2log (problem 3.3.6)

Percentage Accurate: 23.6% → 99.5%

Time: 9.7s

Alternatives: 12

Speedup: 17.3×

Specification

\[N > 1 \land N < 10^{+40}\]

Math

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

FPCore

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

C

double code(double N) {
	return log((N + 1.0)) - log(N);
}

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 23.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double N) {
	return log((N + 1.0)) - log(N);
}

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\mathsf{fma}\left(N, N, N\right)\right)\\ \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 0.0005:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(-\frac{\mathsf{fma}\left(N, \mathsf{fma}\left(N, -0.5, 0.08333333333333333\right), -0.041666666666666664\right)}{N \cdot \left(N \cdot N\right)}, N, N\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{\frac{t\_0}{\log \left(\frac{N}{N + 1}\right)}}{t\_0}}\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (let* ((t_0 (log (fma N N N))))
   (if (<= (- (log (+ N 1.0)) (log N)) 0.0005)
     (/
      1.0
      (fma
       (-
        (/
         (fma N (fma N -0.5 0.08333333333333333) -0.041666666666666664)
         (* N (* N N))))
       N
       N))
     (/ -1.0 (/ (/ t_0 (log (/ N (+ N 1.0)))) t_0)))))

C

double code(double N) {
	double t_0 = log(fma(N, N, N));
	double tmp;
	if ((log((N + 1.0)) - log(N)) <= 0.0005) {
		tmp = 1.0 / fma(-(fma(N, fma(N, -0.5, 0.08333333333333333), -0.041666666666666664) / (N * (N * N))), N, N);
	} else {
		tmp = -1.0 / ((t_0 / log((N / (N + 1.0)))) / t_0);
	}
	return tmp;
}

Julia

function code(N)
	t_0 = log(fma(N, N, N))
	tmp = 0.0
	if (Float64(log(Float64(N + 1.0)) - log(N)) <= 0.0005)
		tmp = Float64(1.0 / fma(Float64(-Float64(fma(N, fma(N, -0.5, 0.08333333333333333), -0.041666666666666664) / Float64(N * Float64(N * N)))), N, N));
	else
		tmp = Float64(-1.0 / Float64(Float64(t_0 / log(Float64(N / Float64(N + 1.0)))) / t_0));
	end
	return tmp
end

Wolfram

code[N_] := Block[{t$95$0 = N[Log[N[(N * N + N), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[Log[N[(N + 1.0), $MachinePrecision]], $MachinePrecision] - N[Log[N], $MachinePrecision]), $MachinePrecision], 0.0005], N[(1.0 / N[((-N[(N[(N * N[(N * -0.5 + 0.08333333333333333), $MachinePrecision] + -0.041666666666666664), $MachinePrecision] / N[(N * N[(N * N), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) * N + N), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(N[(t$95$0 / N[Log[N[(N / N[(N + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 18.9%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip-- N/A

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

      7. lift--.f64 N/A

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

      8. lower-/.f64 18.9

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

      9. lift--.f64 N/A

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

      10. lift-log.f64 N/A

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

      11. lift-log.f64 N/A

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

      12. diff-log N/A

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

      13. lower-log.f64 N/A

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

      14. lower-/.f64 22.1

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

   4. Applied rewrites 22.1%

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

   5. Taylor expanded in N around -inf

      \[\leadsto \frac{1}{\color{blue}{-1 \cdot \left(N \cdot \left(-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N} - 1\right)\right)}} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{1}{\color{blue}{\mathsf{neg}\left(N \cdot \left(-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N} - 1\right)\right)}} \]

      2. lower-neg.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\mathsf{neg}\left(N \cdot \left(-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N} - 1\right)\right)}} \]

      3. sub-neg N/A

         \[\leadsto \frac{1}{\mathsf{neg}\left(N \cdot \color{blue}{\left(-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N} + \left(\mathsf{neg}\left(1\right)\right)\right)}\right)} \]

      4. metadata-eval N/A

         \[\leadsto \frac{1}{\mathsf{neg}\left(N \cdot \left(-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N} + \color{blue}{-1}\right)\right)} \]

      5. distribute-lft-in N/A

         \[\leadsto \frac{1}{\mathsf{neg}\left(\color{blue}{\left(N \cdot \left(-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N}\right) + N \cdot -1\right)}\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{1}{\mathsf{neg}\left(\left(N \cdot \left(-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N}\right) + \color{blue}{-1 \cdot N}\right)\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{1}{\mathsf{neg}\left(\color{blue}{\mathsf{fma}\left(N, -1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N}, -1 \cdot N\right)}\right)} \]

   7. Applied rewrites 99.9%

      \[\leadsto \frac{1}{\color{blue}{-\mathsf{fma}\left(N, \frac{0.5 - \frac{0.08333333333333333 + \frac{-0.041666666666666664}{N}}{N}}{-N}, -N\right)}} \]

   8. Taylor expanded in N around 0

      \[\leadsto \frac{1}{\mathsf{neg}\left(\mathsf{fma}\left(N, \frac{N \cdot \left(\frac{1}{12} + \frac{-1}{2} \cdot N\right) - \frac{1}{24}}{{N}^{3}}, \mathsf{neg}\left(N\right)\right)\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 99.9%

       \[\leadsto \frac{1}{-\mathsf{fma}\left(N, \frac{\mathsf{fma}\left(N, \mathsf{fma}\left(N, -0.5, 0.08333333333333333\right), -0.041666666666666664\right)}{N \cdot \left(N \cdot N\right)}, -N\right)} \]
   11. Step-by-step derivation

   12. Applied rewrites 99.9%

       \[\leadsto \frac{1}{\mathsf{fma}\left(-\frac{\mathsf{fma}\left(N, \mathsf{fma}\left(N, -0.5, 0.08333333333333333\right), -0.041666666666666664\right)}{N \cdot \left(N \cdot N\right)}, \color{blue}{N}, N\right)} \]

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

   1. Initial program 92.2%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip-- N/A

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

      7. lift--.f64 N/A

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

      8. lower-/.f64 92.3

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

      9. lift--.f64 N/A

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

      10. lift-log.f64 N/A

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

      11. lift-log.f64 N/A

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

      12. diff-log N/A

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

      13. lower-log.f64 N/A

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

      14. lower-/.f64 94.8

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

   4. Applied rewrites 94.8%

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

   6. Applied rewrites 95.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. frac-2neg N/A

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

      11. metadata-eval N/A

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

      12. remove-double-div N/A

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

      13. lower-neg.f64 95.8

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

   8. Applied rewrites 95.8%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 0.0005:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(-\frac{\mathsf{fma}\left(N, \mathsf{fma}\left(N, -0.5, 0.08333333333333333\right), -0.041666666666666664\right)}{N \cdot \left(N \cdot N\right)}, N, N\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{\frac{\log \left(\mathsf{fma}\left(N, N, N\right)\right)}{\log \left(\frac{N}{N + 1}\right)}}{\log \left(\mathsf{fma}\left(N, N, N\right)\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\mathsf{fma}\left(N, N, N\right)\right)\\ \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 0.0005:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(-\frac{\mathsf{fma}\left(N, \mathsf{fma}\left(N, -0.5, 0.08333333333333333\right), -0.041666666666666664\right)}{N \cdot \left(N \cdot N\right)}, N, N\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \frac{\log \left(\frac{N}{N + 1}\right)}{-t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (let* ((t_0 (log (fma N N N))))
   (if (<= (- (log (+ N 1.0)) (log N)) 0.0005)
     (/
      1.0
      (fma
       (-
        (/
         (fma N (fma N -0.5 0.08333333333333333) -0.041666666666666664)
         (* N (* N N))))
       N
       N))
     (* t_0 (/ (log (/ N (+ N 1.0))) (- t_0))))))

C

double code(double N) {
	double t_0 = log(fma(N, N, N));
	double tmp;
	if ((log((N + 1.0)) - log(N)) <= 0.0005) {
		tmp = 1.0 / fma(-(fma(N, fma(N, -0.5, 0.08333333333333333), -0.041666666666666664) / (N * (N * N))), N, N);
	} else {
		tmp = t_0 * (log((N / (N + 1.0))) / -t_0);
	}
	return tmp;
}

Julia

function code(N)
	t_0 = log(fma(N, N, N))
	tmp = 0.0
	if (Float64(log(Float64(N + 1.0)) - log(N)) <= 0.0005)
		tmp = Float64(1.0 / fma(Float64(-Float64(fma(N, fma(N, -0.5, 0.08333333333333333), -0.041666666666666664) / Float64(N * Float64(N * N)))), N, N));
	else
		tmp = Float64(t_0 * Float64(log(Float64(N / Float64(N + 1.0))) / Float64(-t_0)));
	end
	return tmp
end

Wolfram

code[N_] := Block[{t$95$0 = N[Log[N[(N * N + N), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[Log[N[(N + 1.0), $MachinePrecision]], $MachinePrecision] - N[Log[N], $MachinePrecision]), $MachinePrecision], 0.0005], N[(1.0 / N[((-N[(N[(N * N[(N * -0.5 + 0.08333333333333333), $MachinePrecision] + -0.041666666666666664), $MachinePrecision] / N[(N * N[(N * N), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) * N + N), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(N[Log[N[(N / N[(N + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 18.9%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip-- N/A

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

      7. lift--.f64 N/A

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

      8. lower-/.f64 18.9

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

      9. lift--.f64 N/A

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

      10. lift-log.f64 N/A

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

      11. lift-log.f64 N/A

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

      12. diff-log N/A

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

      13. lower-log.f64 N/A

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

      14. lower-/.f64 22.1

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

   4. Applied rewrites 22.1%

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

   5. Taylor expanded in N around -inf

      \[\leadsto \frac{1}{\color{blue}{-1 \cdot \left(N \cdot \left(-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N} - 1\right)\right)}} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{1}{\color{blue}{\mathsf{neg}\left(N \cdot \left(-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N} - 1\right)\right)}} \]

      2. lower-neg.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\mathsf{neg}\left(N \cdot \left(-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N} - 1\right)\right)}} \]

      3. sub-neg N/A

         \[\leadsto \frac{1}{\mathsf{neg}\left(N \cdot \color{blue}{\left(-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N} + \left(\mathsf{neg}\left(1\right)\right)\right)}\right)} \]

      4. metadata-eval N/A

         \[\leadsto \frac{1}{\mathsf{neg}\left(N \cdot \left(-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N} + \color{blue}{-1}\right)\right)} \]

      5. distribute-lft-in N/A

         \[\leadsto \frac{1}{\mathsf{neg}\left(\color{blue}{\left(N \cdot \left(-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N}\right) + N \cdot -1\right)}\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{1}{\mathsf{neg}\left(\left(N \cdot \left(-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N}\right) + \color{blue}{-1 \cdot N}\right)\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{1}{\mathsf{neg}\left(\color{blue}{\mathsf{fma}\left(N, -1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N}, -1 \cdot N\right)}\right)} \]

   7. Applied rewrites 99.9%

      \[\leadsto \frac{1}{\color{blue}{-\mathsf{fma}\left(N, \frac{0.5 - \frac{0.08333333333333333 + \frac{-0.041666666666666664}{N}}{N}}{-N}, -N\right)}} \]

   8. Taylor expanded in N around 0

      \[\leadsto \frac{1}{\mathsf{neg}\left(\mathsf{fma}\left(N, \frac{N \cdot \left(\frac{1}{12} + \frac{-1}{2} \cdot N\right) - \frac{1}{24}}{{N}^{3}}, \mathsf{neg}\left(N\right)\right)\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 99.9%

       \[\leadsto \frac{1}{-\mathsf{fma}\left(N, \frac{\mathsf{fma}\left(N, \mathsf{fma}\left(N, -0.5, 0.08333333333333333\right), -0.041666666666666664\right)}{N \cdot \left(N \cdot N\right)}, -N\right)} \]
   11. Step-by-step derivation

   12. Applied rewrites 99.9%

       \[\leadsto \frac{1}{\mathsf{fma}\left(-\frac{\mathsf{fma}\left(N, \mathsf{fma}\left(N, -0.5, 0.08333333333333333\right), -0.041666666666666664\right)}{N \cdot \left(N \cdot N\right)}, \color{blue}{N}, N\right)} \]

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

   1. Initial program 92.2%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip-- N/A

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

      7. lift--.f64 N/A

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

      8. lower-/.f64 92.3

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

      9. lift--.f64 N/A

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

      10. lift-log.f64 N/A

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

      11. lift-log.f64 N/A

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

      12. diff-log N/A

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

      13. lower-log.f64 N/A

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

      14. lower-/.f64 94.8

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

   4. Applied rewrites 94.8%

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

   6. Applied rewrites 95.7%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 0.0005:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(-\frac{\mathsf{fma}\left(N, \mathsf{fma}\left(N, -0.5, 0.08333333333333333\right), -0.041666666666666664\right)}{N \cdot \left(N \cdot N\right)}, N, N\right)}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\mathsf{fma}\left(N, N, N\right)\right) \cdot \frac{\log \left(\frac{N}{N + 1}\right)}{-\log \left(\mathsf{fma}\left(N, N, N\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 0.0005:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(-\frac{\mathsf{fma}\left(N, \mathsf{fma}\left(N, -0.5, 0.08333333333333333\right), -0.041666666666666664\right)}{N \cdot \left(N \cdot N\right)}, N, N\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{-1}{\log \left(\frac{N}{N + 1}\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= (- (log (+ N 1.0)) (log N)) 0.0005)
   (/
    1.0
    (fma
     (-
      (/
       (fma N (fma N -0.5 0.08333333333333333) -0.041666666666666664)
       (* N (* N N))))
     N
     N))
   (/ 1.0 (/ -1.0 (log (/ N (+ N 1.0)))))))

C

double code(double N) {
	double tmp;
	if ((log((N + 1.0)) - log(N)) <= 0.0005) {
		tmp = 1.0 / fma(-(fma(N, fma(N, -0.5, 0.08333333333333333), -0.041666666666666664) / (N * (N * N))), N, N);
	} else {
		tmp = 1.0 / (-1.0 / log((N / (N + 1.0))));
	}
	return tmp;
}

Julia

function code(N)
	tmp = 0.0
	if (Float64(log(Float64(N + 1.0)) - log(N)) <= 0.0005)
		tmp = Float64(1.0 / fma(Float64(-Float64(fma(N, fma(N, -0.5, 0.08333333333333333), -0.041666666666666664) / Float64(N * Float64(N * N)))), N, N));
	else
		tmp = Float64(1.0 / Float64(-1.0 / log(Float64(N / Float64(N + 1.0)))));
	end
	return tmp
end

Wolfram

code[N_] := If[LessEqual[N[(N[Log[N[(N + 1.0), $MachinePrecision]], $MachinePrecision] - N[Log[N], $MachinePrecision]), $MachinePrecision], 0.0005], N[(1.0 / N[((-N[(N[(N * N[(N * -0.5 + 0.08333333333333333), $MachinePrecision] + -0.041666666666666664), $MachinePrecision] / N[(N * N[(N * N), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) * N + N), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(-1.0 / N[Log[N[(N / N[(N + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 18.9%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip-- N/A

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

      7. lift--.f64 N/A

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

      8. lower-/.f64 18.9

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

      9. lift--.f64 N/A

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

      10. lift-log.f64 N/A

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

      11. lift-log.f64 N/A

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

      12. diff-log N/A

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

      13. lower-log.f64 N/A

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

      14. lower-/.f64 22.1

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

   4. Applied rewrites 22.1%

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

   5. Taylor expanded in N around -inf

      \[\leadsto \frac{1}{\color{blue}{-1 \cdot \left(N \cdot \left(-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N} - 1\right)\right)}} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{1}{\color{blue}{\mathsf{neg}\left(N \cdot \left(-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N} - 1\right)\right)}} \]

      2. lower-neg.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\mathsf{neg}\left(N \cdot \left(-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N} - 1\right)\right)}} \]

      3. sub-neg N/A

         \[\leadsto \frac{1}{\mathsf{neg}\left(N \cdot \color{blue}{\left(-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N} + \left(\mathsf{neg}\left(1\right)\right)\right)}\right)} \]

      4. metadata-eval N/A

         \[\leadsto \frac{1}{\mathsf{neg}\left(N \cdot \left(-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N} + \color{blue}{-1}\right)\right)} \]

      5. distribute-lft-in N/A

         \[\leadsto \frac{1}{\mathsf{neg}\left(\color{blue}{\left(N \cdot \left(-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N}\right) + N \cdot -1\right)}\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{1}{\mathsf{neg}\left(\left(N \cdot \left(-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N}\right) + \color{blue}{-1 \cdot N}\right)\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{1}{\mathsf{neg}\left(\color{blue}{\mathsf{fma}\left(N, -1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N}, -1 \cdot N\right)}\right)} \]

   7. Applied rewrites 99.9%

      \[\leadsto \frac{1}{\color{blue}{-\mathsf{fma}\left(N, \frac{0.5 - \frac{0.08333333333333333 + \frac{-0.041666666666666664}{N}}{N}}{-N}, -N\right)}} \]

   8. Taylor expanded in N around 0

      \[\leadsto \frac{1}{\mathsf{neg}\left(\mathsf{fma}\left(N, \frac{N \cdot \left(\frac{1}{12} + \frac{-1}{2} \cdot N\right) - \frac{1}{24}}{{N}^{3}}, \mathsf{neg}\left(N\right)\right)\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 99.9%

       \[\leadsto \frac{1}{-\mathsf{fma}\left(N, \frac{\mathsf{fma}\left(N, \mathsf{fma}\left(N, -0.5, 0.08333333333333333\right), -0.041666666666666664\right)}{N \cdot \left(N \cdot N\right)}, -N\right)} \]
   11. Step-by-step derivation

   12. Applied rewrites 99.9%

       \[\leadsto \frac{1}{\mathsf{fma}\left(-\frac{\mathsf{fma}\left(N, \mathsf{fma}\left(N, -0.5, 0.08333333333333333\right), -0.041666666666666664\right)}{N \cdot \left(N \cdot N\right)}, \color{blue}{N}, N\right)} \]

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

   1. Initial program 92.2%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip-- N/A

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

      7. lift--.f64 N/A

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

      8. lower-/.f64 92.3

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

      9. lift--.f64 N/A

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

      10. lift-log.f64 N/A

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

      11. lift-log.f64 N/A

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

      12. diff-log N/A

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

      13. lower-log.f64 N/A

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

      14. lower-/.f64 94.8

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

   4. Applied rewrites 94.8%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. metadata-eval N/A

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

      4. lift-log.f64 N/A

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

      5. neg-log N/A

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. lift-/.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. lower-/.f64 95.7

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

   6. Applied rewrites 95.7%

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

Alternative 4: 99.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 0.0005:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(-\frac{\mathsf{fma}\left(N, \mathsf{fma}\left(N, -0.5, 0.08333333333333333\right), -0.041666666666666664\right)}{N \cdot \left(N \cdot N\right)}, N, N\right)}\\ \mathbf{else}:\\ \;\;\;\;-\log \left(\frac{N}{N + 1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= (- (log (+ N 1.0)) (log N)) 0.0005)
   (/
    1.0
    (fma
     (-
      (/
       (fma N (fma N -0.5 0.08333333333333333) -0.041666666666666664)
       (* N (* N N))))
     N
     N))
   (- (log (/ N (+ N 1.0))))))

C

double code(double N) {
	double tmp;
	if ((log((N + 1.0)) - log(N)) <= 0.0005) {
		tmp = 1.0 / fma(-(fma(N, fma(N, -0.5, 0.08333333333333333), -0.041666666666666664) / (N * (N * N))), N, N);
	} else {
		tmp = -log((N / (N + 1.0)));
	}
	return tmp;
}

Julia

function code(N)
	tmp = 0.0
	if (Float64(log(Float64(N + 1.0)) - log(N)) <= 0.0005)
		tmp = Float64(1.0 / fma(Float64(-Float64(fma(N, fma(N, -0.5, 0.08333333333333333), -0.041666666666666664) / Float64(N * Float64(N * N)))), N, N));
	else
		tmp = Float64(-log(Float64(N / Float64(N + 1.0))));
	end
	return tmp
end

Wolfram

code[N_] := If[LessEqual[N[(N[Log[N[(N + 1.0), $MachinePrecision]], $MachinePrecision] - N[Log[N], $MachinePrecision]), $MachinePrecision], 0.0005], N[(1.0 / N[((-N[(N[(N * N[(N * -0.5 + 0.08333333333333333), $MachinePrecision] + -0.041666666666666664), $MachinePrecision] / N[(N * N[(N * N), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) * N + N), $MachinePrecision]), $MachinePrecision], (-N[Log[N[(N / N[(N + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 18.9%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip-- N/A

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

      7. lift--.f64 N/A

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

      8. lower-/.f64 18.9

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

      9. lift--.f64 N/A

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

      10. lift-log.f64 N/A

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

      11. lift-log.f64 N/A

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

      12. diff-log N/A

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

      13. lower-log.f64 N/A

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

      14. lower-/.f64 22.1

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

   4. Applied rewrites 22.1%

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

   5. Taylor expanded in N around -inf

      \[\leadsto \frac{1}{\color{blue}{-1 \cdot \left(N \cdot \left(-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N} - 1\right)\right)}} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{1}{\color{blue}{\mathsf{neg}\left(N \cdot \left(-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N} - 1\right)\right)}} \]

      2. lower-neg.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\mathsf{neg}\left(N \cdot \left(-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N} - 1\right)\right)}} \]

      3. sub-neg N/A

         \[\leadsto \frac{1}{\mathsf{neg}\left(N \cdot \color{blue}{\left(-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N} + \left(\mathsf{neg}\left(1\right)\right)\right)}\right)} \]

      4. metadata-eval N/A

         \[\leadsto \frac{1}{\mathsf{neg}\left(N \cdot \left(-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N} + \color{blue}{-1}\right)\right)} \]

      5. distribute-lft-in N/A

         \[\leadsto \frac{1}{\mathsf{neg}\left(\color{blue}{\left(N \cdot \left(-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N}\right) + N \cdot -1\right)}\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{1}{\mathsf{neg}\left(\left(N \cdot \left(-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N}\right) + \color{blue}{-1 \cdot N}\right)\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{1}{\mathsf{neg}\left(\color{blue}{\mathsf{fma}\left(N, -1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N}, -1 \cdot N\right)}\right)} \]

   7. Applied rewrites 99.9%

      \[\leadsto \frac{1}{\color{blue}{-\mathsf{fma}\left(N, \frac{0.5 - \frac{0.08333333333333333 + \frac{-0.041666666666666664}{N}}{N}}{-N}, -N\right)}} \]

   8. Taylor expanded in N around 0

      \[\leadsto \frac{1}{\mathsf{neg}\left(\mathsf{fma}\left(N, \frac{N \cdot \left(\frac{1}{12} + \frac{-1}{2} \cdot N\right) - \frac{1}{24}}{{N}^{3}}, \mathsf{neg}\left(N\right)\right)\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 99.9%

       \[\leadsto \frac{1}{-\mathsf{fma}\left(N, \frac{\mathsf{fma}\left(N, \mathsf{fma}\left(N, -0.5, 0.08333333333333333\right), -0.041666666666666664\right)}{N \cdot \left(N \cdot N\right)}, -N\right)} \]
   11. Step-by-step derivation

   12. Applied rewrites 99.9%

       \[\leadsto \frac{1}{\mathsf{fma}\left(-\frac{\mathsf{fma}\left(N, \mathsf{fma}\left(N, -0.5, 0.08333333333333333\right), -0.041666666666666664\right)}{N \cdot \left(N \cdot N\right)}, \color{blue}{N}, N\right)} \]

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

   1. Initial program 92.2%

      \[\log \left(N + 1\right) - \log N \]
   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. clear-num N/A

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

      6. neg-log N/A

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

      7. diff-log N/A

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

      8. lift-log.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lift-log.f64 N/A

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

      12. lift-log.f64 N/A

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

      13. diff-log N/A

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

      14. lower-log.f64 N/A

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

      15. lower-/.f64 95.6

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

   4. Applied rewrites 95.6%

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

Alternative 5: 99.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 0.0005:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(-\frac{\mathsf{fma}\left(N, \mathsf{fma}\left(N, -0.5, 0.08333333333333333\right), -0.041666666666666664\right)}{N \cdot \left(N \cdot N\right)}, N, N\right)}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= (- (log (+ N 1.0)) (log N)) 0.0005)
   (/
    1.0
    (fma
     (-
      (/
       (fma N (fma N -0.5 0.08333333333333333) -0.041666666666666664)
       (* N (* N N))))
     N
     N))
   (log (/ (+ N 1.0) N))))

C

double code(double N) {
	double tmp;
	if ((log((N + 1.0)) - log(N)) <= 0.0005) {
		tmp = 1.0 / fma(-(fma(N, fma(N, -0.5, 0.08333333333333333), -0.041666666666666664) / (N * (N * N))), N, N);
	} else {
		tmp = log(((N + 1.0) / N));
	}
	return tmp;
}

Julia

function code(N)
	tmp = 0.0
	if (Float64(log(Float64(N + 1.0)) - log(N)) <= 0.0005)
		tmp = Float64(1.0 / fma(Float64(-Float64(fma(N, fma(N, -0.5, 0.08333333333333333), -0.041666666666666664) / Float64(N * Float64(N * N)))), N, N));
	else
		tmp = log(Float64(Float64(N + 1.0) / N));
	end
	return tmp
end

Wolfram

code[N_] := If[LessEqual[N[(N[Log[N[(N + 1.0), $MachinePrecision]], $MachinePrecision] - N[Log[N], $MachinePrecision]), $MachinePrecision], 0.0005], N[(1.0 / N[((-N[(N[(N * N[(N * -0.5 + 0.08333333333333333), $MachinePrecision] + -0.041666666666666664), $MachinePrecision] / N[(N * N[(N * N), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) * N + N), $MachinePrecision]), $MachinePrecision], N[Log[N[(N[(N + 1.0), $MachinePrecision] / N), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 18.9%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip-- N/A

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

      7. lift--.f64 N/A

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

      8. lower-/.f64 18.9

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

      9. lift--.f64 N/A

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

      10. lift-log.f64 N/A

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

      11. lift-log.f64 N/A

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

      12. diff-log N/A

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

      13. lower-log.f64 N/A

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

      14. lower-/.f64 22.1

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

   4. Applied rewrites 22.1%

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

   5. Taylor expanded in N around -inf

      \[\leadsto \frac{1}{\color{blue}{-1 \cdot \left(N \cdot \left(-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N} - 1\right)\right)}} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{1}{\color{blue}{\mathsf{neg}\left(N \cdot \left(-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N} - 1\right)\right)}} \]

      2. lower-neg.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\mathsf{neg}\left(N \cdot \left(-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N} - 1\right)\right)}} \]

      3. sub-neg N/A

         \[\leadsto \frac{1}{\mathsf{neg}\left(N \cdot \color{blue}{\left(-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N} + \left(\mathsf{neg}\left(1\right)\right)\right)}\right)} \]

      4. metadata-eval N/A

         \[\leadsto \frac{1}{\mathsf{neg}\left(N \cdot \left(-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N} + \color{blue}{-1}\right)\right)} \]

      5. distribute-lft-in N/A

         \[\leadsto \frac{1}{\mathsf{neg}\left(\color{blue}{\left(N \cdot \left(-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N}\right) + N \cdot -1\right)}\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{1}{\mathsf{neg}\left(\left(N \cdot \left(-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N}\right) + \color{blue}{-1 \cdot N}\right)\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{1}{\mathsf{neg}\left(\color{blue}{\mathsf{fma}\left(N, -1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N}, -1 \cdot N\right)}\right)} \]

   7. Applied rewrites 99.9%

      \[\leadsto \frac{1}{\color{blue}{-\mathsf{fma}\left(N, \frac{0.5 - \frac{0.08333333333333333 + \frac{-0.041666666666666664}{N}}{N}}{-N}, -N\right)}} \]

   8. Taylor expanded in N around 0

      \[\leadsto \frac{1}{\mathsf{neg}\left(\mathsf{fma}\left(N, \frac{N \cdot \left(\frac{1}{12} + \frac{-1}{2} \cdot N\right) - \frac{1}{24}}{{N}^{3}}, \mathsf{neg}\left(N\right)\right)\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 99.9%

       \[\leadsto \frac{1}{-\mathsf{fma}\left(N, \frac{\mathsf{fma}\left(N, \mathsf{fma}\left(N, -0.5, 0.08333333333333333\right), -0.041666666666666664\right)}{N \cdot \left(N \cdot N\right)}, -N\right)} \]
   11. Step-by-step derivation

   12. Applied rewrites 99.9%

       \[\leadsto \frac{1}{\mathsf{fma}\left(-\frac{\mathsf{fma}\left(N, \mathsf{fma}\left(N, -0.5, 0.08333333333333333\right), -0.041666666666666664\right)}{N \cdot \left(N \cdot N\right)}, \color{blue}{N}, N\right)} \]

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

   1. Initial program 92.2%

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

      1. lift--.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. diff-log N/A

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

      5. lower-log.f64 N/A

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

      6. lower-/.f64 94.7

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

   4. Applied rewrites 94.7%

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

Alternative 6: 96.7% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\mathsf{fma}\left(-\frac{\mathsf{fma}\left(N, \mathsf{fma}\left(N, -0.5, 0.08333333333333333\right), -0.041666666666666664\right)}{N \cdot \left(N \cdot N\right)}, N, N\right)} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (/
  1.0
  (fma
   (-
    (/
     (fma N (fma N -0.5 0.08333333333333333) -0.041666666666666664)
     (* N (* N N))))
   N
   N)))

C

double code(double N) {
	return 1.0 / fma(-(fma(N, fma(N, -0.5, 0.08333333333333333), -0.041666666666666664) / (N * (N * N))), N, N);
}

Julia

function code(N)
	return Float64(1.0 / fma(Float64(-Float64(fma(N, fma(N, -0.5, 0.08333333333333333), -0.041666666666666664) / Float64(N * Float64(N * N)))), N, N))
end

Wolfram

code[N_] := N[(1.0 / N[((-N[(N[(N * N[(N * -0.5 + 0.08333333333333333), $MachinePrecision] + -0.041666666666666664), $MachinePrecision] / N[(N * N[(N * N), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) * N + N), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 25.8%

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

   1. lift--.f64 N/A

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

   2. flip-- N/A

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

   3. clear-num N/A

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

   4. lower-/.f64 N/A

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

   5. clear-num N/A

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

   6. flip-- N/A

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

   7. lift--.f64 N/A

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

   8. lower-/.f64 25.8

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

   9. lift--.f64 N/A

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

   10. lift-log.f64 N/A

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

   11. lift-log.f64 N/A

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

   12. diff-log N/A

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

   13. lower-log.f64 N/A

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

   14. lower-/.f64 28.9

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

4. Applied rewrites 28.9%

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

5. Taylor expanded in N around -inf

   \[\leadsto \frac{1}{\color{blue}{-1 \cdot \left(N \cdot \left(-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N} - 1\right)\right)}} \]
6. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \frac{1}{\color{blue}{\mathsf{neg}\left(N \cdot \left(-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N} - 1\right)\right)}} \]

   2. lower-neg.f64 N/A

      \[\leadsto \frac{1}{\color{blue}{\mathsf{neg}\left(N \cdot \left(-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N} - 1\right)\right)}} \]

   3. sub-neg N/A

      \[\leadsto \frac{1}{\mathsf{neg}\left(N \cdot \color{blue}{\left(-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N} + \left(\mathsf{neg}\left(1\right)\right)\right)}\right)} \]

   4. metadata-eval N/A

      \[\leadsto \frac{1}{\mathsf{neg}\left(N \cdot \left(-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N} + \color{blue}{-1}\right)\right)} \]

   5. distribute-lft-in N/A

      \[\leadsto \frac{1}{\mathsf{neg}\left(\color{blue}{\left(N \cdot \left(-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N}\right) + N \cdot -1\right)}\right)} \]

   6. *-commutative N/A

      \[\leadsto \frac{1}{\mathsf{neg}\left(\left(N \cdot \left(-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N}\right) + \color{blue}{-1 \cdot N}\right)\right)} \]

   7. lower-fma.f64 N/A

      \[\leadsto \frac{1}{\mathsf{neg}\left(\color{blue}{\mathsf{fma}\left(N, -1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N}, -1 \cdot N\right)}\right)} \]

7. Applied rewrites 96.1%

   \[\leadsto \frac{1}{\color{blue}{-\mathsf{fma}\left(N, \frac{0.5 - \frac{0.08333333333333333 + \frac{-0.041666666666666664}{N}}{N}}{-N}, -N\right)}} \]

8. Taylor expanded in N around 0

   \[\leadsto \frac{1}{\mathsf{neg}\left(\mathsf{fma}\left(N, \frac{N \cdot \left(\frac{1}{12} + \frac{-1}{2} \cdot N\right) - \frac{1}{24}}{{N}^{3}}, \mathsf{neg}\left(N\right)\right)\right)} \]
9. Step-by-step derivation

10. Applied rewrites 96.1%

    \[\leadsto \frac{1}{-\mathsf{fma}\left(N, \frac{\mathsf{fma}\left(N, \mathsf{fma}\left(N, -0.5, 0.08333333333333333\right), -0.041666666666666664\right)}{N \cdot \left(N \cdot N\right)}, -N\right)} \]
11. Step-by-step derivation

12. Applied rewrites 96.1%

    \[\leadsto \frac{1}{\mathsf{fma}\left(-\frac{\mathsf{fma}\left(N, \mathsf{fma}\left(N, -0.5, 0.08333333333333333\right), -0.041666666666666664\right)}{N \cdot \left(N \cdot N\right)}, \color{blue}{N}, N\right)} \]
13. Add Preprocessing

Alternative 7: 96.5% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{\mathsf{fma}\left(N, \mathsf{fma}\left(N, N + 0.5, -0.08333333333333333\right), 0.041666666666666664\right)}{N \cdot N}} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (/
  1.0
  (/
   (fma N (fma N (+ N 0.5) -0.08333333333333333) 0.041666666666666664)
   (* N N))))

C

double code(double N) {
	return 1.0 / (fma(N, fma(N, (N + 0.5), -0.08333333333333333), 0.041666666666666664) / (N * N));
}

Julia

function code(N)
	return Float64(1.0 / Float64(fma(N, fma(N, Float64(N + 0.5), -0.08333333333333333), 0.041666666666666664) / Float64(N * N)))
end

Wolfram

code[N_] := N[(1.0 / N[(N[(N * N[(N * N[(N + 0.5), $MachinePrecision] + -0.08333333333333333), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] / N[(N * N), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 25.8%

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

   1. lift--.f64 N/A

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

   2. flip-- N/A

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

   3. clear-num N/A

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

   4. lower-/.f64 N/A

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

   5. clear-num N/A

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

   6. flip-- N/A

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

   7. lift--.f64 N/A

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

   8. lower-/.f64 25.8

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

   9. lift--.f64 N/A

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

   10. lift-log.f64 N/A

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

   11. lift-log.f64 N/A

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

   12. diff-log N/A

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

   13. lower-log.f64 N/A

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

   14. lower-/.f64 28.9

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

4. Applied rewrites 28.9%

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

5. Taylor expanded in N around -inf

   \[\leadsto \frac{1}{\color{blue}{-1 \cdot \left(N \cdot \left(-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N} - 1\right)\right)}} \]
6. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \frac{1}{\color{blue}{\mathsf{neg}\left(N \cdot \left(-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N} - 1\right)\right)}} \]

   2. lower-neg.f64 N/A

      \[\leadsto \frac{1}{\color{blue}{\mathsf{neg}\left(N \cdot \left(-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N} - 1\right)\right)}} \]

   3. sub-neg N/A

      \[\leadsto \frac{1}{\mathsf{neg}\left(N \cdot \color{blue}{\left(-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N} + \left(\mathsf{neg}\left(1\right)\right)\right)}\right)} \]

   4. metadata-eval N/A

      \[\leadsto \frac{1}{\mathsf{neg}\left(N \cdot \left(-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N} + \color{blue}{-1}\right)\right)} \]

   5. distribute-lft-in N/A

      \[\leadsto \frac{1}{\mathsf{neg}\left(\color{blue}{\left(N \cdot \left(-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N}\right) + N \cdot -1\right)}\right)} \]

   6. *-commutative N/A

      \[\leadsto \frac{1}{\mathsf{neg}\left(\left(N \cdot \left(-1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N}\right) + \color{blue}{-1 \cdot N}\right)\right)} \]

   7. lower-fma.f64 N/A

      \[\leadsto \frac{1}{\mathsf{neg}\left(\color{blue}{\mathsf{fma}\left(N, -1 \cdot \frac{\frac{1}{2} + -1 \cdot \frac{\frac{1}{12} - \frac{1}{24} \cdot \frac{1}{N}}{N}}{N}, -1 \cdot N\right)}\right)} \]

7. Applied rewrites 96.1%

   \[\leadsto \frac{1}{\color{blue}{-\mathsf{fma}\left(N, \frac{0.5 - \frac{0.08333333333333333 + \frac{-0.041666666666666664}{N}}{N}}{-N}, -N\right)}} \]

8. Taylor expanded in N around 0

   \[\leadsto \frac{1}{\frac{\frac{1}{24} + N \cdot \left(N \cdot \left(\frac{1}{2} + N\right) - \frac{1}{12}\right)}{\color{blue}{{N}^{2}}}} \]
9. Step-by-step derivation

10. Applied rewrites 95.8%

    \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(N, \mathsf{fma}\left(N, N + 0.5, -0.08333333333333333\right), 0.041666666666666664\right)}{\color{blue}{N \cdot N}}} \]
11. Add Preprocessing

Alternative 8: 94.8% accurate, 5.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 25.8%

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

3. Taylor expanded in N around inf

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

   1. lower-/.f64 N/A

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

5. Applied rewrites 94.2%

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

Alternative 9: 92.9% accurate, 7.1× speedup

Math

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

FPCore

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

C

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

Julia

function code(N)
	return Float64(1.0 / fma(N, Float64(0.5 / N), N))
end

Wolfram

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

Derivation

1. Initial program 25.8%

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

   1. lift--.f64 N/A

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

   2. flip-- N/A

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

   3. clear-num N/A

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

   4. lower-/.f64 N/A

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

   5. clear-num N/A

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

   6. flip-- N/A

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

   7. lift--.f64 N/A

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

   8. lower-/.f64 25.8

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

   9. lift--.f64 N/A

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

   10. lift-log.f64 N/A

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

   11. lift-log.f64 N/A

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

   12. diff-log N/A

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

   13. lower-log.f64 N/A

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

   14. lower-/.f64 28.9

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

4. Applied rewrites 28.9%

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

5. Taylor expanded in N around inf

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. *-rgt-identity N/A

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

   4. lower-fma.f64 N/A

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

   5. associate-*r/ N/A

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

   6. metadata-eval N/A

      \[\leadsto \frac{1}{\mathsf{fma}\left(N, \frac{\color{blue}{\frac{1}{2}}}{N}, N\right)} \]

   7. lower-/.f64 92.2

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

7. Applied rewrites 92.2%

   \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(N, \frac{0.5}{N}, N\right)}} \]
8. Add Preprocessing

Alternative 10: 92.2% accurate, 8.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 25.8%

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

3. Taylor expanded in N around inf

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

   1. lower-/.f64 N/A

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

   2. sub-neg N/A

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

   3. lower-+.f64 N/A

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

   4. associate-*r/ N/A

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

   5. metadata-eval N/A

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

   6. distribute-neg-frac N/A

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

   7. metadata-eval N/A

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

   8. lower-/.f64 91.5

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

5. Applied rewrites 91.5%

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

Alternative 11: 92.0% accurate, 10.4× 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

real(8) function code(n)
    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 25.8%

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

   1. lift--.f64 N/A

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

   2. flip-- N/A

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

   3. clear-num N/A

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

   4. lower-/.f64 N/A

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

   5. clear-num N/A

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

   6. flip-- N/A

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

   7. lift--.f64 N/A

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

   8. lower-/.f64 25.8

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

   9. lift--.f64 N/A

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

   10. lift-log.f64 N/A

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

   11. lift-log.f64 N/A

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

   12. diff-log N/A

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

   13. lower-log.f64 N/A

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

   14. lower-/.f64 28.9

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

4. Applied rewrites 28.9%

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

5. Taylor expanded in N around -inf

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. sub-neg N/A

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

   4. metadata-eval N/A

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

   5. distribute-lft-in N/A

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

   6. neg-mul-1 N/A

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

   7. metadata-eval N/A

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

   8. lower-+.f64 N/A

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

   9. associate-*r/ N/A

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

   10. metadata-eval N/A

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

   11. distribute-neg-frac N/A

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

   12. metadata-eval N/A

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

   13. lower-/.f64 91.5

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

7. Applied rewrites 91.5%

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

8. Taylor expanded in N around 0

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

10. Applied rewrites 91.2%

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

Alternative 12: 84.6% accurate, 17.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(N):
	return 1.0 / N

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 25.8%

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

3. Taylor expanded in N around inf

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

   1. lower-/.f64 83.1

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

5. Applied rewrites 83.1%

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

Developer Target 1: 99.8% accurate, 1.8× 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: 26.2% accurate, 1.8× 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

real(8) function code(n)
    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: 96.1% 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

real(8) function code(n)
    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 2024234 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  :pre (and (> N 1.0) (< N 1e+40))

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

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

  :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)))