2log (problem 3.3.6)

Percentage Accurate: 23.9% → 99.2%

Time: 3.7s

Alternatives: 11

Speedup: 4.1×

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: 23.9% 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: 99.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(N - -1\right)\\ t_1 := {t\_0}^{1.5}\\ \mathbf{if}\;N \leq 820:\\ \;\;\;\;\frac{t\_1 \cdot t\_1 - e^{\log \log N \cdot 3}}{\mathsf{fma}\left(t\_0, t\_0, \mathsf{fma}\left(\log N, \log N, t\_0 \cdot \log N\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{0.5 - \frac{0.3333333333333333 - \frac{0.25}{N}}{N}}{N}}{N}\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (let* ((t_0 (log (- N -1.0))) (t_1 (pow t_0 1.5)))
   (if (<= N 820.0)
     (/
      (- (* t_1 t_1) (exp (* (log (log N)) 3.0)))
      (fma t_0 t_0 (fma (log N) (log N) (* t_0 (log N)))))
     (/ (- 1.0 (/ (- 0.5 (/ (- 0.3333333333333333 (/ 0.25 N)) N)) N)) N))))

C

double code(double N) {
	double t_0 = log((N - -1.0));
	double t_1 = pow(t_0, 1.5);
	double tmp;
	if (N <= 820.0) {
		tmp = ((t_1 * t_1) - exp((log(log(N)) * 3.0))) / fma(t_0, t_0, fma(log(N), log(N), (t_0 * log(N))));
	} else {
		tmp = (1.0 - ((0.5 - ((0.3333333333333333 - (0.25 / N)) / N)) / N)) / N;
	}
	return tmp;
}

Julia

function code(N)
	t_0 = log(Float64(N - -1.0))
	t_1 = t_0 ^ 1.5
	tmp = 0.0
	if (N <= 820.0)
		tmp = Float64(Float64(Float64(t_1 * t_1) - exp(Float64(log(log(N)) * 3.0))) / fma(t_0, t_0, fma(log(N), log(N), Float64(t_0 * log(N)))));
	else
		tmp = Float64(Float64(1.0 - Float64(Float64(0.5 - Float64(Float64(0.3333333333333333 - Float64(0.25 / N)) / N)) / N)) / N);
	end
	return tmp
end

Wolfram

code[N_] := Block[{t$95$0 = N[Log[N[(N - -1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Power[t$95$0, 1.5], $MachinePrecision]}, If[LessEqual[N, 820.0], N[(N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[Exp[N[(N[Log[N[Log[N], $MachinePrecision]], $MachinePrecision] * 3.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * t$95$0 + N[(N[Log[N], $MachinePrecision] * N[Log[N], $MachinePrecision] + N[(t$95$0 * N[Log[N], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(N[(0.5 - N[(N[(0.3333333333333333 - N[(0.25 / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if N < 820

   1. Initial program 92.6%

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

      1. lift-+.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-log1p.f64 92.6

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

   3. Applied rewrites 92.6%

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

      1. lift--.f64 N/A

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

      2. lift-log1p.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. negate-sub N/A

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

      7. flip3-- N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. lift-log.f64 N/A

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

      12. lift--.f64 N/A

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

      13. lower-pow.f64 N/A

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

      14. lift-log.f64 N/A

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

      15. lower-fma.f64 N/A

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

   5. Applied rewrites 92.4%

      \[\leadsto \color{blue}{\frac{{\log \left(N - -1\right)}^{3} - {\log N}^{3}}{\mathsf{fma}\left(\log \left(N - -1\right), \log \left(N - -1\right), \mathsf{fma}\left(\log N, \log N, \log \left(N - -1\right) \cdot \log N\right)\right)}} \]
   6. Step-by-step derivation

      1. lift-log.f64 N/A

         \[\leadsto \frac{{\log \left(N - -1\right)}^{3} - {\color{blue}{\log N}}^{3}}{\mathsf{fma}\left(\log \left(N - -1\right), \log \left(N - -1\right), \mathsf{fma}\left(\log N, \log N, \log \left(N - -1\right) \cdot \log N\right)\right)} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{{\log \left(N - -1\right)}^{3} - \color{blue}{{\log N}^{3}}}{\mathsf{fma}\left(\log \left(N - -1\right), \log \left(N - -1\right), \mathsf{fma}\left(\log N, \log N, \log \left(N - -1\right) \cdot \log N\right)\right)} \]

      3. pow-to-exp N/A

         \[\leadsto \frac{{\log \left(N - -1\right)}^{3} - \color{blue}{e^{\log \log N \cdot 3}}}{\mathsf{fma}\left(\log \left(N - -1\right), \log \left(N - -1\right), \mathsf{fma}\left(\log N, \log N, \log \left(N - -1\right) \cdot \log N\right)\right)} \]

      4. lower-exp.f64 N/A

         \[\leadsto \frac{{\log \left(N - -1\right)}^{3} - \color{blue}{e^{\log \log N \cdot 3}}}{\mathsf{fma}\left(\log \left(N - -1\right), \log \left(N - -1\right), \mathsf{fma}\left(\log N, \log N, \log \left(N - -1\right) \cdot \log N\right)\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{{\log \left(N - -1\right)}^{3} - e^{\color{blue}{\log \log N \cdot 3}}}{\mathsf{fma}\left(\log \left(N - -1\right), \log \left(N - -1\right), \mathsf{fma}\left(\log N, \log N, \log \left(N - -1\right) \cdot \log N\right)\right)} \]

      6. lower-log.f64 N/A

         \[\leadsto \frac{{\log \left(N - -1\right)}^{3} - e^{\color{blue}{\log \log N} \cdot 3}}{\mathsf{fma}\left(\log \left(N - -1\right), \log \left(N - -1\right), \mathsf{fma}\left(\log N, \log N, \log \left(N - -1\right) \cdot \log N\right)\right)} \]

      7. lift-log.f64 91.6

         \[\leadsto \frac{{\log \left(N - -1\right)}^{3} - e^{\log \color{blue}{\log N} \cdot 3}}{\mathsf{fma}\left(\log \left(N - -1\right), \log \left(N - -1\right), \mathsf{fma}\left(\log N, \log N, \log \left(N - -1\right) \cdot \log N\right)\right)} \]

   7. Applied rewrites 91.6%

      \[\leadsto \frac{{\log \left(N - -1\right)}^{3} - \color{blue}{e^{\log \log N \cdot 3}}}{\mathsf{fma}\left(\log \left(N - -1\right), \log \left(N - -1\right), \mathsf{fma}\left(\log N, \log N, \log \left(N - -1\right) \cdot \log N\right)\right)} \]
   8. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\log \left(N - -1\right)}^{3}} - e^{\log \log N \cdot 3}}{\mathsf{fma}\left(\log \left(N - -1\right), \log \left(N - -1\right), \mathsf{fma}\left(\log N, \log N, \log \left(N - -1\right) \cdot \log N\right)\right)} \]

      2. lift--.f64 N/A

         \[\leadsto \frac{{\log \color{blue}{\left(N - -1\right)}}^{3} - e^{\log \log N \cdot 3}}{\mathsf{fma}\left(\log \left(N - -1\right), \log \left(N - -1\right), \mathsf{fma}\left(\log N, \log N, \log \left(N - -1\right) \cdot \log N\right)\right)} \]

      3. lift-log.f64 N/A

         \[\leadsto \frac{{\color{blue}{\log \left(N - -1\right)}}^{3} - e^{\log \log N \cdot 3}}{\mathsf{fma}\left(\log \left(N - -1\right), \log \left(N - -1\right), \mathsf{fma}\left(\log N, \log N, \log \left(N - -1\right) \cdot \log N\right)\right)} \]

      4. sqr-pow N/A

         \[\leadsto \frac{\color{blue}{{\log \left(N - -1\right)}^{\left(\frac{3}{2}\right)} \cdot {\log \left(N - -1\right)}^{\left(\frac{3}{2}\right)}} - e^{\log \log N \cdot 3}}{\mathsf{fma}\left(\log \left(N - -1\right), \log \left(N - -1\right), \mathsf{fma}\left(\log N, \log N, \log \left(N - -1\right) \cdot \log N\right)\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{{\log \left(N - -1\right)}^{\left(\frac{3}{2}\right)} \cdot {\log \left(N - -1\right)}^{\left(\frac{3}{2}\right)}} - e^{\log \log N \cdot 3}}{\mathsf{fma}\left(\log \left(N - -1\right), \log \left(N - -1\right), \mathsf{fma}\left(\log N, \log N, \log \left(N - -1\right) \cdot \log N\right)\right)} \]

      6. metadata-eval N/A

         \[\leadsto \frac{{\log \left(N - -1\right)}^{\color{blue}{\frac{3}{2}}} \cdot {\log \left(N - -1\right)}^{\left(\frac{3}{2}\right)} - e^{\log \log N \cdot 3}}{\mathsf{fma}\left(\log \left(N - -1\right), \log \left(N - -1\right), \mathsf{fma}\left(\log N, \log N, \log \left(N - -1\right) \cdot \log N\right)\right)} \]

      7. lower-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\log \left(N - -1\right)}^{\frac{3}{2}}} \cdot {\log \left(N - -1\right)}^{\left(\frac{3}{2}\right)} - e^{\log \log N \cdot 3}}{\mathsf{fma}\left(\log \left(N - -1\right), \log \left(N - -1\right), \mathsf{fma}\left(\log N, \log N, \log \left(N - -1\right) \cdot \log N\right)\right)} \]

      8. lift-log.f64 N/A

         \[\leadsto \frac{{\color{blue}{\log \left(N - -1\right)}}^{\frac{3}{2}} \cdot {\log \left(N - -1\right)}^{\left(\frac{3}{2}\right)} - e^{\log \log N \cdot 3}}{\mathsf{fma}\left(\log \left(N - -1\right), \log \left(N - -1\right), \mathsf{fma}\left(\log N, \log N, \log \left(N - -1\right) \cdot \log N\right)\right)} \]

      9. lift--.f64 N/A

         \[\leadsto \frac{{\log \color{blue}{\left(N - -1\right)}}^{\frac{3}{2}} \cdot {\log \left(N - -1\right)}^{\left(\frac{3}{2}\right)} - e^{\log \log N \cdot 3}}{\mathsf{fma}\left(\log \left(N - -1\right), \log \left(N - -1\right), \mathsf{fma}\left(\log N, \log N, \log \left(N - -1\right) \cdot \log N\right)\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{{\log \left(N - -1\right)}^{\frac{3}{2}} \cdot {\log \left(N - -1\right)}^{\color{blue}{\frac{3}{2}}} - e^{\log \log N \cdot 3}}{\mathsf{fma}\left(\log \left(N - -1\right), \log \left(N - -1\right), \mathsf{fma}\left(\log N, \log N, \log \left(N - -1\right) \cdot \log N\right)\right)} \]

      11. lower-pow.f64 N/A

          \[\leadsto \frac{{\log \left(N - -1\right)}^{\frac{3}{2}} \cdot \color{blue}{{\log \left(N - -1\right)}^{\frac{3}{2}}} - e^{\log \log N \cdot 3}}{\mathsf{fma}\left(\log \left(N - -1\right), \log \left(N - -1\right), \mathsf{fma}\left(\log N, \log N, \log \left(N - -1\right) \cdot \log N\right)\right)} \]

      12. lift-log.f64 N/A

          \[\leadsto \frac{{\log \left(N - -1\right)}^{\frac{3}{2}} \cdot {\color{blue}{\log \left(N - -1\right)}}^{\frac{3}{2}} - e^{\log \log N \cdot 3}}{\mathsf{fma}\left(\log \left(N - -1\right), \log \left(N - -1\right), \mathsf{fma}\left(\log N, \log N, \log \left(N - -1\right) \cdot \log N\right)\right)} \]

      13. lift--.f64 91.4

          \[\leadsto \frac{{\log \left(N - -1\right)}^{1.5} \cdot {\log \color{blue}{\left(N - -1\right)}}^{1.5} - e^{\log \log N \cdot 3}}{\mathsf{fma}\left(\log \left(N - -1\right), \log \left(N - -1\right), \mathsf{fma}\left(\log N, \log N, \log \left(N - -1\right) \cdot \log N\right)\right)} \]

   9. Applied rewrites 91.4%

      \[\leadsto \frac{\color{blue}{{\log \left(N - -1\right)}^{1.5} \cdot {\log \left(N - -1\right)}^{1.5}} - e^{\log \log N \cdot 3}}{\mathsf{fma}\left(\log \left(N - -1\right), \log \left(N - -1\right), \mathsf{fma}\left(\log N, \log N, \log \left(N - -1\right) \cdot \log N\right)\right)} \]

   if 820 < N

   1. Initial program 18.9%

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

   2. Taylor expanded in N around -inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{-1 \cdot \frac{-1 \cdot \frac{\frac{1}{4} \cdot \frac{1}{N} - \frac{1}{3}}{N} - \frac{1}{2}}{N} - 1}{N}\right) \]

      2. distribute-neg-frac N/A

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

      3. lower-/.f64 N/A

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

   4. Applied rewrites 99.7%

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

Alternative 2: 99.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(N - -1\right)\\ \mathbf{if}\;N \leq 850:\\ \;\;\;\;\frac{{t\_0}^{3} - e^{\log \log N \cdot 3}}{\mathsf{fma}\left(t\_0, t\_0, \mathsf{fma}\left(\log N, \log N, t\_0 \cdot \log N\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{0.5 - \frac{0.3333333333333333 - \frac{0.25}{N}}{N}}{N}}{N}\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (let* ((t_0 (log (- N -1.0))))
   (if (<= N 850.0)
     (/
      (- (pow t_0 3.0) (exp (* (log (log N)) 3.0)))
      (fma t_0 t_0 (fma (log N) (log N) (* t_0 (log N)))))
     (/ (- 1.0 (/ (- 0.5 (/ (- 0.3333333333333333 (/ 0.25 N)) N)) N)) N))))

C

double code(double N) {
	double t_0 = log((N - -1.0));
	double tmp;
	if (N <= 850.0) {
		tmp = (pow(t_0, 3.0) - exp((log(log(N)) * 3.0))) / fma(t_0, t_0, fma(log(N), log(N), (t_0 * log(N))));
	} else {
		tmp = (1.0 - ((0.5 - ((0.3333333333333333 - (0.25 / N)) / N)) / N)) / N;
	}
	return tmp;
}

Julia

function code(N)
	t_0 = log(Float64(N - -1.0))
	tmp = 0.0
	if (N <= 850.0)
		tmp = Float64(Float64((t_0 ^ 3.0) - exp(Float64(log(log(N)) * 3.0))) / fma(t_0, t_0, fma(log(N), log(N), Float64(t_0 * log(N)))));
	else
		tmp = Float64(Float64(1.0 - Float64(Float64(0.5 - Float64(Float64(0.3333333333333333 - Float64(0.25 / N)) / N)) / N)) / N);
	end
	return tmp
end

Wolfram

code[N_] := Block[{t$95$0 = N[Log[N[(N - -1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N, 850.0], N[(N[(N[Power[t$95$0, 3.0], $MachinePrecision] - N[Exp[N[(N[Log[N[Log[N], $MachinePrecision]], $MachinePrecision] * 3.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * t$95$0 + N[(N[Log[N], $MachinePrecision] * N[Log[N], $MachinePrecision] + N[(t$95$0 * N[Log[N], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(N[(0.5 - N[(N[(0.3333333333333333 - N[(0.25 / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if N < 850

   1. Initial program 92.6%

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

      1. lift-+.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-log1p.f64 92.5

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

   3. Applied rewrites 92.5%

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

      1. lift--.f64 N/A

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

      2. lift-log1p.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. negate-sub N/A

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

      7. flip3-- N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. lift-log.f64 N/A

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

      12. lift--.f64 N/A

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

      13. lower-pow.f64 N/A

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

      14. lift-log.f64 N/A

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

      15. lower-fma.f64 N/A

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

   5. Applied rewrites 92.4%

      \[\leadsto \color{blue}{\frac{{\log \left(N - -1\right)}^{3} - {\log N}^{3}}{\mathsf{fma}\left(\log \left(N - -1\right), \log \left(N - -1\right), \mathsf{fma}\left(\log N, \log N, \log \left(N - -1\right) \cdot \log N\right)\right)}} \]
   6. Step-by-step derivation

      1. lift-log.f64 N/A

         \[\leadsto \frac{{\log \left(N - -1\right)}^{3} - {\color{blue}{\log N}}^{3}}{\mathsf{fma}\left(\log \left(N - -1\right), \log \left(N - -1\right), \mathsf{fma}\left(\log N, \log N, \log \left(N - -1\right) \cdot \log N\right)\right)} \]

      2. lift-pow.f64 N/A

         \[\leadsto \frac{{\log \left(N - -1\right)}^{3} - \color{blue}{{\log N}^{3}}}{\mathsf{fma}\left(\log \left(N - -1\right), \log \left(N - -1\right), \mathsf{fma}\left(\log N, \log N, \log \left(N - -1\right) \cdot \log N\right)\right)} \]

      3. pow-to-exp N/A

         \[\leadsto \frac{{\log \left(N - -1\right)}^{3} - \color{blue}{e^{\log \log N \cdot 3}}}{\mathsf{fma}\left(\log \left(N - -1\right), \log \left(N - -1\right), \mathsf{fma}\left(\log N, \log N, \log \left(N - -1\right) \cdot \log N\right)\right)} \]

      4. lower-exp.f64 N/A

         \[\leadsto \frac{{\log \left(N - -1\right)}^{3} - \color{blue}{e^{\log \log N \cdot 3}}}{\mathsf{fma}\left(\log \left(N - -1\right), \log \left(N - -1\right), \mathsf{fma}\left(\log N, \log N, \log \left(N - -1\right) \cdot \log N\right)\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{{\log \left(N - -1\right)}^{3} - e^{\color{blue}{\log \log N \cdot 3}}}{\mathsf{fma}\left(\log \left(N - -1\right), \log \left(N - -1\right), \mathsf{fma}\left(\log N, \log N, \log \left(N - -1\right) \cdot \log N\right)\right)} \]

      6. lower-log.f64 N/A

         \[\leadsto \frac{{\log \left(N - -1\right)}^{3} - e^{\color{blue}{\log \log N} \cdot 3}}{\mathsf{fma}\left(\log \left(N - -1\right), \log \left(N - -1\right), \mathsf{fma}\left(\log N, \log N, \log \left(N - -1\right) \cdot \log N\right)\right)} \]

      7. lift-log.f64 91.6

         \[\leadsto \frac{{\log \left(N - -1\right)}^{3} - e^{\log \color{blue}{\log N} \cdot 3}}{\mathsf{fma}\left(\log \left(N - -1\right), \log \left(N - -1\right), \mathsf{fma}\left(\log N, \log N, \log \left(N - -1\right) \cdot \log N\right)\right)} \]

   7. Applied rewrites 91.6%

      \[\leadsto \frac{{\log \left(N - -1\right)}^{3} - \color{blue}{e^{\log \log N \cdot 3}}}{\mathsf{fma}\left(\log \left(N - -1\right), \log \left(N - -1\right), \mathsf{fma}\left(\log N, \log N, \log \left(N - -1\right) \cdot \log N\right)\right)} \]

   if 850 < N

   1. Initial program 18.9%

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

   2. Taylor expanded in N around -inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{-1 \cdot \frac{-1 \cdot \frac{\frac{1}{4} \cdot \frac{1}{N} - \frac{1}{3}}{N} - \frac{1}{2}}{N} - 1}{N}\right) \]

      2. distribute-neg-frac N/A

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

      3. lower-/.f64 N/A

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

   4. Applied rewrites 99.7%

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

Alternative 3: 99.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(N - -1\right)\\ \mathbf{if}\;N \leq 880:\\ \;\;\;\;\frac{{t\_0}^{3} - {\log N}^{3}}{\mathsf{fma}\left(t\_0, t\_0, \mathsf{fma}\left(\log N, \log N, t\_0 \cdot \log N\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{0.5 - \frac{0.3333333333333333 - \frac{0.25}{N}}{N}}{N}}{N}\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (let* ((t_0 (log (- N -1.0))))
   (if (<= N 880.0)
     (/
      (- (pow t_0 3.0) (pow (log N) 3.0))
      (fma t_0 t_0 (fma (log N) (log N) (* t_0 (log N)))))
     (/ (- 1.0 (/ (- 0.5 (/ (- 0.3333333333333333 (/ 0.25 N)) N)) N)) N))))

C

double code(double N) {
	double t_0 = log((N - -1.0));
	double tmp;
	if (N <= 880.0) {
		tmp = (pow(t_0, 3.0) - pow(log(N), 3.0)) / fma(t_0, t_0, fma(log(N), log(N), (t_0 * log(N))));
	} else {
		tmp = (1.0 - ((0.5 - ((0.3333333333333333 - (0.25 / N)) / N)) / N)) / N;
	}
	return tmp;
}

Julia

function code(N)
	t_0 = log(Float64(N - -1.0))
	tmp = 0.0
	if (N <= 880.0)
		tmp = Float64(Float64((t_0 ^ 3.0) - (log(N) ^ 3.0)) / fma(t_0, t_0, fma(log(N), log(N), Float64(t_0 * log(N)))));
	else
		tmp = Float64(Float64(1.0 - Float64(Float64(0.5 - Float64(Float64(0.3333333333333333 - Float64(0.25 / N)) / N)) / N)) / N);
	end
	return tmp
end

Wolfram

code[N_] := Block[{t$95$0 = N[Log[N[(N - -1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N, 880.0], N[(N[(N[Power[t$95$0, 3.0], $MachinePrecision] - N[Power[N[Log[N], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * t$95$0 + N[(N[Log[N], $MachinePrecision] * N[Log[N], $MachinePrecision] + N[(t$95$0 * N[Log[N], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(N[(0.5 - N[(N[(0.3333333333333333 - N[(0.25 / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if N < 880

   1. Initial program 92.5%

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

      1. lift-+.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-log1p.f64 92.5

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

   3. Applied rewrites 92.5%

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

      1. lift--.f64 N/A

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

      2. lift-log1p.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. negate-sub N/A

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

      7. flip3-- N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. lift-log.f64 N/A

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

      12. lift--.f64 N/A

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

      13. lower-pow.f64 N/A

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

      14. lift-log.f64 N/A

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

      15. lower-fma.f64 N/A

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

   5. Applied rewrites 92.4%

      \[\leadsto \color{blue}{\frac{{\log \left(N - -1\right)}^{3} - {\log N}^{3}}{\mathsf{fma}\left(\log \left(N - -1\right), \log \left(N - -1\right), \mathsf{fma}\left(\log N, \log N, \log \left(N - -1\right) \cdot \log N\right)\right)}} \]

   if 880 < N

   1. Initial program 18.8%

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

   2. Taylor expanded in N around -inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{-1 \cdot \frac{-1 \cdot \frac{\frac{1}{4} \cdot \frac{1}{N} - \frac{1}{3}}{N} - \frac{1}{2}}{N} - 1}{N}\right) \]

      2. distribute-neg-frac N/A

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

      3. lower-/.f64 N/A

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

   4. Applied rewrites 99.7%

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

Alternative 4: 99.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(N - -1\right)\\ \mathbf{if}\;N \leq 880:\\ \;\;\;\;\frac{{t\_0}^{3} - {\log N}^{3}}{\mathsf{fma}\left(\log N, \log N + t\_0, t\_0 \cdot t\_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{0.5 - \frac{0.3333333333333333 - \frac{0.25}{N}}{N}}{N}}{N}\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (let* ((t_0 (log (- N -1.0))))
   (if (<= N 880.0)
     (/
      (- (pow t_0 3.0) (pow (log N) 3.0))
      (fma (log N) (+ (log N) t_0) (* t_0 t_0)))
     (/ (- 1.0 (/ (- 0.5 (/ (- 0.3333333333333333 (/ 0.25 N)) N)) N)) N))))

C

double code(double N) {
	double t_0 = log((N - -1.0));
	double tmp;
	if (N <= 880.0) {
		tmp = (pow(t_0, 3.0) - pow(log(N), 3.0)) / fma(log(N), (log(N) + t_0), (t_0 * t_0));
	} else {
		tmp = (1.0 - ((0.5 - ((0.3333333333333333 - (0.25 / N)) / N)) / N)) / N;
	}
	return tmp;
}

Julia

function code(N)
	t_0 = log(Float64(N - -1.0))
	tmp = 0.0
	if (N <= 880.0)
		tmp = Float64(Float64((t_0 ^ 3.0) - (log(N) ^ 3.0)) / fma(log(N), Float64(log(N) + t_0), Float64(t_0 * t_0)));
	else
		tmp = Float64(Float64(1.0 - Float64(Float64(0.5 - Float64(Float64(0.3333333333333333 - Float64(0.25 / N)) / N)) / N)) / N);
	end
	return tmp
end

Wolfram

code[N_] := Block[{t$95$0 = N[Log[N[(N - -1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N, 880.0], N[(N[(N[Power[t$95$0, 3.0], $MachinePrecision] - N[Power[N[Log[N], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] / N[(N[Log[N], $MachinePrecision] * N[(N[Log[N], $MachinePrecision] + t$95$0), $MachinePrecision] + N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(N[(0.5 - N[(N[(0.3333333333333333 - N[(0.25 / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if N < 880

   1. Initial program 92.5%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. flip3-- N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. metadata-eval N/A

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

      11. negate-sub-reverse N/A

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

      12. lower--.f64 N/A

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

      13. lower-pow.f64 N/A

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

      14. lift-log.f64 N/A

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

      15. +-commutative N/A

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

   3. Applied rewrites 92.4%

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

   if 880 < N

   1. Initial program 18.8%

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

   2. Taylor expanded in N around -inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{-1 \cdot \frac{-1 \cdot \frac{\frac{1}{4} \cdot \frac{1}{N} - \frac{1}{3}}{N} - \frac{1}{2}}{N} - 1}{N}\right) \]

      2. distribute-neg-frac N/A

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

      3. lower-/.f64 N/A

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

   4. Applied rewrites 99.7%

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

Alternative 5: 99.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;N \leq 1200:\\ \;\;\;\;\mathsf{log1p}\left(N\right) - \log N\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{0.5 - \frac{0.3333333333333333 - \frac{0.25}{N}}{N}}{N}}{N}\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= N 1200.0)
   (- (log1p N) (log N))
   (/ (- 1.0 (/ (- 0.5 (/ (- 0.3333333333333333 (/ 0.25 N)) N)) N)) N)))

C

double code(double N) {
	double tmp;
	if (N <= 1200.0) {
		tmp = log1p(N) - log(N);
	} else {
		tmp = (1.0 - ((0.5 - ((0.3333333333333333 - (0.25 / N)) / N)) / N)) / N;
	}
	return tmp;
}

Java

public static double code(double N) {
	double tmp;
	if (N <= 1200.0) {
		tmp = Math.log1p(N) - Math.log(N);
	} else {
		tmp = (1.0 - ((0.5 - ((0.3333333333333333 - (0.25 / N)) / N)) / N)) / N;
	}
	return tmp;
}

Python

def code(N):
	tmp = 0
	if N <= 1200.0:
		tmp = math.log1p(N) - math.log(N)
	else:
		tmp = (1.0 - ((0.5 - ((0.3333333333333333 - (0.25 / N)) / N)) / N)) / N
	return tmp

Julia

function code(N)
	tmp = 0.0
	if (N <= 1200.0)
		tmp = Float64(log1p(N) - log(N));
	else
		tmp = Float64(Float64(1.0 - Float64(Float64(0.5 - Float64(Float64(0.3333333333333333 - Float64(0.25 / N)) / N)) / N)) / N);
	end
	return tmp
end

Wolfram

code[N_] := If[LessEqual[N, 1200.0], N[(N[Log[1 + N], $MachinePrecision] - N[Log[N], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(N[(0.5 - N[(N[(0.3333333333333333 - N[(0.25 / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if N < 1200

   1. Initial program 92.1%

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

      1. lift-+.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-log1p.f64 92.1

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

   3. Applied rewrites 92.1%

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

   if 1200 < N

   1. Initial program 18.6%

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

   2. Taylor expanded in N around -inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{-1 \cdot \frac{-1 \cdot \frac{\frac{1}{4} \cdot \frac{1}{N} - \frac{1}{3}}{N} - \frac{1}{2}}{N} - 1}{N}\right) \]

      2. distribute-neg-frac N/A

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

      3. lower-/.f64 N/A

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

   4. Applied rewrites 99.7%

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

Alternative 6: 98.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;N \leq 4500:\\ \;\;\;\;\mathsf{log1p}\left(N\right) - \log N\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{0.5 - \frac{0.3333333333333333}{N}}{N}}{N}\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= N 4500.0)
   (- (log1p N) (log N))
   (/ (- 1.0 (/ (- 0.5 (/ 0.3333333333333333 N)) N)) N)))

C

double code(double N) {
	double tmp;
	if (N <= 4500.0) {
		tmp = log1p(N) - log(N);
	} else {
		tmp = (1.0 - ((0.5 - (0.3333333333333333 / N)) / N)) / N;
	}
	return tmp;
}

Java

public static double code(double N) {
	double tmp;
	if (N <= 4500.0) {
		tmp = Math.log1p(N) - Math.log(N);
	} else {
		tmp = (1.0 - ((0.5 - (0.3333333333333333 / N)) / N)) / N;
	}
	return tmp;
}

Python

def code(N):
	tmp = 0
	if N <= 4500.0:
		tmp = math.log1p(N) - math.log(N)
	else:
		tmp = (1.0 - ((0.5 - (0.3333333333333333 / N)) / N)) / N
	return tmp

Julia

function code(N)
	tmp = 0.0
	if (N <= 4500.0)
		tmp = Float64(log1p(N) - log(N));
	else
		tmp = Float64(Float64(1.0 - Float64(Float64(0.5 - Float64(0.3333333333333333 / N)) / N)) / N);
	end
	return tmp
end

Wolfram

code[N_] := If[LessEqual[N, 4500.0], N[(N[Log[1 + N], $MachinePrecision] - N[Log[N], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(N[(0.5 - N[(0.3333333333333333 / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if N < 4500

   1. Initial program 90.3%

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

      1. lift-+.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-log1p.f64 90.3

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

   3. Applied rewrites 90.3%

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

   if 4500 < N

   1. Initial program 17.5%

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

   2. Taylor expanded in N around inf

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

      1. negate-sub2 N/A

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

   4. Applied rewrites 99.5%

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

Alternative 7: 98.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;N \leq 4500:\\ \;\;\;\;\log \left(N - -1\right) - \log N\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{0.5 - \frac{0.3333333333333333}{N}}{N}}{N}\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= N 4500.0)
   (- (log (- N -1.0)) (log N))
   (/ (- 1.0 (/ (- 0.5 (/ 0.3333333333333333 N)) N)) N)))

C

double code(double N) {
	double tmp;
	if (N <= 4500.0) {
		tmp = log((N - -1.0)) - log(N);
	} else {
		tmp = (1.0 - ((0.5 - (0.3333333333333333 / N)) / N)) / N;
	}
	return tmp;
}

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
    real(8) :: tmp
    if (n <= 4500.0d0) then
        tmp = log((n - (-1.0d0))) - log(n)
    else
        tmp = (1.0d0 - ((0.5d0 - (0.3333333333333333d0 / n)) / n)) / n
    end if
    code = tmp
end function

Java

public static double code(double N) {
	double tmp;
	if (N <= 4500.0) {
		tmp = Math.log((N - -1.0)) - Math.log(N);
	} else {
		tmp = (1.0 - ((0.5 - (0.3333333333333333 / N)) / N)) / N;
	}
	return tmp;
}

Python

def code(N):
	tmp = 0
	if N <= 4500.0:
		tmp = math.log((N - -1.0)) - math.log(N)
	else:
		tmp = (1.0 - ((0.5 - (0.3333333333333333 / N)) / N)) / N
	return tmp

Julia

function code(N)
	tmp = 0.0
	if (N <= 4500.0)
		tmp = Float64(log(Float64(N - -1.0)) - log(N));
	else
		tmp = Float64(Float64(1.0 - Float64(Float64(0.5 - Float64(0.3333333333333333 / N)) / N)) / N);
	end
	return tmp
end

MATLAB

function tmp_2 = code(N)
	tmp = 0.0;
	if (N <= 4500.0)
		tmp = log((N - -1.0)) - log(N);
	else
		tmp = (1.0 - ((0.5 - (0.3333333333333333 / N)) / N)) / N;
	end
	tmp_2 = tmp;
end

Wolfram

code[N_] := If[LessEqual[N, 4500.0], N[(N[Log[N[(N - -1.0), $MachinePrecision]], $MachinePrecision] - N[Log[N], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(N[(0.5 - N[(0.3333333333333333 / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if N < 4500

   1. Initial program 90.3%

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

      1. lift-+.f64 N/A

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

      2. metadata-eval N/A

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

      3. negate-sub-reverse N/A

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

      4. lower--.f64 90.3

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

   3. Applied rewrites 90.3%

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

   if 4500 < N

   1. Initial program 17.5%

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

   2. Taylor expanded in N around inf

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

      1. negate-sub2 N/A

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

   4. Applied rewrites 99.5%

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

Alternative 8: 95.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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 - ((0.5d0 - (0.3333333333333333d0 / n)) / n)) / n
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 23.9%

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

2. Taylor expanded in N around inf

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

   1. negate-sub2 N/A

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

4. Applied rewrites 95.1%

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

Alternative 9: 92.5% accurate, 1.7× 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

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 - (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 23.9%

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

2. Taylor expanded in N around inf

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

   1. lower-/.f64 N/A

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

   2. lower--.f64 N/A

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

   3. associate-*r/ N/A

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

   4. metadata-eval N/A

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

   5. lower-/.f64 92.5

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

4. Applied rewrites 92.5%

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

Alternative 10: 92.2% accurate, 1.8× speedup

Math

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

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(N - 0.5) / Float64(N * N))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 23.9%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-log.f64 N/A

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

   4. lift-log.f64 N/A

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

   5. flip-- N/A

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

   6. lower-/.f64 N/A

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

3. Applied rewrites 23.9%

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

4. Taylor expanded in N around inf

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

   1. lower-/.f64 N/A

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

   2. lower--.f64 N/A

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

   3. associate-*r/ N/A

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

   4. metadata-eval N/A

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

   5. lower-/.f64 92.5

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

6. Applied rewrites 92.5%

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

7. Taylor expanded in N around 0

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

   1. lower-/.f64 N/A

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

   2. lower--.f64 N/A

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

   3. pow2 N/A

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

   4. lift-*.f64 92.2

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

9. Applied rewrites 92.2%

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

Alternative 11: 84.5% accurate, 4.1× 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 23.9%

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

2. Taylor expanded in N around inf

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

   1. lower-/.f64 84.5

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

4. Applied rewrites 84.5%

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

Developer Target 1: 99.8% accurate, 1.2× 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]

Developer Target 2: 26.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \log \left(1 + \frac{1}{N}\right) \end{array} \]

FPCore

(FPCore (N) :precision binary64 (log (+ 1.0 (/ 1.0 N))))

C

double code(double N) {
	return log((1.0 + (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 = log((1.0d0 + (1.0d0 / n)))
end function

Java

public static double code(double N) {
	return Math.log((1.0 + (1.0 / N)));
}

Python

def code(N):
	return math.log((1.0 + (1.0 / N)))

Julia

function code(N)
	return log(Float64(1.0 + Float64(1.0 / N)))
end

MATLAB

function tmp = code(N)
	tmp = log((1.0 + (1.0 / N)));
end

Wolfram

code[N_] := N[Log[N[(1.0 + N[(1.0 / N), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Developer Target 3: 96.4% accurate, 0.2× 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 2025119 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  :pre (and (> N 1.0) (< N 1e+40))

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

  :alt
  (! :herbie-platform c (log (+ 1 (/ 1 N))))

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

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