2log (problem 3.3.6)

Percentage Accurate: 54.9% → 99.8%

Time: 11.6s

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: 54.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\frac{0.3333333333333333}{{N}^{3}} + \left(\frac{1}{N} - \frac{0.5}{{N}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;-\log \left(\frac{N}{N + 1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= (- (log (+ N 1.0)) (log N)) 5e-8)
   (+ (/ 0.3333333333333333 (pow N 3.0)) (- (/ 1.0 N) (/ 0.5 (pow N 2.0))))
   (- (log (/ N (+ N 1.0))))))

C

double code(double N) {
	double tmp;
	if ((log((N + 1.0)) - log(N)) <= 5e-8) {
		tmp = (0.3333333333333333 / pow(N, 3.0)) + ((1.0 / N) - (0.5 / pow(N, 2.0)));
	} 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)) <= 5d-8) then
        tmp = (0.3333333333333333d0 / (n ** 3.0d0)) + ((1.0d0 / n) - (0.5d0 / (n ** 2.0d0)))
    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)) <= 5e-8) {
		tmp = (0.3333333333333333 / Math.pow(N, 3.0)) + ((1.0 / N) - (0.5 / Math.pow(N, 2.0)));
	} 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)) <= 5e-8:
		tmp = (0.3333333333333333 / math.pow(N, 3.0)) + ((1.0 / N) - (0.5 / math.pow(N, 2.0)))
	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)) <= 5e-8)
		tmp = Float64(Float64(0.3333333333333333 / (N ^ 3.0)) + Float64(Float64(1.0 / N) - Float64(0.5 / (N ^ 2.0))));
	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)) <= 5e-8)
		tmp = (0.3333333333333333 / (N ^ 3.0)) + ((1.0 / N) - (0.5 / (N ^ 2.0)));
	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], 5e-8], N[(N[(0.3333333333333333 / N[Power[N, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / N), $MachinePrecision] - N[(0.5 / N[Power[N, 2.0], $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)) < 4.9999999999999998e-8

   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. Taylor expanded in N around inf 100.0%

      \[\leadsto \color{blue}{\left(0.3333333333333333 \cdot \frac{1}{{N}^{3}} + \frac{1}{N}\right) - 0.5 \cdot \frac{1}{{N}^{2}}} \]
   5. 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. associate-*r/ 100.0%

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

      5. metadata-eval 100.0%

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

   6. Simplified 100.0%

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

   if 4.9999999999999998e-8 < (-.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%

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

      2. exp-diff 99.9%

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

      3. log1p-udef 99.9%

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

      4. add-exp-log 99.9%

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

      5. +-commutative 99.9%

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

      6. 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-div 100.0%

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

      3. metadata-eval 100.0%

         \[\leadsto \color{blue}{0} - \log \left(\frac{N}{N + 1}\right) \]

   7. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{0 - \log \left(\frac{N}{N + 1}\right)} \]
   8. Step-by-step derivation

      1. sub0-neg 100.0%

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

   9. Simplified 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\frac{0.3333333333333333}{{N}^{3}} + \left(\frac{1}{N} - \frac{0.5}{{N}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;-\log \left(\frac{N}{N + 1}\right)\\ \end{array} \]

Alternative 2: 99.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\frac{1 - \frac{0.5}{N}}{N}\\ \mathbf{else}:\\ \;\;\;\;-\log \left(\frac{N}{N + 1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= (- (log (+ N 1.0)) (log N)) 5e-8)
   (/ (- 1.0 (/ 0.5 N)) N)
   (- (log (/ N (+ N 1.0))))))

C

double code(double N) {
	double tmp;
	if ((log((N + 1.0)) - log(N)) <= 5e-8) {
		tmp = (1.0 - (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)) <= 5d-8) then
        tmp = (1.0d0 - (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)) <= 5e-8) {
		tmp = (1.0 - (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)) <= 5e-8:
		tmp = (1.0 - (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)) <= 5e-8)
		tmp = Float64(Float64(1.0 - Float64(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)) <= 5e-8)
		tmp = (1.0 - (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], 5e-8], N[(N[(1.0 - N[(0.5 / N), $MachinePrecision]), $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)) < 4.9999999999999998e-8

   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. Taylor expanded in N around inf 100.0%

      \[\leadsto \color{blue}{\left(0.3333333333333333 \cdot \frac{1}{{N}^{3}} + \frac{1}{N}\right) - 0.5 \cdot \frac{1}{{N}^{2}}} \]
   5. 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. associate-*r/ 100.0%

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

      5. metadata-eval 100.0%

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

   6. Simplified 100.0%

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

      1. frac-sub 38.6%

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

      2. unpow2 38.6%

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

      3. cube-mult 38.5%

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

      4. div-inv 38.5%

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

      5. *-un-lft-identity 38.5%

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

      6. unpow2 38.5%

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

      7. distribute-lft-out-- 38.5%

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

      8. pow-flip 39.2%

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

      9. metadata-eval 39.2%

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

   8. Applied egg-rr 39.2%

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

      1. *-commutative 39.2%

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

      2. sub-neg 39.2%

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

      3. metadata-eval 39.2%

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

   10. Simplified 39.2%

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

       1. +-commutative 39.2%

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

       2. fma-def 39.2%

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

       3. div-inv 39.2%

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

       4. *-commutative 39.2%

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

       5. pow-flip 39.2%

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

       6. metadata-eval 39.2%

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

   12. Applied egg-rr 39.2%

       \[\leadsto \color{blue}{\mathsf{fma}\left({N}^{-3}, N \cdot \left(N + -0.5\right), {N}^{-3} \cdot 0.3333333333333333\right)} \]
   13. Step-by-step derivation

       1. fma-udef 39.2%

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

       2. distribute-lft-out 39.2%

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

   14. Simplified 39.2%

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

   15. Taylor expanded in N around inf 100.0%

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

       1. associate-*r/ 100.0%

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

       2. metadata-eval 100.0%

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

       3. unpow2 100.0%

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

       4. associate-/r* 100.0%

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

       5. div-sub 100.0%

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

   17. Simplified 100.0%

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

   if 4.9999999999999998e-8 < (-.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%

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

      2. exp-diff 99.9%

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

      3. log1p-udef 99.9%

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

      4. add-exp-log 99.9%

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

      5. +-commutative 99.9%

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

      6. 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-div 100.0%

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

      3. metadata-eval 100.0%

         \[\leadsto \color{blue}{0} - \log \left(\frac{N}{N + 1}\right) \]

   7. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{0 - \log \left(\frac{N}{N + 1}\right)} \]
   8. Step-by-step derivation

      1. sub0-neg 100.0%

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

   9. Simplified 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\frac{1 - \frac{0.5}{N}}{N}\\ \mathbf{else}:\\ \;\;\;\;-\log \left(\frac{N}{N + 1}\right)\\ \end{array} \]

Alternative 3: 99.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;N \leq 190000:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{0.5}{N}}{N}\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= N 190000.0) (log (/ (+ N 1.0) N)) (/ (- 1.0 (/ 0.5 N)) N)))

C

double code(double N) {
	double tmp;
	if (N <= 190000.0) {
		tmp = log(((N + 1.0) / N));
	} else {
		tmp = (1.0 - (0.5 / N)) / N;
	}
	return tmp;
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= 190000.0d0) then
        tmp = log(((n + 1.0d0) / n))
    else
        tmp = (1.0d0 - (0.5d0 / n)) / n
    end if
    code = tmp
end function

Java

public static double code(double N) {
	double tmp;
	if (N <= 190000.0) {
		tmp = Math.log(((N + 1.0) / N));
	} else {
		tmp = (1.0 - (0.5 / N)) / N;
	}
	return tmp;
}

Python

def code(N):
	tmp = 0
	if N <= 190000.0:
		tmp = math.log(((N + 1.0) / N))
	else:
		tmp = (1.0 - (0.5 / N)) / N
	return tmp

Julia

function code(N)
	tmp = 0.0
	if (N <= 190000.0)
		tmp = log(Float64(Float64(N + 1.0) / N));
	else
		tmp = Float64(Float64(1.0 - Float64(0.5 / N)) / N);
	end
	return tmp
end

MATLAB

function tmp_2 = code(N)
	tmp = 0.0;
	if (N <= 190000.0)
		tmp = log(((N + 1.0) / N));
	else
		tmp = (1.0 - (0.5 / N)) / N;
	end
	tmp_2 = tmp;
end

Wolfram

code[N_] := If[LessEqual[N, 190000.0], N[Log[N[(N[(N + 1.0), $MachinePrecision] / N), $MachinePrecision]], $MachinePrecision], N[(N[(1.0 - N[(0.5 / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if N < 1.9e5

   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%

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

      2. exp-diff 99.9%

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

      3. log1p-udef 99.9%

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

      4. add-exp-log 99.9%

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

      5. +-commutative 99.9%

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

      6. add-exp-log 99.9%

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

   5. Applied egg-rr 99.9%

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

   if 1.9e5 < N

   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. Taylor expanded in N around inf 100.0%

      \[\leadsto \color{blue}{\left(0.3333333333333333 \cdot \frac{1}{{N}^{3}} + \frac{1}{N}\right) - 0.5 \cdot \frac{1}{{N}^{2}}} \]
   5. 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. associate-*r/ 100.0%

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

      5. metadata-eval 100.0%

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

   6. Simplified 100.0%

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

      1. frac-sub 38.6%

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

      2. unpow2 38.6%

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

      3. cube-mult 38.5%

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

      4. div-inv 38.5%

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

      5. *-un-lft-identity 38.5%

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

      6. unpow2 38.5%

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

      7. distribute-lft-out-- 38.5%

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

      8. pow-flip 39.2%

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

      9. metadata-eval 39.2%

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

   8. Applied egg-rr 39.2%

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

      1. *-commutative 39.2%

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

      2. sub-neg 39.2%

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

      3. metadata-eval 39.2%

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

   10. Simplified 39.2%

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

       1. +-commutative 39.2%

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

       2. fma-def 39.2%

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

       3. div-inv 39.2%

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

       4. *-commutative 39.2%

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

       5. pow-flip 39.2%

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

       6. metadata-eval 39.2%

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

   12. Applied egg-rr 39.2%

       \[\leadsto \color{blue}{\mathsf{fma}\left({N}^{-3}, N \cdot \left(N + -0.5\right), {N}^{-3} \cdot 0.3333333333333333\right)} \]
   13. Step-by-step derivation

       1. fma-udef 39.2%

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

       2. distribute-lft-out 39.2%

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

   14. Simplified 39.2%

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

   15. Taylor expanded in N around inf 100.0%

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

       1. associate-*r/ 100.0%

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

       2. metadata-eval 100.0%

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

       3. unpow2 100.0%

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

       4. associate-/r* 100.0%

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

       5. div-sub 100.0%

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

   17. Simplified 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;N \leq 190000:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{0.5}{N}}{N}\\ \end{array} \]

Alternative 4: 99.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= 0.9d0) then
        tmp = n - log(n)
    else
        tmp = (1.0d0 - (0.5d0 / n)) / n
    end if
    code = tmp
end function

Java

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

Python

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

Julia

function code(N)
	tmp = 0.0
	if (N <= 0.9)
		tmp = Float64(N - log(N));
	else
		tmp = Float64(Float64(1.0 - Float64(0.5 / N)) / N);
	end
	return tmp
end

MATLAB

function tmp_2 = code(N)
	tmp = 0.0;
	if (N <= 0.9)
		tmp = N - log(N);
	else
		tmp = (1.0 - (0.5 / N)) / N;
	end
	tmp_2 = tmp;
end

Wolfram

code[N_] := If[LessEqual[N, 0.9], N[(N - N[Log[N], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(0.5 / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if N < 0.900000000000000022

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. log1p-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in N around 0 99.5%

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

      1. mul-1-neg 99.5%

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

      2. unsub-neg 99.5%

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

   6. Simplified 99.5%

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

   if 0.900000000000000022 < N

   1. Initial program 7.7%

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

      1. +-commutative 7.7%

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

      2. log1p-def 7.7%

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

   3. Simplified 7.7%

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

   4. Taylor expanded in N around inf 99.1%

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

      1. associate--l+ 99.1%

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

      2. associate-*r/ 99.1%

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

      3. metadata-eval 99.1%

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

      4. associate-*r/ 99.1%

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

      5. metadata-eval 99.1%

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

   6. Simplified 99.1%

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

      1. frac-sub 38.7%

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

      2. unpow2 38.7%

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

      3. cube-mult 38.6%

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

      4. div-inv 38.6%

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

      5. *-un-lft-identity 38.6%

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

      6. unpow2 38.6%

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

      7. distribute-lft-out-- 38.6%

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

      8. pow-flip 39.3%

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

      9. metadata-eval 39.3%

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

   8. Applied egg-rr 39.3%

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

      1. *-commutative 39.3%

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

      2. sub-neg 39.3%

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

      3. metadata-eval 39.3%

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

   10. Simplified 39.3%

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

       1. +-commutative 39.3%

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

       2. fma-def 39.3%

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

       3. div-inv 39.3%

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

       4. *-commutative 39.3%

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

       5. pow-flip 39.3%

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

       6. metadata-eval 39.3%

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

   12. Applied egg-rr 39.3%

       \[\leadsto \color{blue}{\mathsf{fma}\left({N}^{-3}, N \cdot \left(N + -0.5\right), {N}^{-3} \cdot 0.3333333333333333\right)} \]
   13. Step-by-step derivation

       1. fma-udef 39.3%

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

       2. distribute-lft-out 39.3%

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

   14. Simplified 39.3%

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

   15. Taylor expanded in N around inf 98.9%

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

       1. associate-*r/ 98.9%

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

       2. metadata-eval 98.9%

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

       3. unpow2 98.9%

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

       4. associate-/r* 98.9%

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

       5. div-sub 98.9%

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

   17. Simplified 98.9%

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

4. Final simplification 99.2%

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

Alternative 5: 98.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if N < 0.680000000000000049

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. log1p-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in N around 0 98.8%

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

      1. mul-1-neg 98.8%

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

   6. Simplified 98.8%

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

   if 0.680000000000000049 < N

   1. Initial program 7.7%

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

      1. +-commutative 7.7%

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

      2. log1p-def 7.7%

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

   3. Simplified 7.7%

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

   4. Taylor expanded in N around inf 99.1%

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

      1. associate--l+ 99.1%

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

      2. associate-*r/ 99.1%

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

      3. metadata-eval 99.1%

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

      4. associate-*r/ 99.1%

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

      5. metadata-eval 99.1%

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

   6. Simplified 99.1%

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

      1. frac-sub 38.7%

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

      2. unpow2 38.7%

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

      3. cube-mult 38.6%

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

      4. div-inv 38.6%

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

      5. *-un-lft-identity 38.6%

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

      6. unpow2 38.6%

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

      7. distribute-lft-out-- 38.6%

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

      8. pow-flip 39.3%

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

      9. metadata-eval 39.3%

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

   8. Applied egg-rr 39.3%

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

      1. *-commutative 39.3%

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

      2. sub-neg 39.3%

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

      3. metadata-eval 39.3%

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

   10. Simplified 39.3%

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

       1. +-commutative 39.3%

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

       2. fma-def 39.3%

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

       3. div-inv 39.3%

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

       4. *-commutative 39.3%

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

       5. pow-flip 39.3%

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

       6. metadata-eval 39.3%

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

   12. Applied egg-rr 39.3%

       \[\leadsto \color{blue}{\mathsf{fma}\left({N}^{-3}, N \cdot \left(N + -0.5\right), {N}^{-3} \cdot 0.3333333333333333\right)} \]
   13. Step-by-step derivation

       1. fma-udef 39.3%

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

       2. distribute-lft-out 39.3%

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

   14. Simplified 39.3%

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

   15. Taylor expanded in N around inf 98.9%

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

       1. associate-*r/ 98.9%

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

       2. metadata-eval 98.9%

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

       3. unpow2 98.9%

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

       4. associate-/r* 98.9%

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

       5. div-sub 98.9%

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

   17. Simplified 98.9%

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

4. Final simplification 98.9%

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

Alternative 6: 50.9% accurate, 68.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(N):
	return 1.0 / N

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.8%

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

   1. +-commutative 53.8%

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

   2. log1p-def 53.8%

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

3. Simplified 53.8%

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

4. Taylor expanded in N around inf 51.8%

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

5. Final simplification 51.8%

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

Alternative 7: 4.6% accurate, 205.0× speedup

Math

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

FPCore

(FPCore (N) :precision binary64 N)

C

double code(double N) {
	return N;
}

Fortran

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

Java

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

Python

def code(N):
	return N

Julia

function code(N)
	return N
end

MATLAB

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

Wolfram

code[N_] := N

Derivation

1. Initial program 53.8%

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

   1. +-commutative 53.8%

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

   2. log1p-def 53.8%

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

3. Simplified 53.8%

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

4. Taylor expanded in N around 0 51.7%

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

   1. mul-1-neg 51.7%

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

   2. unsub-neg 51.7%

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

6. Simplified 51.7%

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

7. Taylor expanded in N around inf 4.6%

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

8. Final simplification 4.6%

   \[\leadsto N \]

Reproduce

herbie shell --seed 2023301 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1.0)) (log N)))