2log (problem 3.3.6)

Percentage Accurate: 24.4% → 99.3%

Time: 6.9s

Alternatives: 9

Speedup: 17.3×

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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(n)
use fmin_fmax_functions
    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: 24.4% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(n)
use fmin_fmax_functions
    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: 99.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;N \leq 1600:\\ \;\;\;\;\log \left(\frac{N - -1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\mathsf{fma}\left(0.3333333333333333, N, -0.25\right)}{N \cdot N} - 0.5}{N} - -1}{N}\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= N 1600.0)
   (log (/ (- N -1.0) N))
   (/ (- (/ (- (/ (fma 0.3333333333333333 N -0.25) (* N N)) 0.5) N) -1.0) N)))

C

double code(double N) {
	double tmp;
	if (N <= 1600.0) {
		tmp = log(((N - -1.0) / N));
	} else {
		tmp = ((((fma(0.3333333333333333, N, -0.25) / (N * N)) - 0.5) / N) - -1.0) / N;
	}
	return tmp;
}

Julia

function code(N)
	tmp = 0.0
	if (N <= 1600.0)
		tmp = log(Float64(Float64(N - -1.0) / N));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(fma(0.3333333333333333, N, -0.25) / Float64(N * N)) - 0.5) / N) - -1.0) / N);
	end
	return tmp
end

Wolfram

code[N_] := If[LessEqual[N, 1600.0], N[Log[N[(N[(N - -1.0), $MachinePrecision] / N), $MachinePrecision]], $MachinePrecision], N[(N[(N[(N[(N[(N[(0.3333333333333333 * N + -0.25), $MachinePrecision] / N[(N * N), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] / N), $MachinePrecision] - -1.0), $MachinePrecision] / N), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if N < 1600

   1. Initial program 91.1%

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

      1. lift--.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. diff-log N/A

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

      5. lower-log.f64 N/A

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

      6. lower-/.f64 93.5

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

      7. lift-+.f64 N/A

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

      8. metadata-eval N/A

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

      9. fp-cancel-sign-sub-inv N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lower--.f64 N/A

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

      14. metadata-eval 93.5

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

   4. Applied rewrites 93.5%

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

   if 1600 < N

   1. Initial program 19.6%

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

   3. Taylor expanded in N around inf

      \[\leadsto \color{blue}{\frac{\left(1 + \frac{\frac{1}{3}}{{N}^{2}}\right) - \left(\frac{1}{2} \cdot \frac{1}{N} + \frac{1}{4} \cdot \frac{1}{{N}^{3}}\right)}{N}} \]

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in N around 0

      \[\leadsto \frac{\frac{\frac{\frac{1}{3} \cdot N - \frac{1}{4}}{{N}^{2}} - \frac{1}{2}}{N} - -1}{N} \]
   7. Step-by-step derivation

   8. Applied rewrites 99.8%

      \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(0.3333333333333333, N, -0.25\right)}{N \cdot N} - 0.5}{N} - -1}{N} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 96.2% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\frac{\mathsf{fma}\left(0.3333333333333333, N, -0.25\right)}{N \cdot N} - 0.5}{N} - -1}{N} \end{array} \]

FPCore

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

C

double code(double N) {
	return ((((fma(0.3333333333333333, N, -0.25) / (N * N)) - 0.5) / N) - -1.0) / N;
}

Julia

function code(N)
	return Float64(Float64(Float64(Float64(Float64(fma(0.3333333333333333, N, -0.25) / Float64(N * N)) - 0.5) / N) - -1.0) / N)
end

Wolfram

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

Derivation

1. Initial program 25.2%

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

3. Taylor expanded in N around inf

   \[\leadsto \color{blue}{\frac{\left(1 + \frac{\frac{1}{3}}{{N}^{2}}\right) - \left(\frac{1}{2} \cdot \frac{1}{N} + \frac{1}{4} \cdot \frac{1}{{N}^{3}}\right)}{N}} \]

5. Applied rewrites 96.4%

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

6. Taylor expanded in N around 0

   \[\leadsto \frac{\frac{\frac{\frac{1}{3} \cdot N - \frac{1}{4}}{{N}^{2}} - \frac{1}{2}}{N} - -1}{N} \]
7. Step-by-step derivation

8. Applied rewrites 96.4%

   \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(0.3333333333333333, N, -0.25\right)}{N \cdot N} - 0.5}{N} - -1}{N} \]
9. Add Preprocessing

Alternative 3: 95.8% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(N - 0.5, N, 0.3333333333333333\right), N, -0.25\right)}{N \cdot N}}{N \cdot N} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (/ (/ (fma (fma (- N 0.5) N 0.3333333333333333) N -0.25) (* N N)) (* N N)))

C

double code(double N) {
	return (fma(fma((N - 0.5), N, 0.3333333333333333), N, -0.25) / (N * N)) / (N * N);
}

Julia

function code(N)
	return Float64(Float64(fma(fma(Float64(N - 0.5), N, 0.3333333333333333), N, -0.25) / Float64(N * N)) / Float64(N * N))
end

Wolfram

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

Derivation

1. Initial program 25.2%

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

3. Taylor expanded in N around inf

   \[\leadsto \color{blue}{\frac{\left(1 + \frac{\frac{1}{3}}{{N}^{2}}\right) - \left(\frac{1}{2} \cdot \frac{1}{N} + \frac{1}{4} \cdot \frac{1}{{N}^{3}}\right)}{N}} \]

5. Applied rewrites 96.4%

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

6. Taylor expanded in N around 0

   \[\leadsto \frac{\frac{\frac{\frac{1}{3} \cdot N - \frac{1}{4}}{{N}^{2}} - \frac{1}{2}}{N} - -1}{N} \]
7. Step-by-step derivation

8. Applied rewrites 96.4%

   \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(0.3333333333333333, N, -0.25\right)}{N \cdot N} - 0.5}{N} - -1}{N} \]
9. Step-by-step derivation

10. Applied rewrites 96.1%

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

11. Taylor expanded in N around 0

    \[\leadsto \frac{\frac{N \cdot \left(\frac{1}{3} + N \cdot \left(N - \frac{1}{2}\right)\right) - \frac{1}{4}}{{N}^{2}}}{\color{blue}{N} \cdot N} \]
12. Step-by-step derivation

13. Applied rewrites 95.9%

    \[\leadsto \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(N - 0.5, N, 0.3333333333333333\right), N, -0.25\right)}{N \cdot N}}{\color{blue}{N} \cdot N} \]
14. Add Preprocessing

Alternative 4: 94.9% accurate, 5.2× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 25.2%

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

3. Taylor expanded in N around inf

   \[\leadsto \color{blue}{\frac{\left(1 + \frac{\frac{1}{3}}{{N}^{2}}\right) - \left(\frac{1}{2} \cdot \frac{1}{N} + \frac{1}{4} \cdot \frac{1}{{N}^{3}}\right)}{N}} \]

5. Applied rewrites 96.4%

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

6. Taylor expanded in N around inf

   \[\leadsto \frac{\frac{\frac{\frac{1}{3}}{N} - \frac{1}{2}}{N} - -1}{N} \]
7. Step-by-step derivation

8. Applied rewrites 95.0%

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

Alternative 5: 94.6% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\mathsf{fma}\left(N - 0.5, N, 0.3333333333333333\right)}{N}}{N \cdot N} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (/ (/ (fma (- N 0.5) N 0.3333333333333333) N) (* N N)))

C

double code(double N) {
	return (fma((N - 0.5), N, 0.3333333333333333) / N) / (N * N);
}

Julia

function code(N)
	return Float64(Float64(fma(Float64(N - 0.5), N, 0.3333333333333333) / N) / Float64(N * N))
end

Wolfram

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

Derivation

1. Initial program 25.2%

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

3. Taylor expanded in N around inf

   \[\leadsto \color{blue}{\frac{\left(1 + \frac{\frac{1}{3}}{{N}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{N}}{N}} \]
4. Step-by-step derivation

5. Applied rewrites 94.7%

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

6. Taylor expanded in N around 0

   \[\leadsto \frac{\frac{\frac{1}{3} + N \cdot \left(N - \frac{1}{2}\right)}{N}}{\color{blue}{N} \cdot N} \]
7. Step-by-step derivation

8. Applied rewrites 94.8%

   \[\leadsto \frac{\frac{\mathsf{fma}\left(N - 0.5, N, 0.3333333333333333\right)}{N}}{\color{blue}{N} \cdot N} \]
9. Add Preprocessing

Alternative 6: 94.6% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \frac{\left(\frac{0.3333333333333333}{N} - 0.5\right) + N}{N \cdot N} \end{array} \]

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 25.2%

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

3. Taylor expanded in N around inf

   \[\leadsto \color{blue}{\frac{\left(1 + \frac{\frac{1}{3}}{{N}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{N}}{N}} \]
4. Step-by-step derivation

5. Applied rewrites 94.7%

   \[\leadsto \color{blue}{\frac{\left(\frac{0.3333333333333333}{N} - 0.5\right) + N}{N \cdot N}} \]
6. Add Preprocessing

Alternative 7: 92.3% accurate, 8.0× 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(n)
use fmin_fmax_functions
    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(-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 25.2%

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

3. Taylor expanded in N around inf

   \[\leadsto \color{blue}{\frac{\left(1 + \frac{\frac{1}{3}}{{N}^{2}}\right) - \left(\frac{1}{2} \cdot \frac{1}{N} + \frac{1}{4} \cdot \frac{1}{{N}^{3}}\right)}{N}} \]

5. Applied rewrites 96.4%

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

6. Taylor expanded in N around inf

   \[\leadsto \frac{\frac{\frac{-1}{2}}{N} - -1}{N} \]
7. Step-by-step derivation

8. Applied rewrites 92.0%

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

Alternative 8: 92.0% accurate, 10.4× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 25.2%

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

3. Taylor expanded in N around inf

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

5. Applied rewrites 91.7%

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

Alternative 9: 84.0% accurate, 17.3× 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(n)
use fmin_fmax_functions
    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. Add Preprocessing

3. Taylor expanded in N around inf

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

5. Applied rewrites 83.1%

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

Developer Target 1: 96.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \left(\left(\frac{1}{N} + \frac{-1}{2 \cdot {N}^{2}}\right) + \frac{1}{3 \cdot {N}^{3}}\right) + \frac{-1}{4 \cdot {N}^{4}} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (+
  (+ (+ (/ 1.0 N) (/ -1.0 (* 2.0 (pow N 2.0)))) (/ 1.0 (* 3.0 (pow N 3.0))))
  (/ -1.0 (* 4.0 (pow N 4.0)))))

C

double code(double N) {
	return (((1.0 / N) + (-1.0 / (2.0 * pow(N, 2.0)))) + (1.0 / (3.0 * pow(N, 3.0)))) + (-1.0 / (4.0 * pow(N, 4.0)));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(n)
use fmin_fmax_functions
    real(8), intent (in) :: n
    code = (((1.0d0 / n) + ((-1.0d0) / (2.0d0 * (n ** 2.0d0)))) + (1.0d0 / (3.0d0 * (n ** 3.0d0)))) + ((-1.0d0) / (4.0d0 * (n ** 4.0d0)))
end function

Java

public static double code(double N) {
	return (((1.0 / N) + (-1.0 / (2.0 * Math.pow(N, 2.0)))) + (1.0 / (3.0 * Math.pow(N, 3.0)))) + (-1.0 / (4.0 * Math.pow(N, 4.0)));
}

Python

def code(N):
	return (((1.0 / N) + (-1.0 / (2.0 * math.pow(N, 2.0)))) + (1.0 / (3.0 * math.pow(N, 3.0)))) + (-1.0 / (4.0 * math.pow(N, 4.0)))

Julia

function code(N)
	return Float64(Float64(Float64(Float64(1.0 / N) + Float64(-1.0 / Float64(2.0 * (N ^ 2.0)))) + Float64(1.0 / Float64(3.0 * (N ^ 3.0)))) + Float64(-1.0 / Float64(4.0 * (N ^ 4.0))))
end

MATLAB

function tmp = code(N)
	tmp = (((1.0 / N) + (-1.0 / (2.0 * (N ^ 2.0)))) + (1.0 / (3.0 * (N ^ 3.0)))) + (-1.0 / (4.0 * (N ^ 4.0)));
end

Wolfram

code[N_] := N[(N[(N[(N[(1.0 / N), $MachinePrecision] + N[(-1.0 / N[(2.0 * N[Power[N, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(3.0 * N[Power[N, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[(4.0 * N[Power[N, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2025019 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  :pre (and (> N 1.0) (< N 1e+40))

  :alt
  (! :herbie-platform default (+ (/ 1 N) (/ -1 (* 2 (pow N 2))) (/ 1 (* 3 (pow N 3))) (/ -1 (* 4 (pow N 4)))))

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