2log (problem 3.3.6)

Average Error: 29.5 → 0.0

Time: 7.1s

Precision: binary64

Cost: 27588

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 0.0001:\\ \;\;\;\;\left(\frac{1}{N} + 0.3333333333333333 \cdot {N}^{-3}\right) + \left(\frac{1}{{N}^{4}} \cdot -0.25 + 0.5 \cdot \left(\frac{1}{N} \cdot \frac{-1}{N}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (N)
 :precision binary64
 (if (<= (- (log (+ N 1.0)) (log N)) 0.0001)
   (+
    (+ (/ 1.0 N) (* 0.3333333333333333 (pow N -3.0)))
    (+ (* (/ 1.0 (pow N 4.0)) -0.25) (* 0.5 (* (/ 1.0 N) (/ -1.0 N)))))
   (log (/ (+ N 1.0) N))))

C

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

↓

double code(double N) {
	double tmp;
	if ((log((N + 1.0)) - log(N)) <= 0.0001) {
		tmp = ((1.0 / N) + (0.3333333333333333 * pow(N, -3.0))) + (((1.0 / pow(N, 4.0)) * -0.25) + (0.5 * ((1.0 / N) * (-1.0 / N))));
	} else {
		tmp = log(((N + 1.0) / N));
	}
	return tmp;
}

Fortran

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

↓

real(8) function code(n)
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((log((n + 1.0d0)) - log(n)) <= 0.0001d0) then
        tmp = ((1.0d0 / n) + (0.3333333333333333d0 * (n ** (-3.0d0)))) + (((1.0d0 / (n ** 4.0d0)) * (-0.25d0)) + (0.5d0 * ((1.0d0 / n) * ((-1.0d0) / n))))
    else
        tmp = log(((n + 1.0d0) / n))
    end if
    code = tmp
end function

Java

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

↓

public static double code(double N) {
	double tmp;
	if ((Math.log((N + 1.0)) - Math.log(N)) <= 0.0001) {
		tmp = ((1.0 / N) + (0.3333333333333333 * Math.pow(N, -3.0))) + (((1.0 / Math.pow(N, 4.0)) * -0.25) + (0.5 * ((1.0 / N) * (-1.0 / N))));
	} else {
		tmp = Math.log(((N + 1.0) / N));
	}
	return tmp;
}

Python

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

↓

def code(N):
	tmp = 0
	if (math.log((N + 1.0)) - math.log(N)) <= 0.0001:
		tmp = ((1.0 / N) + (0.3333333333333333 * math.pow(N, -3.0))) + (((1.0 / math.pow(N, 4.0)) * -0.25) + (0.5 * ((1.0 / N) * (-1.0 / N))))
	else:
		tmp = math.log(((N + 1.0) / N))
	return tmp

Julia

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

↓

function code(N)
	tmp = 0.0
	if (Float64(log(Float64(N + 1.0)) - log(N)) <= 0.0001)
		tmp = Float64(Float64(Float64(1.0 / N) + Float64(0.3333333333333333 * (N ^ -3.0))) + Float64(Float64(Float64(1.0 / (N ^ 4.0)) * -0.25) + Float64(0.5 * Float64(Float64(1.0 / N) * Float64(-1.0 / N)))));
	else
		tmp = log(Float64(Float64(N + 1.0) / N));
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(N)
	tmp = 0.0;
	if ((log((N + 1.0)) - log(N)) <= 0.0001)
		tmp = ((1.0 / N) + (0.3333333333333333 * (N ^ -3.0))) + (((1.0 / (N ^ 4.0)) * -0.25) + (0.5 * ((1.0 / N) * (-1.0 / N))));
	else
		tmp = log(((N + 1.0) / N));
	end
	tmp_2 = tmp;
end

Wolfram

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

↓

code[N_] := If[LessEqual[N[(N[Log[N[(N + 1.0), $MachinePrecision]], $MachinePrecision] - N[Log[N], $MachinePrecision]), $MachinePrecision], 0.0001], N[(N[(N[(1.0 / N), $MachinePrecision] + N[(0.3333333333333333 * N[Power[N, -3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(1.0 / N[Power[N, 4.0], $MachinePrecision]), $MachinePrecision] * -0.25), $MachinePrecision] + N[(0.5 * N[(N[(1.0 / N), $MachinePrecision] * N[(-1.0 / N), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 1)) (log.f64 N)) < 1.00000000000000005e-4

   1. Initial program 59.4

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

   2. Taylor expanded in N around inf 0.0

      \[\leadsto \color{blue}{\left(\frac{1}{N} + 0.3333333333333333 \cdot \frac{1}{{N}^{3}}\right) - \left(0.25 \cdot \frac{1}{{N}^{4}} + 0.5 \cdot \frac{1}{{N}^{2}}\right)} \]

   3. Applied egg-rr 0.0

      \[\leadsto \left(\frac{1}{N} + 0.3333333333333333 \cdot \frac{1}{{N}^{3}}\right) - \left(0.25 \cdot \frac{1}{{N}^{4}} + 0.5 \cdot \color{blue}{\left(\frac{1}{N} \cdot \frac{1}{N}\right)}\right) \]

   4. Applied egg-rr 0.0

      \[\leadsto \left(\frac{1}{N} + 0.3333333333333333 \cdot \color{blue}{{N}^{-3}}\right) - \left(0.25 \cdot \frac{1}{{N}^{4}} + 0.5 \cdot \left(\frac{1}{N} \cdot \frac{1}{N}\right)\right) \]

   if 1.00000000000000005e-4 < (-.f64 (log.f64 (+.f64 N 1)) (log.f64 N))

   1. Initial program 0.1

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

   2. Applied egg-rr 0.1

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

4. Final simplification 0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 0.0001:\\ \;\;\;\;\left(\frac{1}{N} + 0.3333333333333333 \cdot {N}^{-3}\right) + \left(\frac{1}{{N}^{4}} \cdot -0.25 + 0.5 \cdot \left(\frac{1}{N} \cdot \frac{-1}{N}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.0
Cost	19908

\[\begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 0.0001:\\ \;\;\;\;\frac{1}{N} + \left(\frac{-0.5}{N \cdot N} + \frac{\frac{0.3333333333333333 + \frac{-0.25}{N}}{N \cdot N}}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \end{array} \]

Alternative 2
Error	0.4
Cost	6724

\[\begin{array}{l} \mathbf{if}\;N \leq 0.10266196555053207:\\ \;\;\;\;N - \log N\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N} + \left(\frac{-0.5}{N \cdot N} + \frac{\frac{0.3333333333333333 + \frac{-0.25}{N}}{N \cdot N}}{N}\right)\\ \end{array} \]

Alternative 3
Error	0.6
Cost	6660

\[\begin{array}{l} \mathbf{if}\;N \leq 0.10266196555053207:\\ \;\;\;\;-\log N\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N} + \left(\frac{-0.5}{N \cdot N} + \frac{\frac{0.3333333333333333 + \frac{-0.25}{N}}{N \cdot N}}{N}\right)\\ \end{array} \]

Alternative 4
Error	27.6
Cost	320

\[\frac{1}{N + 0.5} \]

Alternative 5
Error	30.9
Cost	192

\[\frac{1}{N} \]

Alternative 6
Error	61.1
Cost	64

\[N \]

Reproduce

herbie shell --seed 2022291 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1.0)) (log N)))