*r/ 99.9%

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

      7. metadata-eval 99.9%

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

      8. associate-*r/ 99.9%

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

      9. metadata-eval 99.9%

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

   4. Simplified 99.9%

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

      1. metadata-eval 99.9%

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

      2. metadata-eval 99.9%

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

      3. pow-prod-up 99.9%

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

      4. pow-prod-down 99.9%

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

      5. pow2 99.9%

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

      6. frac-times 99.9%

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

   6. Applied egg-rr 99.9%

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

      1. associate--r+ 99.9%

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

      2. cancel-sign-sub-inv 99.9%

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

      3. +-commutative 99.9%

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

      4. associate--l+ 99.9%

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

      5. +-commutative 99.9%

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

      6. associate-/r* 99.9%

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

      7. sub-div 99.9%

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

      8. div-inv 99.9%

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

      9. pow-flip 99.9%

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

      10. metadata-eval 99.9%

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

      11. *-commutative 99.9%

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

      12. distribute-neg-frac 99.9%

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

      13. metadata-eval 99.9%

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

   8. Applied egg-rr 99.9%

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

      1. associate-+l+ 99.9%

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

      2. associate-*r/ 99.9%

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

      3. associate-*l/ 99.9%

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

      4. metadata-eval 99.9%

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

   10. Simplified 99.9%

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

       1. metadata-eval 99.9%

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

       2. pow-flip 99.9%

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

       3. div-inv 99.9%

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

       4. unpow3 99.9%

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

       5. associate-/r* 99.9%

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

   12. Applied egg-rr 99.9%

       \[\leadsto \frac{1 - \frac{0.5}{N}}{N} + \left(\color{blue}{\frac{\frac{0.3333333333333333}{N \cdot N}}{N}} + \frac{\frac{-0.25}{N \cdot N}}{N \cdot N}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;N \leq 1400:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{0.5}{N}}{N} + \left(\frac{\frac{-0.25}{N \cdot N}}{N \cdot N} + \frac{\frac{0.3333333333333333}{N \cdot N}}{N}\right)\\ \end{array} \]

Alternative 4: 99.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;N \leq 0.85:\\ \;\;\;\;N - \log N\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{0.5}{N}}{N} + \left(\frac{\frac{-0.25}{N \cdot N}}{N \cdot N} + \frac{\frac{0.3333333333333333}{N \cdot N}}{N}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= N 0.85)
   (- N (log N))
   (+
    (/ (- 1.0 (/ 0.5 N)) N)
    (+ (/ (/ -0.25 (* N N)) (* N N)) (/ (/ 0.3333333333333333 (* N N)) N)))))

C

double code(double N) {
	double tmp;
	if (N <= 0.85) {
		tmp = N - log(N);
	} else {
		tmp = ((1.0 - (0.5 / N)) / N) + (((-0.25 / (N * N)) / (N * N)) + ((0.3333333333333333 / (N * N)) / N));
	}
	return tmp;
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= 0.85d0) then
        tmp = n - log(n)
    else
        tmp = ((1.0d0 - (0.5d0 / n)) / n) + ((((-0.25d0) / (n * n)) / (n * n)) + ((0.3333333333333333d0 / (n * n)) / n))
    end if
    code = tmp
end function

Java

public static double code(double N) {
	double tmp;
	if (N <= 0.85) {
		tmp = N - Math.log(N);
	} else {
		tmp = ((1.0 - (0.5 / N)) / N) + (((-0.25 / (N * N)) / (N * N)) + ((0.3333333333333333 / (N * N)) / N));
	}
	return tmp;
}

Python

def code(N):
	tmp = 0
	if N <= 0.85:
		tmp = N - math.log(N)
	else:
		tmp = ((1.0 - (0.5 / N)) / N) + (((-0.25 / (N * N)) / (N * N)) + ((0.3333333333333333 / (N * N)) / N))
	return tmp

Julia

function code(N)
	tmp = 0.0
	if (N <= 0.85)
		tmp = Float64(N - log(N));
	else
		tmp = Float64(Float64(Float64(1.0 - Float64(0.5 / N)) / N) + Float64(Float64(Float64(-0.25 / Float64(N * N)) / Float64(N * N)) + Float64(Float64(0.3333333333333333 / Float64(N * N)) / N)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(N)
	tmp = 0.0;
	if (N <= 0.85)
		tmp = N - log(N);
	else
		tmp = ((1.0 - (0.5 / N)) / N) + (((-0.25 / (N * N)) / (N * N)) + ((0.3333333333333333 / (N * N)) / N));
	end
	tmp_2 = tmp;
end

Wolfram

code[N_] := If[LessEqual[N, 0.85], N[(N - N[Log[N], $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 - N[(0.5 / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision] + N[(N[(N[(-0.25 / N[(N * N), $MachinePrecision]), $MachinePrecision] / N[(N * N), $MachinePrecision]), $MachinePrecision] + N[(N[(0.3333333333333333 / N[(N * N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if N < 0.849999999999999978

   1. Initial program 100.0%

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

   2. Taylor expanded in N around 0 98.1%

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

      1. neg-mul-1 98.1%

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

      2. unsub-neg 98.1%

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

   4. Simplified 98.1%

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

   if 0.849999999999999978 < N

   1. Initial program 8.8%

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

   2. Taylor expanded in N around inf 99.9%

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

      1. +-commutative 99.9%

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

      2. associate-*r/ 99.9%

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

      3. metadata-eval 99.9%

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

      4. +-commutative 99.9%

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

      5. unpow2 99.9%

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

      6. associate-*r/ 99.9%

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

      7. metadata-eval 99.9%

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

      8. associate-*r/ 99.9%

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

      9. metadata-eval 99.9%

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

   4. Simplified 99.9%

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

      1. metadata-eval 99.9%

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

      2. metadata-eval 99.9%

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

      3. pow-prod-up 99.9%

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

      4. pow-prod-down 99.9%

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

      5. pow2 99.9%

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

      6. frac-times 99.9%

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

   6. Applied egg-rr 99.9%

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

      1. associate--r+ 99.9%

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

      2. cancel-sign-sub-inv 99.9%

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

      3. +-commutative 99.9%

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

      4. associate--l+ 99.9%

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

      5. +-commutative 99.9%

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

      6. associate-/r* 99.9%

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

      7. sub-div 99.9%

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

      8. div-inv 99.9%

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

      9. pow-flip 99.9%

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

      10. metadata-eval 99.9%

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

      11. *-commutative 99.9%

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

      12. distribute-neg-frac 99.9%

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

      13. metadata-eval 99.9%

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

   8. Applied egg-rr 99.9%

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

      1. associate-+l+ 99.9%

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

      2. associate-*r/ 99.9%

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

      3. associate-*l/ 99.9%

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

      4. metadata-eval 99.9%

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

   10. Simplified 99.9%

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

       1. metadata-eval 99.9%

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

       2. pow-flip 99.9%

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

       3. div-inv 99.9%

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

       4. unpow3 99.9%

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

       5. associate-/r* 99.9%

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

   12. Applied egg-rr 99.9%

       \[\leadsto \frac{1 - \frac{0.5}{N}}{N} + \left(\color{blue}{\frac{\frac{0.3333333333333333}{N \cdot N}}{N}} + \frac{\frac{-0.25}{N \cdot N}}{N \cdot N}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;N \leq 0.85:\\ \;\;\;\;N - \log N\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{0.5}{N}}{N} + \left(\frac{\frac{-0.25}{N \cdot N}}{N \cdot N} + \frac{\frac{0.3333333333333333}{N \cdot N}}{N}\right)\\ \end{array} \]

Alternative 5: 98.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;N \leq 0.72:\\ \;\;\;\;-\log N\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{0.5}{N}}{N} + \left(\frac{\frac{-0.25}{N \cdot N}}{N \cdot N} + \frac{\frac{0.3333333333333333}{N \cdot N}}{N}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= N 0.72)
   (- (log N))
   (+
    (/ (- 1.0 (/ 0.5 N)) N)
    (+ (/ (/ -0.25 (* N N)) (* N N)) (/ (/ 0.3333333333333333 (* N N)) N)))))

C

double code(double N) {
	double tmp;
	if (N <= 0.72) {
		tmp = -log(N);
	} else {
		tmp = ((1.0 - (0.5 / N)) / N) + (((-0.25 / (N * N)) / (N * N)) + ((0.3333333333333333 / (N * N)) / N));
	}
	return tmp;
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= 0.72d0) then
        tmp = -log(n)
    else
        tmp = ((1.0d0 - (0.5d0 / n)) / n) + ((((-0.25d0) / (n * n)) / (n * n)) + ((0.3333333333333333d0 / (n * n)) / n))
    end if
    code = tmp
end function

Java

public static double code(double N) {
	double tmp;
	if (N <= 0.72) {
		tmp = -Math.log(N);
	} else {
		tmp = ((1.0 - (0.5 / N)) / N) + (((-0.25 / (N * N)) / (N * N)) + ((0.3333333333333333 / (N * N)) / N));
	}
	return tmp;
}

Python

def code(N):
	tmp = 0
	if N <= 0.72:
		tmp = -math.log(N)
	else:
		tmp = ((1.0 - (0.5 / N)) / N) + (((-0.25 / (N * N)) / (N * N)) + ((0.3333333333333333 / (N * N)) / N))
	return tmp

Julia

function code(N)
	tmp = 0.0
	if (N <= 0.72)
		tmp = Float64(-log(N));
	else
		tmp = Float64(Float64(Float64(1.0 - Float64(0.5 / N)) / N) + Float64(Float64(Float64(-0.25 / Float64(N * N)) / Float64(N * N)) + Float64(Float64(0.3333333333333333 / Float64(N * N)) / N)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(N)
	tmp = 0.0;
	if (N <= 0.72)
		tmp = -log(N);
	else
		tmp = ((1.0 - (0.5 / N)) / N) + (((-0.25 / (N * N)) / (N * N)) + ((0.3333333333333333 / (N * N)) / N));
	end
	tmp_2 = tmp;
end

Wolfram

code[N_] := If[LessEqual[N, 0.72], (-N[Log[N], $MachinePrecision]), N[(N[(N[(1.0 - N[(0.5 / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision] + N[(N[(N[(-0.25 / N[(N * N), $MachinePrecision]), $MachinePrecision] / N[(N * N), $MachinePrecision]), $MachinePrecision] + N[(N[(0.3333333333333333 / N[(N * N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if N < 0.71999999999999997

   1. Initial program 100.0%

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

   2. Taylor expanded in N around 0 96.7%

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

      1. neg-mul-1 96.7%

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

   4. Simplified 96.7%

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

   if 0.71999999999999997 < N

   1. Initial program 8.8%

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

   2. Taylor expanded in N around inf 99.9%

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

      1. +-commutative 99.9%

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

      2. associate-*r/ 99.9%

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

      3. metadata-eval 99.9%

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

      4. +-commutative 99.9%

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

      5. unpow2 99.9%

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

      6. associate-*r/ 99.9%

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

      7. metadata-eval 99.9%

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

      8. associate-*r/ 99.9%

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

      9. metadata-eval 99.9%

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

   4. Simplified 99.9%

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

      1. metadata-eval 99.9%

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

      2. metadata-eval 99.9%

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

      3. pow-prod-up 99.9%

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

      4. pow-prod-down 99.9%

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

      5. pow2 99.9%

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

      6. frac-times 99.9%

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

   6. Applied egg-rr 99.9%

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

      1. associate--r+ 99.9%

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

      2. cancel-sign-sub-inv 99.9%

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

      3. +-commutative 99.9%

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

      4. associate--l+ 99.9%

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

      5. +-commutative 99.9%

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

      6. associate-/r* 99.9%

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

      7. sub-div 99.9%

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

      8. div-inv 99.9%

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

      9. pow-flip 99.9%

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

      10. metadata-eval 99.9%

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

      11. *-commutative 99.9%

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

      12. distribute-neg-frac 99.9%

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

      13. metadata-eval 99.9%

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

   8. Applied egg-rr 99.9%

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

      1. associate-+l+ 99.9%

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

      2. associate-*r/ 99.9%

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

      3. associate-*l/ 99.9%

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

      4. metadata-eval 99.9%

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

   10. Simplified 99.9%

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

       1. metadata-eval 99.9%

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

       2. pow-flip 99.9%

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

       3. div-inv 99.9%

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

       4. unpow3 99.9%

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

       5. associate-/r* 99.9%

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

   12. Applied egg-rr 99.9%

       \[\leadsto \frac{1 - \frac{0.5}{N}}{N} + \left(\color{blue}{\frac{\frac{0.3333333333333333}{N \cdot N}}{N}} + \frac{\frac{-0.25}{N \cdot N}}{N \cdot N}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;N \leq 0.72:\\ \;\;\;\;-\log N\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{0.5}{N}}{N} + \left(\frac{\frac{-0.25}{N \cdot N}}{N \cdot N} + \frac{\frac{0.3333333333333333}{N \cdot N}}{N}\right)\\ \end{array} \]

Alternative 6: 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 54.7%

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

2. Taylor expanded in N around inf 51.3%

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

3. Final simplification 51.3%

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

Alternative 7: 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 54.7%

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

2. Taylor expanded in N around 0 51.3%

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

   1. neg-mul-1 51.3%

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

   2. unsub-neg 51.3%

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

4. Simplified 51.3%

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

5. Taylor expanded in N around inf 4.7%

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

6. Final simplification 4.7%

   \[\leadsto N \]

Reproduce

herbie shell --seed 2023297 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1.0)) (log N)))