2log (problem 3.3.6)

Average Error: 29.3 → 0.2

Time: 7.5s

Precision: binary64

Cost: 19908

Math

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

↓

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

FPCore

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

↓

(FPCore (N)
 :precision binary64
 (if (<= (- (log (+ N 1.0)) (log N)) 5e-10)
   (/ (+ 1.0 (/ -0.5 N)) 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)) <= 5e-10) {
		tmp = (1.0 + (-0.5 / N)) / 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)) <= 5d-10) then
        tmp = (1.0d0 + ((-0.5d0) / n)) / 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)) <= 5e-10) {
		tmp = (1.0 + (-0.5 / N)) / 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)) <= 5e-10:
		tmp = (1.0 + (-0.5 / N)) / 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)) <= 5e-10)
		tmp = Float64(Float64(1.0 + Float64(-0.5 / N)) / 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)) <= 5e-10)
		tmp = (1.0 + (-0.5 / N)) / 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], 5e-10], N[(N[(1.0 + N[(-0.5 / N), $MachinePrecision]), $MachinePrecision] / N), $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)) < 5.00000000000000031e-10

   1. Initial program 60.1

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

   2. Simplified 60.1

      \[\leadsto \color{blue}{\mathsf{log1p}\left(N\right) - \log N} \]
      Proof
      (-.f64 (log1p.f64 N) (log.f64 N)): 0 points increase in error, 0 points decrease in error
      (-.f64 (Rewrite<= log1p-def_binary64 (log.f64 (+.f64 1 N))) (log.f64 N)): 0 points increase in error, 0 points decrease in error
      (-.f64 (log.f64 (Rewrite<= +-commutative_binary64 (+.f64 N 1))) (log.f64 N)): 0 points increase in error, 0 points decrease in error

   3. Taylor expanded in N around inf 0.0

      \[\leadsto \color{blue}{\frac{1}{N} - 0.5 \cdot \frac{1}{{N}^{2}}} \]

   4. Simplified 0.0

      \[\leadsto \color{blue}{\frac{1}{N} - \frac{\frac{0.5}{N}}{N}} \]
      Proof
      (-.f64 (/.f64 1 N) (/.f64 (/.f64 1/2 N) N)): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 1 N) (Rewrite<= associate-/r*_binary64 (/.f64 1/2 (*.f64 N N)))): 11 points increase in error, 13 points decrease in error
      (-.f64 (/.f64 1 N) (/.f64 (Rewrite<= metadata-eval (*.f64 1/2 1)) (*.f64 N N))): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 1 N) (/.f64 (*.f64 1/2 1) (Rewrite<= unpow2_binary64 (pow.f64 N 2)))): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 1 N) (Rewrite<= associate-*r/_binary64 (*.f64 1/2 (/.f64 1 (pow.f64 N 2))))): 0 points increase in error, 0 points decrease in error

   5. Applied egg-rr 0.0

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

   if 5.00000000000000031e-10 < (-.f64 (log.f64 (+.f64 N 1)) (log.f64 N))

   1. Initial program 0.5

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

   2. Simplified 0.5

      \[\leadsto \color{blue}{\mathsf{log1p}\left(N\right) - \log N} \]
      Proof
      (-.f64 (log1p.f64 N) (log.f64 N)): 0 points increase in error, 0 points decrease in error
      (-.f64 (Rewrite<= log1p-def_binary64 (log.f64 (+.f64 1 N))) (log.f64 N)): 0 points increase in error, 0 points decrease in error
      (-.f64 (log.f64 (Rewrite<= +-commutative_binary64 (+.f64 N 1))) (log.f64 N)): 0 points increase in error, 0 points decrease in error

   3. Applied egg-rr 0.4

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

4. Final simplification 0.2

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

Alternatives

Alternative 1
Error	1.0
Cost	6724

\[\begin{array}{l} \mathbf{if}\;N \leq 2.2078869502073876 \cdot 10^{-9}:\\ \;\;\;\;N - \log N\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N} + \frac{\frac{-0.5 + \frac{0.3333333333333333}{N}}{N}}{N}\\ \end{array} \]

Alternative 2
Error	1.1
Cost	6660

\[\begin{array}{l} \mathbf{if}\;N \leq 2.2078869502073876 \cdot 10^{-9}:\\ \;\;\;\;-\log N\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N} + \frac{\frac{-0.5 + \frac{0.3333333333333333}{N}}{N}}{N}\\ \end{array} \]

Alternative 3
Error	31.0
Cost	192

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

Alternative 4
Error	61.1
Cost	64

\[N \]

Reproduce

herbie shell --seed 2022308 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1.0)) (log N)))