2log (problem 3.3.6)

Percentage Accurate: 23.7% → 99.3%

Time: 8.6s

Alternatives: 5

Speedup: 68.3×

Specification

\[N > 1 \land N < 10^{+40}\]

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: 23.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.3% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (-.f64 (log.f64 (+.f64 N #s(literal 1 binary64))) (log.f64 N)) < 5.0000000000000001e-4

   1. Initial program 18.7%

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

      1. +-commutative 18.7%

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

      2. log1p-define 18.7%

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

   3. Simplified 18.7%

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

   5. Taylor expanded in N around inf 99.8%

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

      1. associate-*r/ 99.8%

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

      2. metadata-eval 99.8%

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

      3. associate-*r/ 99.8%

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

      4. metadata-eval 99.8%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in N around -inf 99.8%

      \[\leadsto \frac{\color{blue}{1 + -1 \cdot \frac{0.5 + -1 \cdot \frac{0.3333333333333333 - 0.25 \cdot \frac{1}{N}}{N}}{N}}}{N} \]
   9. Step-by-step derivation

      1. mul-1-neg 99.8%

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

      2. unsub-neg 99.8%

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

      3. mul-1-neg 99.8%

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

      4. unsub-neg 99.8%

         \[\leadsto \frac{1 - \frac{\color{blue}{0.5 - \frac{0.3333333333333333 - 0.25 \cdot \frac{1}{N}}{N}}}{N}}{N} \]

      5. sub-neg 99.8%

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

      6. associate-*r/ 99.8%

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

      7. metadata-eval 99.8%

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

      8. distribute-neg-frac 99.8%

         \[\leadsto \frac{1 - \frac{0.5 - \frac{0.3333333333333333 + \color{blue}{\frac{-0.25}{N}}}{N}}{N}}{N} \]

      9. metadata-eval 99.8%

         \[\leadsto \frac{1 - \frac{0.5 - \frac{0.3333333333333333 + \frac{\color{blue}{-0.25}}{N}}{N}}{N}}{N} \]

   10. Simplified 99.8%

       \[\leadsto \frac{\color{blue}{1 - \frac{0.5 - \frac{0.3333333333333333 + \frac{-0.25}{N}}{N}}{N}}}{N} \]

   if 5.0000000000000001e-4 < (-.f64 (log.f64 (+.f64 N #s(literal 1 binary64))) (log.f64 N))

   1. Initial program 92.6%

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

      1. +-commutative 92.6%

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

      2. log1p-define 92.6%

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

   3. Simplified 92.6%

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

      1. add-log-exp 92.6%

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

      2. log1p-expm1-u 92.6%

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

      3. log1p-undefine 92.6%

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

      4. diff-log 92.6%

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

      5. log1p-undefine 92.6%

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

      6. rem-exp-log 93.0%

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

      7. +-commutative 93.0%

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

      8. add-exp-log 92.8%

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

      9. log1p-undefine 92.8%

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

      10. log1p-expm1-u 92.8%

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

      11. add-exp-log 95.2%

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

   6. Applied egg-rr 95.2%

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

4. Final simplification 99.5%

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

Alternative 2: 96.3% accurate, 13.7× speedup

Math

\[\begin{array}{l} \\ \frac{1 - \frac{0.5 - \frac{0.3333333333333333 + \frac{-0.25}{N}}{N}}{N}}{N} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (/ (- 1.0 (/ (- 0.5 (/ (+ 0.3333333333333333 (/ -0.25 N)) N)) N)) N))

C

double code(double N) {
	return (1.0 - ((0.5 - ((0.3333333333333333 + (-0.25 / N)) / N)) / N)) / N;
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    code = (1.0d0 - ((0.5d0 - ((0.3333333333333333d0 + ((-0.25d0) / n)) / n)) / n)) / n
end function

Java

public static double code(double N) {
	return (1.0 - ((0.5 - ((0.3333333333333333 + (-0.25 / N)) / N)) / N)) / N;
}

Python

def code(N):
	return (1.0 - ((0.5 - ((0.3333333333333333 + (-0.25 / N)) / N)) / N)) / N

Julia

function code(N)
	return Float64(Float64(1.0 - Float64(Float64(0.5 - Float64(Float64(0.3333333333333333 + Float64(-0.25 / N)) / N)) / N)) / N)
end

MATLAB

function tmp = code(N)
	tmp = (1.0 - ((0.5 - ((0.3333333333333333 + (-0.25 / N)) / N)) / N)) / N;
end

Wolfram

code[N_] := N[(N[(1.0 - N[(N[(0.5 - N[(N[(0.3333333333333333 + N[(-0.25 / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision]

Derivation

1. Initial program 23.9%

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

   1. +-commutative 23.9%

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

   2. log1p-define 23.9%

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

3. Simplified 23.9%

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

5. Taylor expanded in N around inf 96.4%

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

   1. associate-*r/ 96.4%

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

   2. metadata-eval 96.4%

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

   3. associate-*r/ 96.4%

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

   4. metadata-eval 96.4%

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

7. Simplified 96.4%

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

8. Taylor expanded in N around -inf 96.4%

   \[\leadsto \frac{\color{blue}{1 + -1 \cdot \frac{0.5 + -1 \cdot \frac{0.3333333333333333 - 0.25 \cdot \frac{1}{N}}{N}}{N}}}{N} \]
9. Step-by-step derivation

   1. mul-1-neg 96.4%

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

   2. unsub-neg 96.4%

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

   3. mul-1-neg 96.4%

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

   4. unsub-neg 96.4%

      \[\leadsto \frac{1 - \frac{\color{blue}{0.5 - \frac{0.3333333333333333 - 0.25 \cdot \frac{1}{N}}{N}}}{N}}{N} \]

   5. sub-neg 96.4%

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

   6. associate-*r/ 96.4%

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

   7. metadata-eval 96.4%

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

   8. distribute-neg-frac 96.4%

      \[\leadsto \frac{1 - \frac{0.5 - \frac{0.3333333333333333 + \color{blue}{\frac{-0.25}{N}}}{N}}{N}}{N} \]

   9. metadata-eval 96.4%

      \[\leadsto \frac{1 - \frac{0.5 - \frac{0.3333333333333333 + \frac{\color{blue}{-0.25}}{N}}{N}}{N}}{N} \]

10. Simplified 96.4%

    \[\leadsto \frac{\color{blue}{1 - \frac{0.5 - \frac{0.3333333333333333 + \frac{-0.25}{N}}{N}}{N}}}{N} \]

11. Final simplification 96.4%

    \[\leadsto \frac{1 - \frac{0.5 - \frac{0.3333333333333333 + \frac{-0.25}{N}}{N}}{N}}{N} \]
12. Add Preprocessing

Alternative 3: 95.1% accurate, 18.6× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{-0.5 + \frac{0.3333333333333333}{N}}{N} + 1}{N} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (/ (+ (/ (+ -0.5 (/ 0.3333333333333333 N)) N) 1.0) N))

C

double code(double N) {
	return (((-0.5 + (0.3333333333333333 / N)) / N) + 1.0) / N;
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    code = ((((-0.5d0) + (0.3333333333333333d0 / n)) / n) + 1.0d0) / n
end function

Java

public static double code(double N) {
	return (((-0.5 + (0.3333333333333333 / N)) / N) + 1.0) / N;
}

Python

def code(N):
	return (((-0.5 + (0.3333333333333333 / N)) / N) + 1.0) / N

Julia

function code(N)
	return Float64(Float64(Float64(Float64(-0.5 + Float64(0.3333333333333333 / N)) / N) + 1.0) / N)
end

MATLAB

function tmp = code(N)
	tmp = (((-0.5 + (0.3333333333333333 / N)) / N) + 1.0) / N;
end

Wolfram

code[N_] := N[(N[(N[(N[(-0.5 + N[(0.3333333333333333 / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision] + 1.0), $MachinePrecision] / N), $MachinePrecision]

Derivation

1. Initial program 23.9%

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

   1. +-commutative 23.9%

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

   2. log1p-define 23.9%

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

3. Simplified 23.9%

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

5. Taylor expanded in N around inf 95.3%

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

   1. associate--l+ 95.3%

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

   2. unpow2 95.3%

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

   3. associate-/r* 95.3%

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

   4. metadata-eval 95.3%

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

   5. associate-*r/ 95.3%

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

   6. associate-*r/ 95.3%

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

   7. metadata-eval 95.3%

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

   8. div-sub 95.3%

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

   9. sub-neg 95.3%

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

   10. metadata-eval 95.3%

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

   11. +-commutative 95.3%

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

   12. associate-*r/ 95.3%

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

   13. metadata-eval 95.3%

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

7. Simplified 95.3%

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

8. Final simplification 95.3%

   \[\leadsto \frac{\frac{-0.5 + \frac{0.3333333333333333}{N}}{N} + 1}{N} \]
9. Add Preprocessing

Alternative 4: 92.5% accurate, 29.3× speedup

Math

\[\begin{array}{l} \\ \frac{1 - \frac{0.5}{N}}{N} \end{array} \]

FPCore

(FPCore (N) :precision binary64 (/ (- 1.0 (/ 0.5 N)) N))

C

double code(double N) {
	return (1.0 - (0.5 / N)) / N;
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    code = (1.0d0 - (0.5d0 / n)) / n
end function

Java

public static double code(double N) {
	return (1.0 - (0.5 / N)) / N;
}

Python

def code(N):
	return (1.0 - (0.5 / N)) / N

Julia

function code(N)
	return Float64(Float64(1.0 - Float64(0.5 / N)) / N)
end

MATLAB

function tmp = code(N)
	tmp = (1.0 - (0.5 / N)) / N;
end

Wolfram

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

Derivation

1. Initial program 23.9%

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

   1. +-commutative 23.9%

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

   2. log1p-define 23.9%

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

3. Simplified 23.9%

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

5. Taylor expanded in N around inf 92.3%

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

   1. associate-*r/ 92.3%

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

   2. metadata-eval 92.3%

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

7. Simplified 92.3%

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

8. Final simplification 92.3%

   \[\leadsto \frac{1 - \frac{0.5}{N}}{N} \]
9. Add Preprocessing

Alternative 5: 84.5% 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 23.9%

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

   1. +-commutative 23.9%

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

   2. log1p-define 23.9%

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

3. Simplified 23.9%

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

5. Taylor expanded in N around inf 84.5%

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

6. Final simplification 84.5%

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

Developer target: 99.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{log1p}\left(\frac{1}{N}\right) \end{array} \]

FPCore

(FPCore (N) :precision binary64 (log1p (/ 1.0 N)))

C

double code(double N) {
	return log1p((1.0 / N));
}

Java

public static double code(double N) {
	return Math.log1p((1.0 / N));
}

Python

def code(N):
	return math.log1p((1.0 / N))

Julia

function code(N)
	return log1p(Float64(1.0 / N))
end

Wolfram

code[N_] := N[Log[1 + N[(1.0 / N), $MachinePrecision]], $MachinePrecision]

Reproduce

herbie shell --seed 2024078 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  :pre (and (> N 1.0) (< N 1e+40))

  :alt
  (log1p (/ 1.0 N))

  (- (log (+ N 1.0)) (log N)))