2log (problem 3.3.6)

Average Error: 29.2 → 0.1

Time: 19.0s

Precision: binary64

Cost: 46404

Math

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

↓

\[\begin{array}{l} t_0 := \log \left(N + 1\right)\\ \mathbf{if}\;t_0 - \log N \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\frac{1}{N} + \left(\frac{0.3333333333333333}{{N}^{3}} - \frac{0.5}{{N}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t_0 + \frac{\log N}{-2}\right) + \left(t_0 - \left(t_0 + \frac{\log N}{2}\right)\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (N)
 :precision binary64
 (let* ((t_0 (log (+ N 1.0))))
   (if (<= (- t_0 (log N)) 5e-6)
     (+ (/ 1.0 N) (- (/ 0.3333333333333333 (pow N 3.0)) (/ 0.5 (pow N 2.0))))
     (+ (+ t_0 (/ (log N) -2.0)) (- t_0 (+ t_0 (/ (log N) 2.0)))))))

C

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

↓

double code(double N) {
	double t_0 = log((N + 1.0));
	double tmp;
	if ((t_0 - log(N)) <= 5e-6) {
		tmp = (1.0 / N) + ((0.3333333333333333 / pow(N, 3.0)) - (0.5 / pow(N, 2.0)));
	} else {
		tmp = (t_0 + (log(N) / -2.0)) + (t_0 - (t_0 + (log(N) / 2.0)));
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = log((n + 1.0d0))
    if ((t_0 - log(n)) <= 5d-6) then
        tmp = (1.0d0 / n) + ((0.3333333333333333d0 / (n ** 3.0d0)) - (0.5d0 / (n ** 2.0d0)))
    else
        tmp = (t_0 + (log(n) / (-2.0d0))) + (t_0 - (t_0 + (log(n) / 2.0d0)))
    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 t_0 = Math.log((N + 1.0));
	double tmp;
	if ((t_0 - Math.log(N)) <= 5e-6) {
		tmp = (1.0 / N) + ((0.3333333333333333 / Math.pow(N, 3.0)) - (0.5 / Math.pow(N, 2.0)));
	} else {
		tmp = (t_0 + (Math.log(N) / -2.0)) + (t_0 - (t_0 + (Math.log(N) / 2.0)));
	}
	return tmp;
}

Python

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

↓

def code(N):
	t_0 = math.log((N + 1.0))
	tmp = 0
	if (t_0 - math.log(N)) <= 5e-6:
		tmp = (1.0 / N) + ((0.3333333333333333 / math.pow(N, 3.0)) - (0.5 / math.pow(N, 2.0)))
	else:
		tmp = (t_0 + (math.log(N) / -2.0)) + (t_0 - (t_0 + (math.log(N) / 2.0)))
	return tmp

Julia

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

↓

function code(N)
	t_0 = log(Float64(N + 1.0))
	tmp = 0.0
	if (Float64(t_0 - log(N)) <= 5e-6)
		tmp = Float64(Float64(1.0 / N) + Float64(Float64(0.3333333333333333 / (N ^ 3.0)) - Float64(0.5 / (N ^ 2.0))));
	else
		tmp = Float64(Float64(t_0 + Float64(log(N) / -2.0)) + Float64(t_0 - Float64(t_0 + Float64(log(N) / 2.0))));
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(N)
	t_0 = log((N + 1.0));
	tmp = 0.0;
	if ((t_0 - log(N)) <= 5e-6)
		tmp = (1.0 / N) + ((0.3333333333333333 / (N ^ 3.0)) - (0.5 / (N ^ 2.0)));
	else
		tmp = (t_0 + (log(N) / -2.0)) + (t_0 - (t_0 + (log(N) / 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

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

↓

code[N_] := Block[{t$95$0 = N[Log[N[(N + 1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(t$95$0 - N[Log[N], $MachinePrecision]), $MachinePrecision], 5e-6], N[(N[(1.0 / N), $MachinePrecision] + N[(N[(0.3333333333333333 / N[Power[N, 3.0], $MachinePrecision]), $MachinePrecision] - N[(0.5 / N[Power[N, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 + N[(N[Log[N], $MachinePrecision] / -2.0), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 - N[(t$95$0 + N[(N[Log[N], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (log.f64 (+.f64 N 1)) (log.f64 N)) < 5.00000000000000041e-6

   1. Initial program 59.8

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

   2. Taylor expanded in N around inf 0.0

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

   3. Simplified 0.0

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

      [Start] 0.0	\[ \left(\frac{1}{N} + 0.3333333333333333 \cdot \frac{1}{{N}^{3}}\right) - 0.5 \cdot \frac{1}{{N}^{2}} \]
      rational_best-simplify-59 [=>] 0.0	\[ \color{blue}{\left(0.3333333333333333 \cdot \frac{1}{{N}^{3}} - \left(-\frac{1}{N}\right)\right)} - 0.5 \cdot \frac{1}{{N}^{2}} \]
      rational_best-simplify-52 [=>] 0.0	\[ \color{blue}{0.3333333333333333 \cdot \frac{1}{{N}^{3}} - \left(0.5 \cdot \frac{1}{{N}^{2}} + \left(-\frac{1}{N}\right)\right)} \]
      rational_best-simplify-55 [=>] 0.0	\[ \color{blue}{1 \cdot \frac{0.3333333333333333}{{N}^{3}}} - \left(0.5 \cdot \frac{1}{{N}^{2}} + \left(-\frac{1}{N}\right)\right) \]
      rational_best-simplify-1 [=>] 0.0	\[ \color{blue}{\frac{0.3333333333333333}{{N}^{3}} \cdot 1} - \left(0.5 \cdot \frac{1}{{N}^{2}} + \left(-\frac{1}{N}\right)\right) \]
      rational_best-simplify-7 [=>] 0.0	\[ \color{blue}{\frac{0.3333333333333333}{{N}^{3}}} - \left(0.5 \cdot \frac{1}{{N}^{2}} + \left(-\frac{1}{N}\right)\right) \]
      rational_best-simplify-55 [=>] 0.0	\[ \frac{0.3333333333333333}{{N}^{3}} - \left(\color{blue}{1 \cdot \frac{0.5}{{N}^{2}}} + \left(-\frac{1}{N}\right)\right) \]
      rational_best-simplify-1 [=>] 0.0	\[ \frac{0.3333333333333333}{{N}^{3}} - \left(\color{blue}{\frac{0.5}{{N}^{2}} \cdot 1} + \left(-\frac{1}{N}\right)\right) \]
      rational_best-simplify-7 [=>] 0.0	\[ \frac{0.3333333333333333}{{N}^{3}} - \left(\color{blue}{\frac{0.5}{{N}^{2}}} + \left(-\frac{1}{N}\right)\right) \]
      rational_best-simplify-13 [=>] 0.0	\[ \frac{0.3333333333333333}{{N}^{3}} - \left(\frac{0.5}{{N}^{2}} + \color{blue}{\frac{\frac{1}{N}}{-1}}\right) \]
      rational_best-simplify-49 [=>] 0.0	\[ \frac{0.3333333333333333}{{N}^{3}} - \left(\frac{0.5}{{N}^{2}} + \color{blue}{\frac{\frac{1}{-1}}{N}}\right) \]
      metadata-eval [=>] 0.0	\[ \frac{0.3333333333333333}{{N}^{3}} - \left(\frac{0.5}{{N}^{2}} + \frac{\color{blue}{-1}}{N}\right) \]

   4. Taylor expanded in N around 0 0.0

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

   5. Simplified 0.0

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

      [Start] 0.0	\[ \left(0.3333333333333333 \cdot \frac{1}{{N}^{3}} + \frac{1}{N}\right) - 0.5 \cdot \frac{1}{{N}^{2}} \]
      rational_best-simplify-3 [<=] 0.0	\[ \color{blue}{\left(\frac{1}{N} + 0.3333333333333333 \cdot \frac{1}{{N}^{3}}\right)} - 0.5 \cdot \frac{1}{{N}^{2}} \]
      rational_best-simplify-59 [=>] 0.0	\[ \color{blue}{\left(0.3333333333333333 \cdot \frac{1}{{N}^{3}} - \left(-\frac{1}{N}\right)\right)} - 0.5 \cdot \frac{1}{{N}^{2}} \]
      rational_best-simplify-55 [<=] 0.0	\[ \left(\color{blue}{1 \cdot \frac{0.3333333333333333}{{N}^{3}}} - \left(-\frac{1}{N}\right)\right) - 0.5 \cdot \frac{1}{{N}^{2}} \]
      rational_best-simplify-1 [=>] 0.0	\[ \left(\color{blue}{\frac{0.3333333333333333}{{N}^{3}} \cdot 1} - \left(-\frac{1}{N}\right)\right) - 0.5 \cdot \frac{1}{{N}^{2}} \]
      rational_best-simplify-7 [=>] 0.0	\[ \left(\color{blue}{\frac{0.3333333333333333}{{N}^{3}}} - \left(-\frac{1}{N}\right)\right) - 0.5 \cdot \frac{1}{{N}^{2}} \]
      rational_best-simplify-55 [=>] 0.0	\[ \left(\frac{0.3333333333333333}{{N}^{3}} - \left(-\frac{1}{N}\right)\right) - \color{blue}{1 \cdot \frac{0.5}{{N}^{2}}} \]
      rational_best-simplify-1 [<=] 0.0	\[ \left(\frac{0.3333333333333333}{{N}^{3}} - \left(-\frac{1}{N}\right)\right) - \color{blue}{\frac{0.5}{{N}^{2}} \cdot 1} \]
      rational_best-simplify-7 [=>] 0.0	\[ \left(\frac{0.3333333333333333}{{N}^{3}} - \left(-\frac{1}{N}\right)\right) - \color{blue}{\frac{0.5}{{N}^{2}}} \]
      rational_best-simplify-48 [=>] 0.0	\[ \color{blue}{\left(\frac{0.3333333333333333}{{N}^{3}} - \frac{0.5}{{N}^{2}}\right) - \left(-\frac{1}{N}\right)} \]
      rational_best-simplify-12 [<=] 0.0	\[ \left(\frac{0.3333333333333333}{{N}^{3}} - \frac{0.5}{{N}^{2}}\right) - \color{blue}{\frac{\frac{1}{N}}{-1}} \]
      rational_best-simplify-49 [=>] 0.0	\[ \left(\frac{0.3333333333333333}{{N}^{3}} - \frac{0.5}{{N}^{2}}\right) - \color{blue}{\frac{\frac{1}{-1}}{N}} \]
      metadata-eval [=>] 0.0	\[ \left(\frac{0.3333333333333333}{{N}^{3}} - \frac{0.5}{{N}^{2}}\right) - \frac{\color{blue}{-1}}{N} \]
      rational_best-simplify-52 [=>] 0.0	\[ \color{blue}{\frac{0.3333333333333333}{{N}^{3}} - \left(\frac{-1}{N} + \frac{0.5}{{N}^{2}}\right)} \]
      rational_best-simplify-7 [<=] 0.0	\[ \color{blue}{\frac{0.3333333333333333}{{N}^{3}} \cdot 1} - \left(\frac{-1}{N} + \frac{0.5}{{N}^{2}}\right) \]
      metadata-eval [<=] 0.0	\[ \frac{0.3333333333333333}{{N}^{3}} \cdot 1 - \left(\frac{\color{blue}{\frac{1}{-1}}}{N} + \frac{0.5}{{N}^{2}}\right) \]
      rational_best-simplify-49 [<=] 0.0	\[ \frac{0.3333333333333333}{{N}^{3}} \cdot 1 - \left(\color{blue}{\frac{\frac{1}{N}}{-1}} + \frac{0.5}{{N}^{2}}\right) \]
      metadata-eval [<=] 0.0	\[ \frac{0.3333333333333333}{{N}^{3}} \cdot 1 - \left(\frac{\frac{1}{N}}{-1} + \frac{\color{blue}{\frac{-0.5}{-1}}}{{N}^{2}}\right) \]
      rational_best-simplify-49 [<=] 0.0	\[ \frac{0.3333333333333333}{{N}^{3}} \cdot 1 - \left(\frac{\frac{1}{N}}{-1} + \color{blue}{\frac{\frac{-0.5}{{N}^{2}}}{-1}}\right) \]
      rational_best-simplify-65 [<=] 0.0	\[ \frac{0.3333333333333333}{{N}^{3}} \cdot 1 - \color{blue}{\frac{\frac{1}{N} + \frac{-0.5}{{N}^{2}}}{-1}} \]
      rational_best-simplify-12 [=>] 0.0	\[ \frac{0.3333333333333333}{{N}^{3}} \cdot 1 - \color{blue}{\left(-\left(\frac{1}{N} + \frac{-0.5}{{N}^{2}}\right)\right)} \]

   if 5.00000000000000041e-6 < (-.f64 (log.f64 (+.f64 N 1)) (log.f64 N))

   1. Initial program 0.2

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

   2. Applied egg-rr 0.2

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\frac{1}{N} + \left(\frac{0.3333333333333333}{{N}^{3}} - \frac{0.5}{{N}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log \left(N + 1\right) + \frac{\log N}{-2}\right) + \left(\log \left(N + 1\right) - \left(\log \left(N + 1\right) + \frac{\log N}{2}\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.1
Cost	26820

\[\begin{array}{l} t_0 := \log \left(N + 1\right) - \log N\\ \mathbf{if}\;t_0 \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\frac{1}{N} + \left(\frac{0.3333333333333333}{{N}^{3}} - \frac{0.5}{{N}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	0.1
Cost	13252

\[\begin{array}{l} \mathbf{if}\;N \leq 78000:\\ \;\;\;\;\log \left(N + 1\right) - \log N\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{N} + \left(\frac{-0.5}{{N}^{2}} + \frac{0.5}{N}\right)\\ \end{array} \]

Alternative 3
Error	0.7
Cost	7300

\[\begin{array}{l} \mathbf{if}\;N \leq 0.9:\\ \;\;\;\;N + \log \left(\frac{1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{N} + \left(\frac{-0.5}{{N}^{2}} + \frac{0.5}{N}\right)\\ \end{array} \]

Alternative 4
Error	0.7
Cost	7044

\[\begin{array}{l} \mathbf{if}\;N \leq 0.9:\\ \;\;\;\;N + \log \left(\frac{1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N} - \frac{0.5}{{N}^{2}}\\ \end{array} \]

Alternative 5
Error	1.0
Cost	6852

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

Alternative 6
Error	1.0
Cost	6724

\[\begin{array}{l} \mathbf{if}\;N \leq 1:\\ \;\;\;\;N - \log N\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N}\\ \end{array} \]

Alternative 7
Error	1.3
Cost	6660

\[\begin{array}{l} \mathbf{if}\;N \leq 0.53:\\ \;\;\;\;-\log N\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N}\\ \end{array} \]

Alternative 8
Error	31.1
Cost	192

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

Alternative 9
Error	61.0
Cost	64

\[N \]

Reproduce

herbie shell --seed 2023100 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1.0)) (log N)))