2log (problem 3.3.6)

Average Error: 29.0 → 0.2

Time: 8.3s

Precision: binary64

Cost: 19908

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 10^{-7}:\\ \;\;\;\;\frac{1}{N} + \frac{\frac{\frac{0.3333333333333333}{N} + -0.5}{N}}{N}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (N)
 :precision binary64
 (if (<= (- (log (+ N 1.0)) (log N)) 1e-7)
   (+ (/ 1.0 N) (/ (/ (+ (/ 0.3333333333333333 N) -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)) <= 1e-7) {
		tmp = (1.0 / N) + ((((0.3333333333333333 / N) + -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)) <= 1d-7) then
        tmp = (1.0d0 / n) + ((((0.3333333333333333d0 / n) + (-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)) <= 1e-7) {
		tmp = (1.0 / N) + ((((0.3333333333333333 / N) + -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)) <= 1e-7:
		tmp = (1.0 / N) + ((((0.3333333333333333 / N) + -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)) <= 1e-7)
		tmp = Float64(Float64(1.0 / N) + Float64(Float64(Float64(Float64(0.3333333333333333 / N) + -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)) <= 1e-7)
		tmp = (1.0 / N) + ((((0.3333333333333333 / N) + -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], 1e-7], N[(N[(1.0 / N), $MachinePrecision] + N[(N[(N[(N[(0.3333333333333333 / N), $MachinePrecision] + -0.5), $MachinePrecision] / N), $MachinePrecision] / 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 1)) (log.f64 N)) < 9.9999999999999995e-8

   1. Initial program 60.0

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

   2. Simplified 60.0

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

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

   4. Simplified 0.0

      \[\leadsto \color{blue}{\frac{1}{N} + \frac{1}{N \cdot N} \cdot \left(\frac{0.3333333333333333}{N} - 0.5\right)} \]
      Proof
      (+.f64 (/.f64 1 N) (*.f64 (/.f64 1 (*.f64 N N)) (-.f64 (/.f64 1/3 N) 1/2))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 1 N) (*.f64 (/.f64 1 (Rewrite<= unpow2_binary64 (pow.f64 N 2))) (-.f64 (/.f64 1/3 N) 1/2))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 1 N) (Rewrite<= distribute-rgt-out--_binary64 (-.f64 (*.f64 (/.f64 1/3 N) (/.f64 1 (pow.f64 N 2))) (*.f64 1/2 (/.f64 1 (pow.f64 N 2)))))): 1 points increase in error, 2 points decrease in error
      (+.f64 (/.f64 1 N) (-.f64 (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 1/3 (/.f64 1 (pow.f64 N 2))) N)) (*.f64 1/2 (/.f64 1 (pow.f64 N 2))))): 4 points increase in error, 8 points decrease in error
      (+.f64 (/.f64 1 N) (-.f64 (Rewrite<= associate-*r/_binary64 (*.f64 1/3 (/.f64 (/.f64 1 (pow.f64 N 2)) N))) (*.f64 1/2 (/.f64 1 (pow.f64 N 2))))): 11 points increase in error, 5 points decrease in error
      (+.f64 (/.f64 1 N) (-.f64 (*.f64 1/3 (Rewrite<= associate-/r*_binary64 (/.f64 1 (*.f64 (pow.f64 N 2) N)))) (*.f64 1/2 (/.f64 1 (pow.f64 N 2))))): 8 points increase in error, 8 points decrease in error
      (+.f64 (/.f64 1 N) (-.f64 (*.f64 1/3 (/.f64 1 (*.f64 (Rewrite=> unpow2_binary64 (*.f64 N N)) N))) (*.f64 1/2 (/.f64 1 (pow.f64 N 2))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 1 N) (-.f64 (*.f64 1/3 (/.f64 1 (Rewrite<= unpow3_binary64 (pow.f64 N 3)))) (*.f64 1/2 (/.f64 1 (pow.f64 N 2))))): 1 points increase in error, 5 points decrease in error
      (Rewrite<= associate--l+_binary64 (-.f64 (+.f64 (/.f64 1 N) (*.f64 1/3 (/.f64 1 (pow.f64 N 3)))) (*.f64 1/2 (/.f64 1 (pow.f64 N 2))))): 0 points increase in error, 1 points decrease in error

   5. Applied egg-rr 0.0

      \[\leadsto \frac{1}{N} + \color{blue}{\frac{\frac{\frac{0.3333333333333333}{N} + -0.5}{N}}{N}} \]

   if 9.9999999999999995e-8 < (-.f64 (log.f64 (+.f64 N 1)) (log.f64 N))

   1. Initial program 0.3

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

   2. Simplified 0.3

      \[\leadsto \color{blue}{\mathsf{log1p}\left(N\right) - \log N} \]
      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 0.3

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

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 10^{-7}:\\ \;\;\;\;\frac{1}{N} + \frac{\frac{\frac{0.3333333333333333}{N} + -0.5}{N}}{N}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.5
Cost	6724

\[\begin{array}{l} \mathbf{if}\;N \leq 0.008572902970753304:\\ \;\;\;\;N - \log N\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N} + \frac{\frac{\frac{0.3333333333333333}{N} + -0.5}{N}}{N}\\ \end{array} \]

Alternative 2
Error	0.8
Cost	6660

\[\begin{array}{l} \mathbf{if}\;N \leq 0.008572902970753304:\\ \;\;\;\;-\log N\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N} + \frac{\frac{\frac{0.3333333333333333}{N} + -0.5}{N}}{N}\\ \end{array} \]

Alternative 3
Error	28.4
Cost	324

\[\begin{array}{l} \mathbf{if}\;N \leq 0.008572902970753304:\\ \;\;\;\;2\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N}\\ \end{array} \]

Alternative 4
Error	28.0
Cost	320

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

Alternative 5
Error	57.6
Cost	64

\[2 \]

Reproduce

herbie shell --seed 2022290 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1.0)) (log N)))