2log (problem 3.3.6)

Percentage Accurate: 23.4% → 99.4%

Time: 9.1s

Alternatives: 8

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: 23.4% 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: 99.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

double code(double N) {
	double tmp;
	if ((log((N + 1.0)) - log(N)) <= 0.0005) {
		tmp = (1.0 - (((((0.25 / N) + -0.3333333333333333) / N) - -0.5) / N)) / N;
	} else {
		tmp = -log((N / (N + 1.0)));
	}
	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 - (((((0.25d0 / n) + (-0.3333333333333333d0)) / n) - (-0.5d0)) / n)) / n
    else
        tmp = -log((n / (n + 1.0d0)))
    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 - (((((0.25 / N) + -0.3333333333333333) / N) - -0.5) / N)) / N;
	} else {
		tmp = -Math.log((N / (N + 1.0)));
	}
	return tmp;
}

Python

def code(N):
	tmp = 0
	if (math.log((N + 1.0)) - math.log(N)) <= 0.0005:
		tmp = (1.0 - (((((0.25 / N) + -0.3333333333333333) / N) - -0.5) / N)) / N
	else:
		tmp = -math.log((N / (N + 1.0)))
	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(Float64(Float64(Float64(0.25 / N) + -0.3333333333333333) / N) - -0.5) / N)) / N);
	else
		tmp = Float64(-log(Float64(N / Float64(N + 1.0))));
	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 - (((((0.25 / N) + -0.3333333333333333) / N) - -0.5) / N)) / N;
	else
		tmp = -log((N / (N + 1.0)));
	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[(N[(N[(N[(0.25 / N), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] / N), $MachinePrecision] - -0.5), $MachinePrecision] / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision], (-N[Log[N[(N / N[(N + 1.0), $MachinePrecision]), $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 17.7%

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

      1. +-commutative 17.7%

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

      2. log1p-define 17.7%

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

   3. Simplified 17.7%

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

   5. Taylor expanded in N around -inf 99.8%

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

      1. mul-1-neg 99.8%

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

      2. distribute-neg-frac2 99.8%

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

   7. Simplified 99.8%

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

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

   1. Initial program 92.1%

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

      1. +-commutative 92.1%

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

      2. log1p-define 92.1%

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

   3. Simplified 92.1%

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

      1. expm1-log1p-u 91.4%

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

      2. expm1-undefine 91.7%

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

      3. log1p-undefine 91.8%

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

      4. rem-exp-log 92.2%

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

   6. Applied egg-rr 92.2%

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

      1. log1p-undefine 92.1%

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

      2. +-commutative 92.1%

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

      3. add-exp-log 91.7%

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

      4. expm1-define 91.5%

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

      5. log1p-define 91.3%

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

      6. expm1-log1p-u 92.1%

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

      7. diff-log 94.4%

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

      8. clear-num 94.3%

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

      9. log-rec 95.0%

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

   8. Applied egg-rr 95.0%

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

4. Final simplification 99.3%

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

Alternative 2: 99.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;N \leq 1350:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{\frac{\frac{0.25}{N} + -0.3333333333333333}{N} - -0.5}{N}}{N}\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= N 1350.0)
   (log (/ (+ N 1.0) N))
   (/ (- 1.0 (/ (- (/ (+ (/ 0.25 N) -0.3333333333333333) N) -0.5) N)) N)))

C

double code(double N) {
	double tmp;
	if (N <= 1350.0) {
		tmp = log(((N + 1.0) / N));
	} else {
		tmp = (1.0 - (((((0.25 / N) + -0.3333333333333333) / N) - -0.5) / N)) / N;
	}
	return tmp;
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= 1350.0d0) then
        tmp = log(((n + 1.0d0) / n))
    else
        tmp = (1.0d0 - (((((0.25d0 / n) + (-0.3333333333333333d0)) / n) - (-0.5d0)) / n)) / n
    end if
    code = tmp
end function

Java

public static double code(double N) {
	double tmp;
	if (N <= 1350.0) {
		tmp = Math.log(((N + 1.0) / N));
	} else {
		tmp = (1.0 - (((((0.25 / N) + -0.3333333333333333) / N) - -0.5) / N)) / N;
	}
	return tmp;
}

Python

def code(N):
	tmp = 0
	if N <= 1350.0:
		tmp = math.log(((N + 1.0) / N))
	else:
		tmp = (1.0 - (((((0.25 / N) + -0.3333333333333333) / N) - -0.5) / N)) / N
	return tmp

Julia

function code(N)
	tmp = 0.0
	if (N <= 1350.0)
		tmp = log(Float64(Float64(N + 1.0) / N));
	else
		tmp = Float64(Float64(1.0 - Float64(Float64(Float64(Float64(Float64(0.25 / N) + -0.3333333333333333) / N) - -0.5) / N)) / N);
	end
	return tmp
end

MATLAB

function tmp_2 = code(N)
	tmp = 0.0;
	if (N <= 1350.0)
		tmp = log(((N + 1.0) / N));
	else
		tmp = (1.0 - (((((0.25 / N) + -0.3333333333333333) / N) - -0.5) / N)) / N;
	end
	tmp_2 = tmp;
end

Wolfram

code[N_] := If[LessEqual[N, 1350.0], N[Log[N[(N[(N + 1.0), $MachinePrecision] / N), $MachinePrecision]], $MachinePrecision], N[(N[(1.0 - N[(N[(N[(N[(N[(0.25 / N), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] / N), $MachinePrecision] - -0.5), $MachinePrecision] / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if N < 1350

   1. Initial program 92.1%

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

      1. +-commutative 92.1%

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

      2. log1p-define 92.1%

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

   3. Simplified 92.1%

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

      1. add-log-exp 92.1%

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

      2. log1p-expm1-u 92.1%

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

      3. log1p-undefine 92.1%

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

      4. diff-log 91.9%

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

      5. log1p-undefine 92.0%

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

      6. rem-exp-log 92.5%

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

      7. +-commutative 92.5%

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

      8. add-exp-log 92.3%

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

      9. log1p-undefine 92.3%

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

      10. log1p-expm1-u 92.3%

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

      11. add-exp-log 94.4%

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

   6. Applied egg-rr 94.4%

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

   if 1350 < N

   1. Initial program 17.7%

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

      1. +-commutative 17.7%

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

      2. log1p-define 17.7%

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

   3. Simplified 17.7%

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

   5. Taylor expanded in N around -inf 99.8%

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

      1. mul-1-neg 99.8%

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

      2. distribute-neg-frac2 99.8%

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

   7. Simplified 99.8%

      \[\leadsto \color{blue}{\frac{-1 - \frac{-0.5 - \frac{\frac{0.25}{N} + -0.3333333333333333}{N}}{N}}{-N}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;N \leq 1350:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{\frac{\frac{0.25}{N} + -0.3333333333333333}{N} - -0.5}{N}}{N}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.4% accurate, 13.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 25.2%

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

   1. +-commutative 25.2%

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

   2. log1p-define 25.3%

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

3. Simplified 25.3%

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

5. Taylor expanded in N around -inf 95.0%

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

   1. mul-1-neg 95.0%

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

   2. distribute-neg-frac2 95.0%

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

7. Simplified 95.0%

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

8. Final simplification 95.0%

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

Alternative 4: 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 25.2%

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

   1. +-commutative 25.2%

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

   2. log1p-define 25.3%

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

3. Simplified 25.3%

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

5. Taylor expanded in N around inf 93.7%

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

   1. associate--l+ 93.7%

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

   2. unpow2 93.7%

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

   3. associate-/r* 93.7%

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

   4. metadata-eval 93.7%

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

   5. associate-*r/ 93.7%

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

   6. associate-*r/ 93.7%

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

   7. metadata-eval 93.7%

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

   8. div-sub 93.7%

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

   9. sub-neg 93.7%

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

   10. metadata-eval 93.7%

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

   11. +-commutative 93.7%

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

   12. associate-*r/ 93.7%

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

   13. metadata-eval 93.7%

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

7. Simplified 93.7%

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

8. Final simplification 93.7%

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

Alternative 5: 92.6% 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 25.2%

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

   1. +-commutative 25.2%

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

   2. log1p-define 25.3%

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

3. Simplified 25.3%

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

5. Taylor expanded in N around inf 90.9%

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

   1. associate-*r/ 90.9%

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

   2. metadata-eval 90.9%

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

7. Simplified 90.9%

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

   1. clear-num 90.9%

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

   2. inv-pow 90.9%

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

9. Applied egg-rr 90.9%

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

    1. unpow-1 90.9%

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

    2. sub-neg 90.9%

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

    3. distribute-neg-frac 90.9%

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

    4. metadata-eval 90.9%

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

11. Simplified 90.9%

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

12. Final simplification 90.9%

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

Alternative 6: 92.6% 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 25.2%

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

   1. +-commutative 25.2%

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

   2. log1p-define 25.3%

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

3. Simplified 25.3%

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

5. Taylor expanded in N around inf 90.9%

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

   1. associate-*r/ 90.9%

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

   2. metadata-eval 90.9%

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

7. Simplified 90.9%

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

Alternative 7: 84.8% 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 25.2%

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

   1. +-commutative 25.2%

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

   2. log1p-define 25.3%

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

3. Simplified 25.3%

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

5. Taylor expanded in N around inf 83.4%

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

Alternative 8: 8.4% accurate, 205.0× speedup

Math

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

FPCore

(FPCore (N) :precision binary64 N)

C

double code(double N) {
	return N;
}

Fortran

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

Java

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

Python

def code(N):
	return N

Julia

function code(N)
	return N
end

MATLAB

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

Wolfram

code[N_] := N

Derivation

1. Initial program 25.2%

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

   1. +-commutative 25.2%

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

   2. log1p-define 25.3%

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

3. Simplified 25.3%

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

5. Taylor expanded in N around inf 83.4%

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

   1. add-exp-log 79.9%

      \[\leadsto \frac{1}{\color{blue}{e^{\log N}}} \]

   2. rec-exp 79.9%

      \[\leadsto \color{blue}{e^{-\log N}} \]

   3. add-sqr-sqrt 79.0%

      \[\leadsto e^{-\color{blue}{\sqrt{\log N} \cdot \sqrt{\log N}}} \]

   4. distribute-rgt-neg-in 79.0%

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

   5. add-sqr-sqrt 0.0%

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

   6. sqrt-unprod 8.6%

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

   7. sqr-neg 8.6%

      \[\leadsto e^{\sqrt{\log N} \cdot \sqrt{\color{blue}{\sqrt{\log N} \cdot \sqrt{\log N}}}} \]

   8. add-sqr-sqrt 8.6%

      \[\leadsto e^{\sqrt{\log N} \cdot \sqrt{\color{blue}{\log N}}} \]

   9. add-sqr-sqrt 8.6%

      \[\leadsto e^{\color{blue}{\log N}} \]

   10. add-exp-log 8.6%

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

   11. add-sqr-sqrt 8.6%

       \[\leadsto \color{blue}{\sqrt{N} \cdot \sqrt{N}} \]

   12. sqrt-unprod 8.6%

       \[\leadsto \color{blue}{\sqrt{N \cdot N}} \]

   13. sqr-neg 8.6%

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

   14. sqrt-unprod 0.0%

       \[\leadsto \color{blue}{\sqrt{-N} \cdot \sqrt{-N}} \]

   15. add-sqr-sqrt 1.6%

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

   16. neg-sub0 1.6%

       \[\leadsto \color{blue}{0 - N} \]

   17. sub-neg 1.6%

       \[\leadsto \color{blue}{0 + \left(-N\right)} \]

   18. add-sqr-sqrt 0.0%

       \[\leadsto 0 + \color{blue}{\sqrt{-N} \cdot \sqrt{-N}} \]

   19. sqrt-unprod 8.6%

       \[\leadsto 0 + \color{blue}{\sqrt{\left(-N\right) \cdot \left(-N\right)}} \]

   20. sqr-neg 8.6%

       \[\leadsto 0 + \sqrt{\color{blue}{N \cdot N}} \]

   21. sqrt-unprod 8.6%

       \[\leadsto 0 + \color{blue}{\sqrt{N} \cdot \sqrt{N}} \]

   22. add-sqr-sqrt 8.6%

       \[\leadsto 0 + \color{blue}{N} \]

7. Applied egg-rr 8.6%

   \[\leadsto \color{blue}{0 + N} \]
8. Step-by-step derivation

   1. +-lft-identity 8.6%

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

9. Simplified 8.6%

   \[\leadsto \color{blue}{N} \]
10. 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 2024139 
(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)))