2log (problem 3.3.6)

Percentage Accurate: 23.5% → 99.8%

Time: 13.6s

Alternatives: 3

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: 23.5% 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, 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]

Derivation

1. Initial program 24.3%

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

   1. sub-neg 24.3%

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

   2. +-commutative 24.3%

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

   3. log1p-udef 24.3%

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

4. Applied egg-rr 24.3%

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

   1. +-commutative 24.3%

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

   2. log-rec 24.3%

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

   3. log1p-def 24.3%

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

   4. +-commutative 24.3%

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

   5. log-prod 26.5%

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

   6. distribute-rgt-in 26.5%

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

   7. rgt-mult-inverse 26.7%

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

   8. *-lft-identity 26.7%

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

   9. log1p-def 99.8%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\frac{1}{N}\right)} \]

6. Simplified 99.8%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(\frac{1}{N}\right)} \]

7. Final simplification 99.8%

   \[\leadsto \mathsf{log1p}\left(\frac{1}{N}\right) \]
8. Add Preprocessing

Alternative 2: 92.9% accurate, 29.3× speedup

Math

\[\begin{array}{l} \\ \frac{N}{N \cdot \left(N + 0.5\right)} \end{array} \]

FPCore

(FPCore (N) :precision binary64 (/ N (* N (+ N 0.5))))

C

double code(double N) {
	return N / (N * (N + 0.5));
}

Fortran

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

Java

public static double code(double N) {
	return N / (N * (N + 0.5));
}

Python

def code(N):
	return N / (N * (N + 0.5))

Julia

function code(N)
	return Float64(N / Float64(N * Float64(N + 0.5)))
end

MATLAB

function tmp = code(N)
	tmp = N / (N * (N + 0.5));
end

Wolfram

code[N_] := N[(N / N[(N * N[(N + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 24.3%

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

3. Taylor expanded in N around inf 92.1%

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

   1. associate-*r/ 92.1%

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

   2. metadata-eval 92.1%

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

5. Simplified 92.1%

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

   1. add-cube-cbrt 90.3%

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

   2. associate-*l* 90.3%

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

   3. frac-sub 90.4%

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

   4. unpow2 90.4%

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

   5. cube-mult 90.4%

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

   6. cbrt-div 90.6%

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

   7. *-un-lft-identity 90.6%

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

   8. unpow2 90.6%

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

   9. distribute-lft-out-- 90.6%

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

   10. rem-cbrt-cube 90.7%

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

   11. pow2 90.7%

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

7. Applied egg-rr 90.6%

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

   1. unpow2 90.6%

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

   2. cube-mult 90.6%

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

   3. cube-div 90.7%

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

   4. rem-cube-cbrt 91.9%

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

   5. associate-/l* 91.8%

      \[\leadsto \color{blue}{\frac{N}{\frac{{N}^{3}}{N - 0.5}}} \]

   6. sub-neg 91.8%

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

   7. metadata-eval 91.8%

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

9. Simplified 91.8%

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

10. Taylor expanded in N around inf 92.4%

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

    1. unpow2 92.4%

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

    2. distribute-rgt-out 92.4%

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

    3. +-commutative 92.4%

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

12. Simplified 92.4%

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

13. Final simplification 92.4%

    \[\leadsto \frac{N}{N \cdot \left(N + 0.5\right)} \]
14. Add Preprocessing

Alternative 3: 84.7% 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 24.3%

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

3. Taylor expanded in N around inf 84.3%

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

4. Final simplification 84.3%

   \[\leadsto \frac{1}{N} \]
5. 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 2024101 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  :pre (and (> N 1.0) (< N 1e+40))

  :herbie-target
  (log1p (/ 1.0 N))

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