2log (problem 3.3.6)

Percentage Accurate: 23.7% → 99.8%

Time: 9.3s

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: 23.7% 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: 99.8% 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 23.6%

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

   1. sub-neg 23.6%

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

   2. +-commutative 23.6%

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

   3. log1p-udef 23.5%

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

4. Applied egg-rr 23.5%

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

   1. +-commutative 23.5%

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

   2. log-rec 23.6%

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

   3. log1p-def 23.6%

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

   4. +-commutative 23.6%

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

   5. log-prod 25.9%

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

   6. distribute-lft-in 25.8%

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

   7. lft-mult-inverse 26.2%

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

   8. *-rgt-identity 26.2%

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

6. Simplified 26.2%

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

   1. *-un-lft-identity 26.2%

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

   2. log-prod 26.2%

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

   3. metadata-eval 26.2%

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

   4. log1p-def 99.8%

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

8. Applied egg-rr 99.8%

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

   1. +-lft-identity 99.8%

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

10. Simplified 99.8%

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

11. Final simplification 99.8%

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

Alternative 2: 92.3% accurate, 25.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 23.6%

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

3. Taylor expanded in N around inf 92.4%

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

   1. associate-*r/ 92.4%

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

   2. metadata-eval 92.4%

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

5. Simplified 92.4%

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

   1. frac-2neg 92.4%

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

   2. metadata-eval 92.4%

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

   3. frac-sub 92.2%

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

   4. *-un-lft-identity 92.2%

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

7. Applied egg-rr 92.2%

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

8. Taylor expanded in N around 0 92.2%

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

   1. +-commutative 92.2%

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

   2. fma-def 92.2%

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

   3. mul-1-neg 92.2%

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

   4. fma-neg 92.2%

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

   5. unpow2 92.2%

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

   6. distribute-rgt-out-- 92.1%

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

10. Simplified 92.1%

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

    1. times-frac 92.1%

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

    2. *-inverses 92.1%

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

    3. associate-*r/ 92.1%

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

    4. neg-mul-1 92.1%

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

    5. times-frac 92.1%

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

    6. metadata-eval 92.1%

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

    7. add-sqr-sqrt 92.1%

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

    8. sqrt-unprod 92.1%

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

    9. sqr-neg 92.1%

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

    10. sqrt-unprod 0.0%

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

    11. add-sqr-sqrt 1.7%

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

    12. div-inv 1.7%

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

    13. metadata-eval 1.7%

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

    14. add-sqr-sqrt 0.0%

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

    15. sqrt-unprod 92.0%

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

    16. sqr-neg 92.0%

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

    17. sqrt-unprod 92.0%

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

    18. add-sqr-sqrt 92.0%

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

    19. unpow2 92.0%

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

    20. frac-times 92.1%

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

    21. inv-pow 92.1%

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

    22. inv-pow 92.1%

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

    23. pow-prod-up 92.1%

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

    24. metadata-eval 92.1%

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

12. Applied egg-rr 92.1%

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

    1. neg-mul-1 92.1%

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

    2. *-commutative 92.1%

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

    3. metadata-eval 92.1%

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

    4. pow-sqr 92.1%

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

    5. unpow-1 92.1%

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

    6. unpow-1 92.1%

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

    7. associate-*l* 92.2%

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

    8. associate-*l/ 92.4%

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

    9. *-lft-identity 92.4%

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

    10. associate-*l/ 92.5%

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

    11. *-lft-identity 92.5%

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

    12. div-sub 92.5%

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

    13. metadata-eval 92.5%

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

    14. associate-*r/ 92.5%

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

    15. sub-neg 92.5%

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

    16. associate-*r/ 92.5%

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

    17. metadata-eval 92.5%

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

    18. *-inverses 92.5%

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

    19. metadata-eval 92.5%

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

14. Simplified 92.5%

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

15. Final simplification 92.5%

    \[\leadsto \frac{\frac{-0.5}{N} - -1}{N} \]
16. 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 23.6%

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

3. Taylor expanded in N around inf 84.5%

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

4. Final simplification 84.5%

   \[\leadsto \frac{1}{N} \]
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 2024031 
(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)))