2log (problem 3.3.6)

Percentage Accurate: 53.2% → 99.9%

Time: 10.2s

Alternatives: 7

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 53.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((log((n + 1.0d0)) - log(n)) <= 0.0002d0) then
        tmp = (0.3333333333333333d0 / (n ** 3.0d0)) + (((1.0d0 / n) * ((n + (-0.5d0)) / n)) + ((-0.25d0) / (n ** 4.0d0)))
    else
        tmp = log(((n + 1.0d0) / n))
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

function tmp_2 = code(N)
	tmp = 0.0;
	if ((log((N + 1.0)) - log(N)) <= 0.0002)
		tmp = (0.3333333333333333 / (N ^ 3.0)) + (((1.0 / N) * ((N + -0.5) / N)) + (-0.25 / (N ^ 4.0)));
	else
		tmp = log(((N + 1.0) / N));
	end
	tmp_2 = tmp;
end

Wolfram

code[N_] := If[LessEqual[N[(N[Log[N[(N + 1.0), $MachinePrecision]], $MachinePrecision] - N[Log[N], $MachinePrecision]), $MachinePrecision], 0.0002], N[(N[(0.3333333333333333 / N[Power[N, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(1.0 / N), $MachinePrecision] * N[(N[(N + -0.5), $MachinePrecision] / N), $MachinePrecision]), $MachinePrecision] + N[(-0.25 / N[Power[N, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[N[(N[(N + 1.0), $MachinePrecision] / N), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 7.4%

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

      1. +-commutative 7.4%

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

      2. log1p-def 7.4%

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

   3. Simplified 7.4%

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

      1. add-log-exp 7.4%

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

      2. log1p-expm1-u 7.4%

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

      3. log1p-udef 7.4%

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

      4. diff-log 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 5.8%

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

      7. +-commutative 5.8%

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

      8. add-exp-log 5.8%

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

      9. log1p-udef 5.8%

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

      10. log1p-expm1-u 5.8%

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

      11. add-exp-log 7.7%

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

   6. Applied egg-rr 7.7%

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

   7. Taylor expanded in N around inf 100.0%

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

   9. Simplified 54.6%

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

       1. *-un-lft-identity 54.6%

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

       2. unpow2 54.6%

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

       3. times-frac 100.0%

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

   11. Applied egg-rr 100.0%

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

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

   1. Initial program 100.0%

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

      1. +-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. Add Preprocessing
   5. Step-by-step derivation

      1. add-log-exp 100.0%

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

      2. log1p-expm1-u 10.1%

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

      3. log1p-udef 10.1%

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

      4. diff-log 10.1%

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

      5. log1p-udef 10.1%

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

      6. rem-exp-log 10.1%

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

      7. +-commutative 10.1%

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

      8. add-exp-log 10.1%

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

      9. log1p-udef 10.1%

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

      10. log1p-expm1-u 100.0%

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

      11. add-exp-log 100.0%

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

   6. Applied egg-rr 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 99.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\frac{0.3333333333333333}{{N}^{3}} + \frac{1}{N} \cdot \frac{N + -0.5}{N}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= (- (log (+ N 1.0)) (log N)) 2e-6)
   (+ (/ 0.3333333333333333 (pow N 3.0)) (* (/ 1.0 N) (/ (+ N -0.5) N)))
   (log (/ (+ N 1.0) N))))

C

double code(double N) {
	double tmp;
	if ((log((N + 1.0)) - log(N)) <= 2e-6) {
		tmp = (0.3333333333333333 / pow(N, 3.0)) + ((1.0 / N) * ((N + -0.5) / 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)) <= 2d-6) then
        tmp = (0.3333333333333333d0 / (n ** 3.0d0)) + ((1.0d0 / n) * ((n + (-0.5d0)) / 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)) <= 2e-6) {
		tmp = (0.3333333333333333 / Math.pow(N, 3.0)) + ((1.0 / N) * ((N + -0.5) / 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)) <= 2e-6:
		tmp = (0.3333333333333333 / math.pow(N, 3.0)) + ((1.0 / N) * ((N + -0.5) / 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)) <= 2e-6)
		tmp = Float64(Float64(0.3333333333333333 / (N ^ 3.0)) + Float64(Float64(1.0 / N) * Float64(Float64(N + -0.5) / 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)) <= 2e-6)
		tmp = (0.3333333333333333 / (N ^ 3.0)) + ((1.0 / N) * ((N + -0.5) / 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], 2e-6], N[(N[(0.3333333333333333 / N[Power[N, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / N), $MachinePrecision] * N[(N[(N + -0.5), $MachinePrecision] / N), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[N[(N[(N + 1.0), $MachinePrecision] / N), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (log.f64 (+.f64 N 1)) (log.f64 N)) < 1.99999999999999991e-6

   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. Add Preprocessing
   5. Step-by-step derivation

      1. add-log-exp 6.8%

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

      2. log1p-expm1-u 6.8%

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

      3. log1p-udef 6.8%

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

      4. diff-log 6.7%

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

      5. log1p-udef 6.7%

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

      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 7.1%

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

   6. Applied egg-rr 7.1%

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

   7. Taylor expanded in N around inf 100.0%

      \[\leadsto \color{blue}{\left(0.3333333333333333 \cdot \frac{1}{{N}^{3}} + \frac{1}{N}\right) - 0.5 \cdot \frac{1}{{N}^{2}}} \]
   8. 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. *-inverses 51.0%

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

      5. associate-/r* 34.6%

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

      6. *-commutative 34.6%

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

      7. *-rgt-identity 34.6%

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

      8. associate-*r/ 34.6%

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

      9. metadata-eval 34.6%

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

      10. *-inverses 34.6%

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

      11. times-frac 34.7%

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

      12. neg-mul-1 34.7%

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

      13. associate-*r* 34.7%

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

      14. metadata-eval 34.7%

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

      15. *-commutative 34.7%

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

      16. *-commutative 34.7%

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

      17. distribute-lft-neg-out 34.7%

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

      18. distribute-rgt-neg-out 34.7%

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

   9. Simplified 54.3%

      \[\leadsto \color{blue}{\frac{0.3333333333333333}{{N}^{3}} + \frac{N + -0.5}{{N}^{2}}} \]
   10. Step-by-step derivation

       1. *-un-lft-identity 54.3%

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

       2. unpow2 54.3%

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

       3. times-frac 100.0%

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

   11. Applied egg-rr 100.0%

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

   if 1.99999999999999991e-6 < (-.f64 (log.f64 (+.f64 N 1)) (log.f64 N))

   1. Initial program 99.8%

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

      1. +-commutative 99.8%

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

      2. log1p-def 99.8%

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

   3. Simplified 99.8%

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

      1. add-log-exp 99.8%

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

      2. log1p-expm1-u 10.7%

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

      3. log1p-udef 10.7%

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

      4. diff-log 10.7%

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

      5. log1p-udef 10.7%

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

      6. rem-exp-log 10.7%

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

      7. +-commutative 10.7%

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

      8. add-exp-log 10.7%

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

      9. log1p-udef 10.7%

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

      10. log1p-expm1-u 99.8%

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

      11. add-exp-log 99.9%

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

   6. Applied egg-rr 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\frac{0.3333333333333333}{{N}^{3}} + \frac{1}{N} \cdot \frac{N + -0.5}{N}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;N \leq 160000:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{N + -0.5}{N}}{N}\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= N 160000.0) (log (/ (+ N 1.0) N)) (/ (/ (+ N -0.5) N) N)))

C

double code(double N) {
	double tmp;
	if (N <= 160000.0) {
		tmp = log(((N + 1.0) / N));
	} else {
		tmp = ((N + -0.5) / N) / N;
	}
	return tmp;
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= 160000.0d0) then
        tmp = log(((n + 1.0d0) / n))
    else
        tmp = ((n + (-0.5d0)) / n) / n
    end if
    code = tmp
end function

Java

public static double code(double N) {
	double tmp;
	if (N <= 160000.0) {
		tmp = Math.log(((N + 1.0) / N));
	} else {
		tmp = ((N + -0.5) / N) / N;
	}
	return tmp;
}

Python

def code(N):
	tmp = 0
	if N <= 160000.0:
		tmp = math.log(((N + 1.0) / N))
	else:
		tmp = ((N + -0.5) / N) / N
	return tmp

Julia

function code(N)
	tmp = 0.0
	if (N <= 160000.0)
		tmp = log(Float64(Float64(N + 1.0) / N));
	else
		tmp = Float64(Float64(Float64(N + -0.5) / N) / N);
	end
	return tmp
end

MATLAB

function tmp_2 = code(N)
	tmp = 0.0;
	if (N <= 160000.0)
		tmp = log(((N + 1.0) / N));
	else
		tmp = ((N + -0.5) / N) / N;
	end
	tmp_2 = tmp;
end

Wolfram

code[N_] := If[LessEqual[N, 160000.0], N[Log[N[(N[(N + 1.0), $MachinePrecision] / N), $MachinePrecision]], $MachinePrecision], N[(N[(N[(N + -0.5), $MachinePrecision] / N), $MachinePrecision] / N), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if N < 1.6e5

   1. Initial program 99.8%

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

      1. +-commutative 99.8%

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

      2. log1p-def 99.8%

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

   3. Simplified 99.8%

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

      1. add-log-exp 99.8%

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

      2. log1p-expm1-u 10.7%

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

      3. log1p-udef 10.7%

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

      4. diff-log 10.7%

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

      5. log1p-udef 10.7%

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

      6. rem-exp-log 10.7%

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

      7. +-commutative 10.7%

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

      8. add-exp-log 10.7%

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

      9. log1p-udef 10.7%

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

      10. log1p-expm1-u 99.8%

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

      11. add-exp-log 99.9%

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

   6. Applied egg-rr 99.9%

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

   if 1.6e5 < 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. Add Preprocessing

   5. Taylor expanded in N around inf 99.7%

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

      1. associate-*r/ 99.7%

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

      2. metadata-eval 99.7%

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

   7. Simplified 99.7%

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

      1. frac-sub 34.4%

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

      2. *-un-lft-identity 34.4%

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

      3. unpow2 34.4%

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

      4. distribute-lft-out-- 34.4%

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

      5. unpow2 34.4%

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

      6. cube-unmult 34.3%

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

   9. Applied egg-rr 34.3%

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

       1. *-un-lft-identity 34.3%

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

       2. unpow3 34.4%

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

       3. unpow2 34.4%

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

       4. times-frac 50.5%

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

       5. pow-flip 50.6%

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

       6. metadata-eval 50.6%

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

       7. sub-neg 50.6%

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

       8. metadata-eval 50.6%

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

   11. Applied egg-rr 50.6%

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

       1. associate-/l* 54.8%

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

   13. Simplified 54.8%

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

       1. associate-*r/ 54.8%

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

       2. pow-plus 99.7%

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

       3. metadata-eval 99.7%

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

       4. inv-pow 99.7%

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

       5. associate-/l/ 99.7%

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

       6. associate-/r* 99.7%

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

       7. clear-num 99.7%

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

   15. Applied egg-rr 99.7%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;N \leq 160000:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{N + -0.5}{N}}{N}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if N < 0.900000000000000022

   1. Initial program 100.0%

      \[\log \left(N + 1\right) - \log N \]
   2. 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. Add Preprocessing

   5. Taylor expanded in N around 0 97.2%

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

      1. neg-mul-1 97.2%

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

      2. unsub-neg 97.2%

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

   7. Simplified 97.2%

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

   if 0.900000000000000022 < N

   1. Initial program 7.4%

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

      1. +-commutative 7.4%

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

      2. log1p-def 7.4%

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

   3. Simplified 7.4%

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

   5. Taylor expanded in N around inf 99.5%

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

      1. associate-*r/ 99.5%

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

      2. metadata-eval 99.5%

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

   7. Simplified 99.5%

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

      1. frac-sub 34.6%

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

      2. *-un-lft-identity 34.6%

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

      3. unpow2 34.6%

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

      4. distribute-lft-out-- 34.6%

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

      5. unpow2 34.6%

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

      6. cube-unmult 34.5%

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

   9. Applied egg-rr 34.5%

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

       1. *-un-lft-identity 34.5%

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

       2. unpow3 34.6%

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

       3. unpow2 34.6%

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

       4. times-frac 50.6%

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

       5. pow-flip 50.7%

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

       6. metadata-eval 50.7%

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

       7. sub-neg 50.7%

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

       8. metadata-eval 50.7%

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

   11. Applied egg-rr 50.7%

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

       1. associate-/l* 54.8%

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

   13. Simplified 54.8%

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

       1. associate-*r/ 54.8%

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

       2. pow-plus 99.5%

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

       3. metadata-eval 99.5%

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

       4. inv-pow 99.5%

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

       5. associate-/l/ 99.5%

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

       6. associate-/r* 99.5%

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

       7. clear-num 99.5%

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

   15. Applied egg-rr 99.5%

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

4. Final simplification 98.4%

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

Alternative 5: 98.6% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(N):
	tmp = 0
	if N <= 0.68:
		tmp = -math.log(N)
	else:
		tmp = ((N + -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(Float64(N + -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 = ((N + -0.5) / N) / N;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if N < 0.680000000000000049

   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. Add Preprocessing

   5. Taylor expanded in N around 0 95.6%

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

      1. neg-mul-1 95.6%

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

   7. Simplified 95.6%

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

   if 0.680000000000000049 < N

   1. Initial program 7.4%

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

      1. +-commutative 7.4%

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

      2. log1p-def 7.4%

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

   3. Simplified 7.4%

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

   5. Taylor expanded in N around inf 99.5%

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

      1. associate-*r/ 99.5%

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

      2. metadata-eval 99.5%

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

   7. Simplified 99.5%

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

      1. frac-sub 34.6%

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

      2. *-un-lft-identity 34.6%

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

      3. unpow2 34.6%

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

      4. distribute-lft-out-- 34.6%

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

      5. unpow2 34.6%

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

      6. cube-unmult 34.5%

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

   9. Applied egg-rr 34.5%

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

       1. *-un-lft-identity 34.5%

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

       2. unpow3 34.6%

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

       3. unpow2 34.6%

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

       4. times-frac 50.6%

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

       5. pow-flip 50.7%

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

       6. metadata-eval 50.7%

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

       7. sub-neg 50.7%

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

       8. metadata-eval 50.7%

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

   11. Applied egg-rr 50.7%

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

       1. associate-/l* 54.8%

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

   13. Simplified 54.8%

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

       1. associate-*r/ 54.8%

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

       2. pow-plus 99.5%

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

       3. metadata-eval 99.5%

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

       4. inv-pow 99.5%

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

       5. associate-/l/ 99.5%

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

       6. associate-/r* 99.5%

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

       7. clear-num 99.5%

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

   15. Applied egg-rr 99.5%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;N \leq 0.68:\\ \;\;\;\;-\log N\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{N + -0.5}{N}}{N}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 52.6% accurate, 68.3× speedup

Math

\[\begin{array}{l} \\ \frac{1}{N} \end{array} \]

FPCore

(FPCore (N) :precision binary64 (/ 1.0 N))

C

double code(double N) {
	return 1.0 / N;
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    code = 1.0d0 / n
end function

Java

public static double code(double N) {
	return 1.0 / N;
}

Python

def code(N):
	return 1.0 / N

Julia

function code(N)
	return Float64(1.0 / N)
end

MATLAB

function tmp = code(N)
	tmp = 1.0 / N;
end

Wolfram

code[N_] := N[(1.0 / N), $MachinePrecision]

Derivation

1. Initial program 48.6%

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

   1. +-commutative 48.6%

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

   2. log1p-def 48.6%

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

3. Simplified 48.6%

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

5. Taylor expanded in N around inf 57.2%

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

6. Final simplification 57.2%

   \[\leadsto \frac{1}{N} \]
7. Add Preprocessing

Alternative 7: 4.5% accurate, 205.0× speedup

Math

\[\begin{array}{l} \\ N \end{array} \]

FPCore

(FPCore (N) :precision binary64 N)

C

double code(double N) {
	return N;
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    code = n
end function

Java

public static double code(double N) {
	return N;
}

Python

def code(N):
	return N

Julia

function code(N)
	return N
end

MATLAB

function tmp = code(N)
	tmp = N;
end

Wolfram

code[N_] := N

Derivation

1. Initial program 48.6%

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

   1. +-commutative 48.6%

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

   2. log1p-def 48.6%

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

3. Simplified 48.6%

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

5. Taylor expanded in N around 0 45.4%

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

   1. neg-mul-1 45.4%

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

   2. unsub-neg 45.4%

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

7. Simplified 45.4%

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

8. Taylor expanded in N around inf 4.6%

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

9. Final simplification 4.6%

   \[\leadsto N \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024020 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1.0)) (log N)))