2log (problem 3.3.6)

Average Error: 29.0 → 0.1

Time: 3.8s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \sqrt{\mathsf{log1p}\left(N\right)}\\ \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 0.0002:\\ \;\;\;\;\frac{1}{N} + \left(\frac{0.3333333333333333}{{N}^{3}} + \left(\frac{-0.25}{{N}^{4}} + \frac{\frac{-0.5}{N}}{N}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t_0, t_0, -\log N\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (N)
 :precision binary64
 (let* ((t_0 (sqrt (log1p N))))
   (if (<= (- (log (+ N 1.0)) (log N)) 0.0002)
     (+
      (/ 1.0 N)
      (+
       (/ 0.3333333333333333 (pow N 3.0))
       (+ (/ -0.25 (pow N 4.0)) (/ (/ -0.5 N) N))))
     (fma t_0 t_0 (- (log N))))))

C

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

↓

double code(double N) {
	double t_0 = sqrt(log1p(N));
	double tmp;
	if ((log((N + 1.0)) - log(N)) <= 0.0002) {
		tmp = (1.0 / N) + ((0.3333333333333333 / pow(N, 3.0)) + ((-0.25 / pow(N, 4.0)) + ((-0.5 / N) / N)));
	} else {
		tmp = fma(t_0, t_0, -log(N));
	}
	return tmp;
}

Julia

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

↓

function code(N)
	t_0 = sqrt(log1p(N))
	tmp = 0.0
	if (Float64(log(Float64(N + 1.0)) - log(N)) <= 0.0002)
		tmp = Float64(Float64(1.0 / N) + Float64(Float64(0.3333333333333333 / (N ^ 3.0)) + Float64(Float64(-0.25 / (N ^ 4.0)) + Float64(Float64(-0.5 / N) / N))));
	else
		tmp = fma(t_0, t_0, Float64(-log(N)));
	end
	return 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[Sqrt[N[Log[1 + N], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[Log[N[(N + 1.0), $MachinePrecision]], $MachinePrecision] - N[Log[N], $MachinePrecision]), $MachinePrecision], 0.0002], N[(N[(1.0 / N), $MachinePrecision] + N[(N[(0.3333333333333333 / N[Power[N, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(-0.25 / N[Power[N, 4.0], $MachinePrecision]), $MachinePrecision] + N[(N[(-0.5 / N), $MachinePrecision] / N), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * t$95$0 + (-N[Log[N], $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (log.f64 (+.f64 N 1)) (log.f64 N)) < 2.0000000000000001e-4

   1. Initial program 59.5

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

   2. Simplified 59.5

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

   3. Applied egg-rr 59.2

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

   4. Taylor expanded in N around inf 0.0

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

   5. Simplified 0.0

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

   if 2.0000000000000001e-4 < (-.f64 (log.f64 (+.f64 N 1)) (log.f64 N))

   1. Initial program 0.1

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

   2. Simplified 0.1

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

   3. Applied egg-rr 0.1

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\mathsf{log1p}\left(N\right)}, \sqrt{\mathsf{log1p}\left(N\right)}, -\log N\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 0.0002:\\ \;\;\;\;\frac{1}{N} + \left(\frac{0.3333333333333333}{{N}^{3}} + \left(\frac{-0.25}{{N}^{4}} + \frac{\frac{-0.5}{N}}{N}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{\mathsf{log1p}\left(N\right)}, \sqrt{\mathsf{log1p}\left(N\right)}, -\log N\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022210 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1.0)) (log N)))