2log (problem 3.3.6)

Percentage Accurate: 54.3% → 100.0%

Time: 6.4s

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: 54.3% 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: 100.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 0.0005:\\ \;\;\;\;\frac{0.3333333333333333}{{N}^{3}} + \left(\frac{\frac{N + -0.5}{N}}{N} + \frac{-0.25}{{N}^{4}}\right)\\ \mathbf{else}:\\ \;\;\;\;-\log \left(\frac{N}{N + 1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= (- (log (+ N 1.0)) (log N)) 0.0005)
   (+
    (/ 0.3333333333333333 (pow N 3.0))
    (+ (/ (/ (+ N -0.5) N) N) (/ -0.25 (pow N 4.0))))
   (- (log (/ N (+ N 1.0))))))

C

double code(double N) {
	double tmp;
	if ((log((N + 1.0)) - log(N)) <= 0.0005) {
		tmp = (0.3333333333333333 / pow(N, 3.0)) + ((((N + -0.5) / N) / N) + (-0.25 / pow(N, 4.0)));
	} else {
		tmp = -log((N / (N + 1.0)));
	}
	return tmp;
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((log((n + 1.0d0)) - log(n)) <= 0.0005d0) then
        tmp = (0.3333333333333333d0 / (n ** 3.0d0)) + ((((n + (-0.5d0)) / n) / n) + ((-0.25d0) / (n ** 4.0d0)))
    else
        tmp = -log((n / (n + 1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double N) {
	double tmp;
	if ((Math.log((N + 1.0)) - Math.log(N)) <= 0.0005) {
		tmp = (0.3333333333333333 / Math.pow(N, 3.0)) + ((((N + -0.5) / N) / N) + (-0.25 / Math.pow(N, 4.0)));
	} else {
		tmp = -Math.log((N / (N + 1.0)));
	}
	return tmp;
}

Python

def code(N):
	tmp = 0
	if (math.log((N + 1.0)) - math.log(N)) <= 0.0005:
		tmp = (0.3333333333333333 / math.pow(N, 3.0)) + ((((N + -0.5) / N) / N) + (-0.25 / math.pow(N, 4.0)))
	else:
		tmp = -math.log((N / (N + 1.0)))
	return tmp

Julia

function code(N)
	tmp = 0.0
	if (Float64(log(Float64(N + 1.0)) - log(N)) <= 0.0005)
		tmp = Float64(Float64(0.3333333333333333 / (N ^ 3.0)) + Float64(Float64(Float64(Float64(N + -0.5) / N) / N) + Float64(-0.25 / (N ^ 4.0))));
	else
		tmp = Float64(-log(Float64(N / Float64(N + 1.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(N)
	tmp = 0.0;
	if ((log((N + 1.0)) - log(N)) <= 0.0005)
		tmp = (0.3333333333333333 / (N ^ 3.0)) + ((((N + -0.5) / N) / N) + (-0.25 / (N ^ 4.0)));
	else
		tmp = -log((N / (N + 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[N_] := If[LessEqual[N[(N[Log[N[(N + 1.0), $MachinePrecision]], $MachinePrecision] - N[Log[N], $MachinePrecision]), $MachinePrecision], 0.0005], N[(N[(0.3333333333333333 / N[Power[N, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(N + -0.5), $MachinePrecision] / N), $MachinePrecision] / N), $MachinePrecision] + N[(-0.25 / N[Power[N, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-N[Log[N[(N / N[(N + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 6.3%

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

      1. +-commutative 6.3%

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

      2. log1p-def 6.3%

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

   3. Simplified 6.3%

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

      1. add-log-exp 6.3%

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

      2. log1p-expm1-u 6.3%

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

      3. log1p-udef 6.3%

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

      4. diff-log 5.8%

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

      5. log1p-udef 5.8%

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

      6. rem-exp-log 4.6%

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

      7. +-commutative 4.6%

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

      8. add-exp-log 4.6%

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

      9. log1p-udef 4.6%

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

      10. log1p-expm1-u 4.6%

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

      11. add-exp-log 6.6%

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

   5. Applied egg-rr 6.6%

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

   6. Taylor expanded in N around inf 99.9%

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

   8. Simplified 52.3%

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

      1. *-un-lft-identity 52.3%

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

      2. unpow2 52.3%

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

      3. times-frac 99.9%

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

   10. Applied egg-rr 99.9%

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

       1. associate-*l/ 100.0%

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

       2. *-un-lft-identity 100.0%

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

   12. Applied egg-rr 100.0%

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

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

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. log1p-def 99.9%

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

   3. Simplified 99.9%

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

      1. add-log-exp 99.9%

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

      2. log1p-expm1-u 8.2%

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

      3. log1p-udef 8.2%

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

      4. diff-log 8.2%

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

      5. log1p-udef 8.2%

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

      6. rem-exp-log 8.3%

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

      7. +-commutative 8.3%

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

      8. add-exp-log 8.3%

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

      9. log1p-udef 8.3%

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

      10. log1p-expm1-u 99.9%

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

      11. add-exp-log 99.9%

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

   5. Applied egg-rr 99.9%

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

      1. clear-num 99.9%

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

      2. log-rec 100.0%

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

   7. Applied egg-rr 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 0.0005:\\ \;\;\;\;\frac{0.3333333333333333}{{N}^{3}} + \left(\frac{\frac{N + -0.5}{N}}{N} + \frac{-0.25}{{N}^{4}}\right)\\ \mathbf{else}:\\ \;\;\;\;-\log \left(\frac{N}{N + 1}\right)\\ \end{array} \]

Alternative 2: 99.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 0:\\ \;\;\;\;\frac{\frac{N + -0.5}{N}}{N}\\ \mathbf{else}:\\ \;\;\;\;-\log \left(\frac{N}{N + 1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= (- (log (+ N 1.0)) (log N)) 0.0)
   (/ (/ (+ N -0.5) N) N)
   (- (log (/ N (+ N 1.0))))))

C

double code(double N) {
	double tmp;
	if ((log((N + 1.0)) - log(N)) <= 0.0) {
		tmp = ((N + -0.5) / N) / N;
	} else {
		tmp = -log((N / (N + 1.0)));
	}
	return tmp;
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((log((n + 1.0d0)) - log(n)) <= 0.0d0) then
        tmp = ((n + (-0.5d0)) / n) / n
    else
        tmp = -log((n / (n + 1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double N) {
	double tmp;
	if ((Math.log((N + 1.0)) - Math.log(N)) <= 0.0) {
		tmp = ((N + -0.5) / N) / N;
	} else {
		tmp = -Math.log((N / (N + 1.0)));
	}
	return tmp;
}

Python

def code(N):
	tmp = 0
	if (math.log((N + 1.0)) - math.log(N)) <= 0.0:
		tmp = ((N + -0.5) / N) / N
	else:
		tmp = -math.log((N / (N + 1.0)))
	return tmp

Julia

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

MATLAB

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

Wolfram

code[N_] := If[LessEqual[N[(N[Log[N[(N + 1.0), $MachinePrecision]], $MachinePrecision] - N[Log[N], $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(N[(N + -0.5), $MachinePrecision] / N), $MachinePrecision] / N), $MachinePrecision], (-N[Log[N[(N / N[(N + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (log.f64 (+.f64 N 1)) (log.f64 N)) < 0.0

   1. Initial program 5.7%

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

      1. +-commutative 5.7%

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

      2. log1p-def 5.7%

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

   3. Simplified 5.7%

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

      1. add-log-exp 5.7%

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

      2. log1p-expm1-u 5.7%

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

      3. log1p-udef 5.7%

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

      4. diff-log 5.2%

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

      5. log1p-udef 5.2%

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

      6. rem-exp-log 4.0%

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

      7. +-commutative 4.0%

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

      8. add-exp-log 4.0%

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

      9. log1p-udef 4.0%

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

      10. log1p-expm1-u 4.0%

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

      11. add-exp-log 6.0%

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

   5. Applied egg-rr 6.0%

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

   6. Taylor expanded in N around inf 100.0%

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

      1. *-rgt-identity 100.0%

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

      2. *-rgt-identity 100.0%

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

      3. *-inverses 48.3%

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

      4. associate-/r* 35.6%

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

      5. *-commutative 35.6%

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

      6. *-lft-identity 35.6%

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

      7. *-inverses 35.6%

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

      8. associate-*r/ 35.6%

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

      9. metadata-eval 35.6%

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

      10. times-frac 35.7%

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

      11. distribute-lft-neg-out 35.7%

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

      12. distribute-rgt-neg-out 35.7%

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

      13. metadata-eval 35.7%

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

      14. distribute-rgt-neg-in 35.7%

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

      15. distribute-lft-neg-out 35.7%

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

      16. remove-double-neg 35.7%

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

   8. Simplified 52.0%

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

      1. *-un-lft-identity 52.0%

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

      2. unpow2 52.0%

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

      3. times-frac 100.0%

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

   10. Applied egg-rr 100.0%

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

       1. associate-*l/ 100.0%

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

       2. *-un-lft-identity 100.0%

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

   12. Applied egg-rr 100.0%

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

   if 0.0 < (-.f64 (log.f64 (+.f64 N 1)) (log.f64 N))

   1. Initial program 99.7%

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

      1. +-commutative 99.7%

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

      2. log1p-def 99.7%

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

   3. Simplified 99.7%

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

      1. add-log-exp 99.7%

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

      2. log1p-expm1-u 8.8%

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

      3. log1p-udef 8.8%

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

      4. diff-log 8.8%

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

      5. log1p-udef 8.8%

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

      6. rem-exp-log 8.8%

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

      7. +-commutative 8.8%

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

      8. add-exp-log 8.8%

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

      9. log1p-udef 8.8%

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

      10. log1p-expm1-u 99.8%

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

      11. add-exp-log 99.8%

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

   5. Applied egg-rr 99.8%

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

      1. clear-num 99.8%

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

      2. log-rec 99.9%

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

   7. Applied egg-rr 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 0:\\ \;\;\;\;\frac{\frac{N + -0.5}{N}}{N}\\ \mathbf{else}:\\ \;\;\;\;-\log \left(\frac{N}{N + 1}\right)\\ \end{array} \]

Alternative 3: 99.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;N \leq 205000:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{N + -0.5}{N}}{N}\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= N 205000.0) (log (/ (+ N 1.0) N)) (/ (/ (+ N -0.5) N) N)))

C

double code(double N) {
	double tmp;
	if (N <= 205000.0) {
		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 <= 205000.0d0) 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 <= 205000.0) {
		tmp = Math.log(((N + 1.0) / N));
	} else {
		tmp = ((N + -0.5) / N) / N;
	}
	return tmp;
}

Python

def code(N):
	tmp = 0
	if N <= 205000.0:
		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 <= 205000.0)
		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 <= 205000.0)
		tmp = log(((N + 1.0) / N));
	else
		tmp = ((N + -0.5) / N) / N;
	end
	tmp_2 = tmp;
end

Wolfram

code[N_] := If[LessEqual[N, 205000.0], 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 < 205000

   1. Initial program 99.7%

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

      1. +-commutative 99.7%

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

      2. log1p-def 99.7%

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

   3. Simplified 99.7%

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

      1. add-log-exp 99.7%

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

      2. log1p-expm1-u 8.8%

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

      3. log1p-udef 8.8%

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

      4. diff-log 8.8%

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

      5. log1p-udef 8.8%

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

      6. rem-exp-log 8.8%

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

      7. +-commutative 8.8%

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

      8. add-exp-log 8.8%

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

      9. log1p-udef 8.8%

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

      10. log1p-expm1-u 99.8%

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

      11. add-exp-log 99.8%

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

   5. Applied egg-rr 99.8%

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

   if 205000 < N

   1. Initial program 5.7%

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

      1. +-commutative 5.7%

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

      2. log1p-def 5.7%

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

   3. Simplified 5.7%

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

      1. add-log-exp 5.7%

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

      2. log1p-expm1-u 5.7%

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

      3. log1p-udef 5.7%

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

      4. diff-log 5.2%

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

      5. log1p-udef 5.2%

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

      6. rem-exp-log 4.0%

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

      7. +-commutative 4.0%

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

      8. add-exp-log 4.0%

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

      9. log1p-udef 4.0%

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

      10. log1p-expm1-u 4.0%

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

      11. add-exp-log 6.0%

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

   5. Applied egg-rr 6.0%

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

   6. Taylor expanded in N around inf 100.0%

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

      1. *-rgt-identity 100.0%

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

      2. *-rgt-identity 100.0%

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

      3. *-inverses 48.3%

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

      4. associate-/r* 35.6%

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

      5. *-commutative 35.6%

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

      6. *-lft-identity 35.6%

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

      7. *-inverses 35.6%

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

      8. associate-*r/ 35.6%

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

      9. metadata-eval 35.6%

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

      10. times-frac 35.7%

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

      11. distribute-lft-neg-out 35.7%

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

      12. distribute-rgt-neg-out 35.7%

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

      13. metadata-eval 35.7%

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

      14. distribute-rgt-neg-in 35.7%

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

      15. distribute-lft-neg-out 35.7%

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

      16. remove-double-neg 35.7%

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

   8. Simplified 52.0%

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

      1. *-un-lft-identity 52.0%

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

      2. unpow2 52.0%

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

      3. times-frac 100.0%

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

   10. Applied egg-rr 100.0%

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

       1. associate-*l/ 100.0%

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

       2. *-un-lft-identity 100.0%

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

   12. Applied egg-rr 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;N \leq 205000:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{N + -0.5}{N}}{N}\\ \end{array} \]

Alternative 4: 99.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if N < 0.900000000000000022

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. log1p-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in N around 0 98.7%

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

      1. neg-mul-1 98.7%

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

      2. unsub-neg 98.7%

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

   6. Simplified 98.7%

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

   if 0.900000000000000022 < N

   1. Initial program 6.9%

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

      1. +-commutative 6.9%

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

      2. log1p-def 6.9%

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

   3. Simplified 6.9%

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

      1. add-log-exp 6.9%

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

      2. log1p-expm1-u 6.9%

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

      3. log1p-udef 6.9%

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

      4. diff-log 6.5%

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

      5. log1p-udef 6.4%

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

      6. rem-exp-log 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 7.3%

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

   5. Applied egg-rr 7.3%

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

   6. Taylor expanded in N around inf 99.2%

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

      1. *-rgt-identity 99.2%

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

      2. *-rgt-identity 99.2%

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

      3. *-inverses 48.4%

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

      4. associate-/r* 35.9%

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

      5. *-commutative 35.9%

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

      6. *-lft-identity 35.9%

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

      7. *-inverses 35.9%

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

      8. associate-*r/ 35.9%

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

      9. metadata-eval 35.9%

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

      10. times-frac 35.9%

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

      11. distribute-lft-neg-out 35.9%

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

      12. distribute-rgt-neg-out 35.9%

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

      13. metadata-eval 35.9%

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

      14. distribute-rgt-neg-in 35.9%

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

      15. distribute-lft-neg-out 35.9%

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

      16. remove-double-neg 35.9%

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

   8. Simplified 51.9%

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

      1. *-un-lft-identity 52.4%

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

      2. unpow2 52.4%

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

      3. times-frac 99.6%

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

   10. Applied egg-rr 99.2%

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

       1. associate-*l/ 99.6%

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

       2. *-un-lft-identity 99.6%

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

   12. Applied egg-rr 99.2%

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

4. Final simplification 98.9%

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

Alternative 5: 98.7% accurate, 2.0× 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. Taylor expanded in N around 0 97.4%

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

      1. neg-mul-1 97.4%

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

   6. Simplified 97.4%

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

   if 0.680000000000000049 < N

   1. Initial program 6.9%

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

      1. +-commutative 6.9%

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

      2. log1p-def 6.9%

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

   3. Simplified 6.9%

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

      1. add-log-exp 6.9%

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

      2. log1p-expm1-u 6.9%

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

      3. log1p-udef 6.9%

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

      4. diff-log 6.5%

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

      5. log1p-udef 6.4%

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

      6. rem-exp-log 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 7.3%

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

   5. Applied egg-rr 7.3%

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

   6. Taylor expanded in N around inf 99.2%

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

      1. *-rgt-identity 99.2%

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

      2. *-rgt-identity 99.2%

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

      3. *-inverses 48.4%

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

      4. associate-/r* 35.9%

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

      5. *-commutative 35.9%

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

      6. *-lft-identity 35.9%

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

      7. *-inverses 35.9%

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

      8. associate-*r/ 35.9%

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

      9. metadata-eval 35.9%

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

      10. times-frac 35.9%

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

      11. distribute-lft-neg-out 35.9%

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

      12. distribute-rgt-neg-out 35.9%

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

      13. metadata-eval 35.9%

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

      14. distribute-rgt-neg-in 35.9%

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

      15. distribute-lft-neg-out 35.9%

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

      16. remove-double-neg 35.9%

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

   8. Simplified 51.9%

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

      1. *-un-lft-identity 52.4%

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

      2. unpow2 52.4%

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

      3. times-frac 99.6%

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

   10. Applied egg-rr 99.2%

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

       1. associate-*l/ 99.6%

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

       2. *-un-lft-identity 99.6%

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

   12. Applied egg-rr 99.2%

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

4. Final simplification 98.3%

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

Alternative 6: 51.4% 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 54.9%

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

   1. +-commutative 54.9%

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

   2. log1p-def 54.9%

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

3. Simplified 54.9%

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

4. Taylor expanded in N around inf 50.9%

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

5. Final simplification 50.9%

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

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 54.9%

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

   1. +-commutative 54.9%

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

   2. log1p-def 54.9%

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

3. Simplified 54.9%

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

4. Taylor expanded in N around 0 52.7%

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

   1. neg-mul-1 52.7%

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

   2. unsub-neg 52.7%

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

6. Simplified 52.7%

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

7. Taylor expanded in N around inf 4.7%

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

8. Final simplification 4.7%

   \[\leadsto N \]

Reproduce

herbie shell --seed 2023322 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1.0)) (log N)))