2log (problem 3.3.6)

Percentage Accurate: 24.1% → 99.8%

Time: 7.9s

Alternatives: 9

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: 24.1% 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. diff-log 24.1%

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

4. Applied egg-rr 24.1%

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

   1. *-lft-identity 24.1%

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

   2. associate-*l/ 23.7%

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

   3. distribute-lft-in 23.8%

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

   4. lft-mult-inverse 24.0%

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

   5. *-rgt-identity 24.0%

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

   6. log1p-define 99.9%

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

6. Simplified 99.9%

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

Alternative 2: 96.2% accurate, 12.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[N_] := N[(N[(1.0 / N), $MachinePrecision] + N[(N[(N[(-0.5 - N[(N[(N[(0.25 / N), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision] / N), $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 96.1%

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

   1. mul-1-neg 96.1%

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

   2. distribute-neg-frac2 96.1%

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

5. Simplified 96.1%

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

   1. div-sub 96.1%

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

   2. metadata-eval 96.1%

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

   3. frac-2neg 96.1%

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

   4. sub-neg 96.1%

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

   5. add-sqr-sqrt 0.0%

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

   6. sqrt-unprod 85.4%

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

   7. sqr-neg 85.4%

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

   8. sqrt-unprod 85.4%

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

   9. add-sqr-sqrt 85.4%

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

   10. distribute-frac-neg2 85.4%

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

   11. frac-2neg 85.4%

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

7. Applied egg-rr 92.6%

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

   1. add-sqr-sqrt 92.6%

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

   2. sqrt-unprod 92.6%

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

   3. sqr-neg 92.6%

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

   4. sqrt-unprod 0.0%

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

   5. add-sqr-sqrt 96.1%

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

   6. distribute-neg-frac2 96.1%

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

9. Applied egg-rr 96.1%

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

    1. unsub-neg 96.1%

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

11. Applied egg-rr 96.1%

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

Alternative 3: 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 21.6%

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

   1. diff-log 24.1%

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

4. Applied egg-rr 24.1%

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

   1. *-lft-identity 24.1%

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

   2. associate-*l/ 23.7%

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

   3. distribute-lft-in 23.8%

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

   4. lft-mult-inverse 24.0%

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

   5. *-rgt-identity 24.0%

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

   6. log1p-define 99.9%

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

6. Simplified 99.9%

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

7. Taylor expanded in N around inf 96.1%

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

9. Simplified 96.1%

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

Alternative 4: 95.0% accurate, 15.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[N_] := N[(1.0 / N[(N / N[(1.0 + N[(N[(-0.5 + N[(0.3333333333333333 / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision]), $MachinePrecision]), $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 95.0%

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

   1. associate--l+ 95.0%

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

   2. unpow2 95.0%

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

   3. associate-/r* 95.0%

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

   4. metadata-eval 95.0%

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

   5. associate-*r/ 95.0%

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

   6. associate-*r/ 95.0%

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

   7. metadata-eval 95.0%

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

   8. div-sub 95.0%

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

   9. sub-neg 95.0%

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

   10. metadata-eval 95.0%

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

   11. +-commutative 95.0%

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

   12. associate-*r/ 95.0%

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

   13. metadata-eval 95.0%

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

5. Simplified 95.0%

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

   1. expm1-log1p-u 95.0%

      \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 + \frac{-0.5 + \frac{0.3333333333333333}{N}}{N}\right)\right)}}{N} \]

   2. expm1-undefine 94.9%

      \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(1 + \frac{-0.5 + \frac{0.3333333333333333}{N}}{N}\right)} - 1}}{N} \]

7. Applied egg-rr 94.9%

   \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(1 + \frac{\frac{0.3333333333333333}{N} + -0.5}{N}\right)} - 1}}{N} \]
8. Step-by-step derivation

   1. sub-neg 94.9%

      \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(1 + \frac{\frac{0.3333333333333333}{N} + -0.5}{N}\right)} + \left(-1\right)}}{N} \]

   2. log1p-undefine 94.9%

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

   3. rem-exp-log 94.9%

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

   4. associate-+r+ 95.0%

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

   5. metadata-eval 95.0%

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

   6. +-commutative 95.0%

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

   7. metadata-eval 95.0%

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

9. Simplified 95.0%

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

    1. clear-num 95.0%

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

    2. inv-pow 95.0%

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

    3. associate-+l+ 95.0%

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

    4. +-commutative 95.0%

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

11. Applied egg-rr 95.0%

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

    1. unpow-1 95.0%

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

    2. +-commutative 95.0%

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

    3. associate-+r+ 95.1%

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

    4. metadata-eval 95.1%

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

    5. +-commutative 95.1%

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

13. Simplified 95.1%

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

Alternative 5: 95.0% accurate, 18.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[N_] := N[(N[(1.0 + N[(N[(-0.5 + N[(0.3333333333333333 / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision]), $MachinePrecision] / 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 95.0%

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

   1. associate--l+ 95.0%

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

   2. unpow2 95.0%

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

   3. associate-/r* 95.0%

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

   4. metadata-eval 95.0%

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

   5. associate-*r/ 95.0%

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

   6. associate-*r/ 95.0%

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

   7. metadata-eval 95.0%

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

   8. div-sub 95.0%

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

   9. sub-neg 95.0%

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

   10. metadata-eval 95.0%

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

   11. +-commutative 95.0%

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

   12. associate-*r/ 95.0%

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

   13. metadata-eval 95.0%

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

5. Simplified 95.0%

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

Alternative 6: 92.4% accurate, 22.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[N_] := N[(1.0 / N[(N / N[(1.0 - N[(0.5 / N), $MachinePrecision]), $MachinePrecision]), $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 92.9%

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

   1. associate-*r/ 92.9%

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

   2. metadata-eval 92.9%

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

5. Simplified 92.9%

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

6. Taylor expanded in N around 0 92.9%

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

   1. clear-num 92.9%

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

   2. inv-pow 92.9%

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

   3. div-sub 92.9%

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

   4. pow1 92.9%

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

   5. pow1 92.9%

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

   6. pow-div 92.9%

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

   7. metadata-eval 92.9%

      \[\leadsto {\left(\frac{N}{{N}^{\color{blue}{0}} - \frac{0.5}{N}}\right)}^{-1} \]

   8. metadata-eval 92.9%

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

8. Applied egg-rr 92.9%

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

   1. unpow-1 92.9%

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

10. Simplified 92.9%

    \[\leadsto \color{blue}{\frac{1}{\frac{N}{1 - \frac{0.5}{N}}}} \]
11. Add Preprocessing

Alternative 7: 92.3% accurate, 22.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[N_] := N[(N[(1.0 / N), $MachinePrecision] - N[(N[(0.5 / N), $MachinePrecision] / N), $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 92.9%

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

   1. associate-*r/ 92.9%

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

   2. metadata-eval 92.9%

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

5. Simplified 92.9%

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

6. Taylor expanded in N around 0 92.9%

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

   1. div-sub 92.9%

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

   2. pow1 92.9%

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

   3. pow1 92.9%

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

   4. pow-div 92.9%

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

   5. metadata-eval 92.9%

      \[\leadsto \frac{{N}^{\color{blue}{0}} - \frac{0.5}{N}}{N} \]

   6. metadata-eval 92.9%

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

   7. div-sub 92.9%

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

8. Applied egg-rr 92.9%

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

Alternative 8: 92.4% 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 21.6%

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

3. Taylor expanded in N around inf 92.9%

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

   1. associate-*r/ 92.9%

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

   2. metadata-eval 92.9%

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

5. Simplified 92.9%

   \[\leadsto \color{blue}{\frac{1 - \frac{0.5}{N}}{N}} \]
6. Add Preprocessing

Alternative 9: 84.3% 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 85.9%

   \[\leadsto \color{blue}{\frac{1}{N}} \]
4. Add Preprocessing

Developer Target 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]

Reproduce

herbie shell --seed 2024128 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  :pre (and (> N 1.0) (< N 1e+40))

  :alt
  (! :herbie-platform default (log1p (/ 1 N)))

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