2log (problem 3.3.6)

Percentage Accurate: 23.8% → 99.8%

Time: 9.7s

Alternatives: 4

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: 23.8% 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 21.6%

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

   1. add-cbrt-cube 21.6%

      \[\leadsto \color{blue}{\sqrt[3]{\left(\left(\log \left(N + 1\right) - \log N\right) \cdot \left(\log \left(N + 1\right) - \log N\right)\right) \cdot \left(\log \left(N + 1\right) - \log N\right)}} \]

   2. pow1/3 21.6%

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

   3. add-sqr-sqrt 21.6%

      \[\leadsto {\left(\left(\left(\log \left(N + 1\right) - \log N\right) \cdot \left(\log \left(N + 1\right) - \log N\right)\right) \cdot \color{blue}{\left(\sqrt{\log \left(N + 1\right) - \log N} \cdot \sqrt{\log \left(N + 1\right) - \log N}\right)}\right)}^{0.3333333333333333} \]

   4. unswap-sqr 21.6%

      \[\leadsto {\color{blue}{\left(\left(\left(\log \left(N + 1\right) - \log N\right) \cdot \sqrt{\log \left(N + 1\right) - \log N}\right) \cdot \left(\left(\log \left(N + 1\right) - \log N\right) \cdot \sqrt{\log \left(N + 1\right) - \log N}\right)\right)}}^{0.3333333333333333} \]

   5. add-sqr-sqrt 21.6%

      \[\leadsto {\left(\left(\color{blue}{\left(\sqrt{\log \left(N + 1\right) - \log N} \cdot \sqrt{\log \left(N + 1\right) - \log N}\right)} \cdot \sqrt{\log \left(N + 1\right) - \log N}\right) \cdot \left(\left(\log \left(N + 1\right) - \log N\right) \cdot \sqrt{\log \left(N + 1\right) - \log N}\right)\right)}^{0.3333333333333333} \]

   6. add-sqr-sqrt 21.6%

      \[\leadsto {\left(\left(\left(\sqrt{\log \left(N + 1\right) - \log N} \cdot \sqrt{\log \left(N + 1\right) - \log N}\right) \cdot \sqrt{\log \left(N + 1\right) - \log N}\right) \cdot \left(\color{blue}{\left(\sqrt{\log \left(N + 1\right) - \log N} \cdot \sqrt{\log \left(N + 1\right) - \log N}\right)} \cdot \sqrt{\log \left(N + 1\right) - \log N}\right)\right)}^{0.3333333333333333} \]

4. Applied egg-rr 21.6%

   \[\leadsto \color{blue}{{\left({\left(\mathsf{log1p}\left(N\right) - \log N\right)}^{1.5}\right)}^{0.3333333333333333} \cdot {\left({\left(\mathsf{log1p}\left(N\right) - \log N\right)}^{1.5}\right)}^{0.3333333333333333}} \]
5. Step-by-step derivation

   1. pow-sqr 21.6%

      \[\leadsto \color{blue}{{\left({\left(\mathsf{log1p}\left(N\right) - \log N\right)}^{1.5}\right)}^{\left(2 \cdot 0.3333333333333333\right)}} \]

   2. sqr-pow 21.6%

      \[\leadsto {\color{blue}{\left({\left(\mathsf{log1p}\left(N\right) - \log N\right)}^{\left(\frac{1.5}{2}\right)} \cdot {\left(\mathsf{log1p}\left(N\right) - \log N\right)}^{\left(\frac{1.5}{2}\right)}\right)}}^{\left(2 \cdot 0.3333333333333333\right)} \]

   3. fabs-sqr 21.6%

      \[\leadsto {\color{blue}{\left(\left|{\left(\mathsf{log1p}\left(N\right) - \log N\right)}^{\left(\frac{1.5}{2}\right)} \cdot {\left(\mathsf{log1p}\left(N\right) - \log N\right)}^{\left(\frac{1.5}{2}\right)}\right|\right)}}^{\left(2 \cdot 0.3333333333333333\right)} \]

   4. sqr-pow 21.6%

      \[\leadsto {\left(\left|\color{blue}{{\left(\mathsf{log1p}\left(N\right) - \log N\right)}^{1.5}}\right|\right)}^{\left(2 \cdot 0.3333333333333333\right)} \]

   5. rem-sqrt-square 21.6%

      \[\leadsto {\color{blue}{\left(\sqrt{{\left(\mathsf{log1p}\left(N\right) - \log N\right)}^{1.5} \cdot {\left(\mathsf{log1p}\left(N\right) - \log N\right)}^{1.5}}\right)}}^{\left(2 \cdot 0.3333333333333333\right)} \]

   6. pow-sqr 21.6%

      \[\leadsto {\left(\sqrt{\color{blue}{{\left(\mathsf{log1p}\left(N\right) - \log N\right)}^{\left(2 \cdot 1.5\right)}}}\right)}^{\left(2 \cdot 0.3333333333333333\right)} \]

   7. metadata-eval 21.6%

      \[\leadsto {\left(\sqrt{{\left(\mathsf{log1p}\left(N\right) - \log N\right)}^{\color{blue}{3}}}\right)}^{\left(2 \cdot 0.3333333333333333\right)} \]

   8. pow-sqr 21.6%

      \[\leadsto \color{blue}{{\left(\sqrt{{\left(\mathsf{log1p}\left(N\right) - \log N\right)}^{3}}\right)}^{0.3333333333333333} \cdot {\left(\sqrt{{\left(\mathsf{log1p}\left(N\right) - \log N\right)}^{3}}\right)}^{0.3333333333333333}} \]

6. Simplified 24.4%

   \[\leadsto \color{blue}{\sqrt[3]{{\log \left(1 + \frac{1}{N}\right)}^{1.5}} \cdot \sqrt[3]{{\log \left(1 + \frac{1}{N}\right)}^{1.5}}} \]
7. Step-by-step derivation

   1. expm1-log1p-u 24.4%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{{\log \left(1 + \frac{1}{N}\right)}^{1.5}}\right)\right)} \cdot \sqrt[3]{{\log \left(1 + \frac{1}{N}\right)}^{1.5}} \]

   2. expm1-udef 24.4%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt[3]{{\log \left(1 + \frac{1}{N}\right)}^{1.5}}\right)} - 1\right)} \cdot \sqrt[3]{{\log \left(1 + \frac{1}{N}\right)}^{1.5}} \]

   3. pow1/3 24.4%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{{\left({\log \left(1 + \frac{1}{N}\right)}^{1.5}\right)}^{0.3333333333333333}}\right)} - 1\right) \cdot \sqrt[3]{{\log \left(1 + \frac{1}{N}\right)}^{1.5}} \]

   4. pow-pow 24.4%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{{\log \left(1 + \frac{1}{N}\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}}\right)} - 1\right) \cdot \sqrt[3]{{\log \left(1 + \frac{1}{N}\right)}^{1.5}} \]

   5. metadata-eval 24.4%

      \[\leadsto \left(e^{\mathsf{log1p}\left({\log \left(1 + \frac{1}{N}\right)}^{\color{blue}{0.5}}\right)} - 1\right) \cdot \sqrt[3]{{\log \left(1 + \frac{1}{N}\right)}^{1.5}} \]

   6. pow1/2 24.4%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\sqrt{\log \left(1 + \frac{1}{N}\right)}}\right)} - 1\right) \cdot \sqrt[3]{{\log \left(1 + \frac{1}{N}\right)}^{1.5}} \]

   7. log1p-def 25.0%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\mathsf{log1p}\left(\frac{1}{N}\right)}}\right)} - 1\right) \cdot \sqrt[3]{{\log \left(1 + \frac{1}{N}\right)}^{1.5}} \]

8. Applied egg-rr 25.0%

   \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\mathsf{log1p}\left(\frac{1}{N}\right)}\right)} - 1\right)} \cdot \sqrt[3]{{\log \left(1 + \frac{1}{N}\right)}^{1.5}} \]
9. Step-by-step derivation

   1. expm1-def 25.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\mathsf{log1p}\left(\frac{1}{N}\right)}\right)\right)} \cdot \sqrt[3]{{\log \left(1 + \frac{1}{N}\right)}^{1.5}} \]

   2. expm1-log1p 25.0%

      \[\leadsto \color{blue}{\sqrt{\mathsf{log1p}\left(\frac{1}{N}\right)}} \cdot \sqrt[3]{{\log \left(1 + \frac{1}{N}\right)}^{1.5}} \]

10. Simplified 25.0%

    \[\leadsto \color{blue}{\sqrt{\mathsf{log1p}\left(\frac{1}{N}\right)}} \cdot \sqrt[3]{{\log \left(1 + \frac{1}{N}\right)}^{1.5}} \]
11. Step-by-step derivation

    1. pow1/3 25.0%

       \[\leadsto \sqrt{\mathsf{log1p}\left(\frac{1}{N}\right)} \cdot \color{blue}{{\left({\log \left(1 + \frac{1}{N}\right)}^{1.5}\right)}^{0.3333333333333333}} \]

    2. log1p-udef 95.4%

       \[\leadsto \sqrt{\mathsf{log1p}\left(\frac{1}{N}\right)} \cdot {\left({\color{blue}{\left(\mathsf{log1p}\left(\frac{1}{N}\right)\right)}}^{1.5}\right)}^{0.3333333333333333} \]

    3. pow-pow 99.2%

       \[\leadsto \sqrt{\mathsf{log1p}\left(\frac{1}{N}\right)} \cdot \color{blue}{{\left(\mathsf{log1p}\left(\frac{1}{N}\right)\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} \]

    4. metadata-eval 99.2%

       \[\leadsto \sqrt{\mathsf{log1p}\left(\frac{1}{N}\right)} \cdot {\left(\mathsf{log1p}\left(\frac{1}{N}\right)\right)}^{\color{blue}{0.5}} \]

    5. pow1/2 99.2%

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

    6. add-sqr-sqrt 99.9%

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

    7. log1p-udef 24.4%

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

    8. *-un-lft-identity 24.4%

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

    9. log-prod 24.4%

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

    10. metadata-eval 24.4%

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

    11. log1p-udef 99.9%

        \[\leadsto 0 + \color{blue}{\mathsf{log1p}\left(\frac{1}{N}\right)} \]

12. Applied egg-rr 99.9%

    \[\leadsto \color{blue}{0 + \mathsf{log1p}\left(\frac{1}{N}\right)} \]
13. Step-by-step derivation

    1. +-lft-identity 99.9%

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

14. Simplified 99.9%

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

15. Final simplification 99.9%

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

Alternative 2: 93.2% accurate, 41.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 21.6%

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

3. Taylor expanded in N around inf 93.6%

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

   1. associate-*r/ 93.6%

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

   2. metadata-eval 93.6%

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

5. Simplified 93.6%

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

   1. frac-sub 93.4%

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

   2. unpow2 93.4%

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

   3. cube-mult 93.2%

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

   4. clear-num 93.4%

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

   5. *-un-lft-identity 93.4%

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

   6. unpow2 93.4%

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

   7. distribute-lft-out-- 93.4%

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

7. Applied egg-rr 93.4%

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

8. Taylor expanded in N around inf 94.2%

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

   1. +-commutative 94.2%

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

10. Simplified 94.2%

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

11. Final simplification 94.2%

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

Alternative 3: 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 21.6%

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

3. Taylor expanded in N around inf 86.1%

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

4. Final simplification 86.1%

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

Alternative 4: 9.9% accurate, 205.0× speedup

Math

\[\begin{array}{l} \\ 2 \end{array} \]

FPCore

(FPCore (N) :precision binary64 2.0)

C

double code(double N) {
	return 2.0;
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    code = 2.0d0
end function

Java

public static double code(double N) {
	return 2.0;
}

Python

def code(N):
	return 2.0

Julia

function code(N)
	return 2.0
end

MATLAB

function tmp = code(N)
	tmp = 2.0;
end

Wolfram

code[N_] := 2.0

Derivation

1. Initial program 21.6%

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

3. Taylor expanded in N around inf 93.6%

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

   1. associate-*r/ 93.6%

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

   2. metadata-eval 93.6%

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

5. Simplified 93.6%

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

   1. frac-sub 93.4%

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

   2. unpow2 93.4%

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

   3. cube-mult 93.2%

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

   4. clear-num 93.4%

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

   5. *-un-lft-identity 93.4%

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

   6. unpow2 93.4%

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

   7. distribute-lft-out-- 93.4%

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

7. Applied egg-rr 93.4%

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

8. Taylor expanded in N around inf 94.2%

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

   1. +-commutative 94.2%

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

10. Simplified 94.2%

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

11. Taylor expanded in N around 0 9.7%

    \[\leadsto \color{blue}{2} \]

12. Final simplification 9.7%

    \[\leadsto 2 \]
13. 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 2024029 
(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)))