2log (problem 3.3.6)

Percentage Accurate: 24.1% → 99.8%

Time: 6.3s

Alternatives: 13

Speedup: 9.4×

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: 24.1% 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

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: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{log1p}\left({N}^{-1}\right) \end{array} \]

FPCore

(FPCore (N) :precision binary64 (log1p (pow N -1.0)))

C

double code(double N) {
	return log1p(pow(N, -1.0));
}

Java

public static double code(double N) {
	return Math.log1p(Math.pow(N, -1.0));
}

Python

def code(N):
	return math.log1p(math.pow(N, -1.0))

Julia

function code(N)
	return log1p((N ^ -1.0))
end

Wolfram

code[N_] := N[Log[1 + N[Power[N, -1.0], $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 20.9%

   \[\log \left(N + 1\right) - \log N \]
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. lift-+.f64 N/A

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

   6. div-add N/A

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

   7. *-inverses N/A

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

   8. lower-log1p.f64 N/A

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

   9. inv-pow N/A

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

   10. lower-pow.f64 99.9

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

4. Applied rewrites 99.9%

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

   1. lift-pow.f64 N/A

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

   2. inv-pow N/A

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

   3. lift-/.f64 99.9

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

6. Applied rewrites 99.9%

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

7. Final simplification 99.9%

   \[\leadsto \mathsf{log1p}\left({N}^{-1}\right) \]
8. Add Preprocessing

Alternative 2: 84.3% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (N) :precision binary64 (pow N -1.0))

C

double code(double N) {
	return pow(N, -1.0);
}

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 = n ** (-1.0d0)
end function

Java

public static double code(double N) {
	return Math.pow(N, -1.0);
}

Python

def code(N):
	return math.pow(N, -1.0)

Julia

function code(N)
	return N ^ -1.0
end

MATLAB

function tmp = code(N)
	tmp = N ^ -1.0;
end

Wolfram

code[N_] := N[Power[N, -1.0], $MachinePrecision]

Derivation

1. Initial program 20.9%

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

3. Taylor expanded in N around inf

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

   1. lower-/.f64 86.6

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

5. Applied rewrites 86.6%

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

6. Final simplification 86.6%

   \[\leadsto {N}^{-1} \]
7. Add Preprocessing

Alternative 3: 96.1% accurate, 3.8× speedup

Math

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

FPCore

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

C

double code(double N) {
	return ((((((-0.25 / N) + 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 = (((((((-0.25d0) / n) + 0.3333333333333333d0) / n) - 0.5d0) / n) - (-1.0d0)) / n
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 20.9%

   \[\log \left(N + 1\right) - \log N \]
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 96.6%

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

Alternative 4: 96.0% accurate, 4.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 20.9%

   \[\log \left(N + 1\right) - \log N \]
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. lift-+.f64 N/A

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

   6. div-add N/A

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

   7. *-inverses N/A

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

   8. lower-log1p.f64 N/A

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

   9. inv-pow N/A

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

   10. lower-pow.f64 99.9

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

4. Applied rewrites 99.9%

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

5. Taylor expanded in N around inf

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

7. Applied rewrites 96.5%

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

Alternative 5: 95.8% accurate, 4.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 20.9%

   \[\log \left(N + 1\right) - \log N \]
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. lift-+.f64 N/A

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

   6. div-add N/A

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

   7. *-inverses N/A

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

   8. lower-log1p.f64 N/A

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

   9. inv-pow N/A

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

   10. lower-pow.f64 99.9

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

4. Applied rewrites 99.9%

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

5. Taylor expanded in N around inf

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

7. Applied rewrites 96.5%

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

9. Applied rewrites 96.2%

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

10. Final simplification 96.2%

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

Alternative 6: 94.8% accurate, 4.6× speedup

Math

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

FPCore

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

C

double code(double N) {
	return (((0.3333333333333333 / N) - 0.5) / (N * 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 * n)) - ((-1.0d0) / n)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 20.9%

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

3. Taylor expanded in N around inf

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

5. Applied rewrites 95.4%

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

7. Applied rewrites 95.4%

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

9. Applied rewrites 95.4%

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

Alternative 7: 94.8% 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 20.9%

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

3. Taylor expanded in N around inf

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

5. Applied rewrites 95.4%

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

Alternative 8: 94.8% accurate, 5.2× speedup

Math

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

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(Float64(0.3333333333333333 / N) - 0.5) + N) / 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[(N[(0.3333333333333333 / N), $MachinePrecision] - 0.5), $MachinePrecision] + N), $MachinePrecision] / N), $MachinePrecision] / N), $MachinePrecision]

Derivation

1. Initial program 20.9%

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

3. Taylor expanded in N around inf

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

5. Applied rewrites 95.4%

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

6. Taylor expanded in N around inf

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

8. Applied rewrites 93.2%

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

9. Taylor expanded in N around 0

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

11. Applied rewrites 95.4%

    \[\leadsto \frac{\frac{\left(\frac{0.3333333333333333}{N} - 0.5\right) + N}{N}}{N} \]
12. Add Preprocessing

Alternative 9: 94.7% accurate, 5.6× speedup

Math

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

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) / Float64(N * N)) / N)
end

Wolfram

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

Derivation

1. Initial program 20.9%

   \[\log \left(N + 1\right) - \log N \]
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. lift-+.f64 N/A

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

   6. div-add N/A

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

   7. *-inverses N/A

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

   8. lower-log1p.f64 N/A

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

   9. inv-pow N/A

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

   10. lower-pow.f64 99.9

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

4. Applied rewrites 99.9%

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

5. Taylor expanded in N around inf

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

7. Applied rewrites 96.5%

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

8. Taylor expanded in N around inf

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

10. Applied rewrites 95.4%

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

Alternative 10: 94.5% accurate, 5.8× speedup

Math

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

FPCore

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

C

double code(double N) {
	return ((0.5 - (0.3333333333333333 / N)) - 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 - (0.3333333333333333d0 / n)) - n) / (-n * n)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 20.9%

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

3. Taylor expanded in N around inf

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

5. Applied rewrites 95.4%

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

7. Applied rewrites 95.1%

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

8. Taylor expanded in N around 0

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

10. Applied rewrites 95.1%

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

11. Final simplification 95.1%

    \[\leadsto \frac{\left(0.5 - \frac{0.3333333333333333}{N}\right) - N}{\left(-N\right) \cdot N} \]
12. Add Preprocessing

Alternative 11: 92.2% 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 20.9%

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

3. Taylor expanded in N around -inf

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

   1. mul-1-neg N/A

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

   2. *-lft-identity N/A

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

   3. associate-*r/ N/A

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

   4. distribute-neg-frac2 N/A

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

   5. *-lft-identity N/A

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

   6. mul-1-neg N/A

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

   7. associate-/r* N/A

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

   8. lower-/.f64 N/A

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

   9. div-sub N/A

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

   10. metadata-eval N/A

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

   11. distribute-neg-frac2 N/A

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

   12. /-rgt-identity N/A

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

   13. *-commutative N/A

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

   14. *-commutative N/A

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

   15. distribute-lft-neg-in N/A

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

   16. metadata-eval N/A

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

   17. lower--.f64 N/A

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

   18. metadata-eval N/A

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

   19. associate-*r/ N/A

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

   20. metadata-eval N/A

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

   21. lower-/.f64 93.2

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

5. Applied rewrites 93.2%

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

Alternative 12: 92.2% accurate, 8.0× speedup

Math

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

FPCore

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

C

double code(double N) {
	return ((N - 0.5) / 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 = ((n - 0.5d0) / n) / n
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 20.9%

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

3. Taylor expanded in N around inf

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

5. Applied rewrites 95.4%

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

6. Taylor expanded in N around inf

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

8. Applied rewrites 93.2%

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

Alternative 13: 91.9% accurate, 9.4× speedup

Math

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

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(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 20.9%

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

3. Taylor expanded in N around inf

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

5. Applied rewrites 95.4%

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

7. Applied rewrites 95.1%

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

8. Taylor expanded in N around inf

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

10. Applied rewrites 92.9%

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

11. Final simplification 92.9%

    \[\leadsto \frac{0.5 - N}{\left(-N\right) \cdot N} \]
12. Add Preprocessing

Developer Target 1: 96.1% 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 2024346 
(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)))