2log (problem 3.3.6)

Average Error: 29.6 → 0.0

Time: 7.8s

Precision: binary64

Cost: 27204

Math

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

↓

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

FPCore

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

↓

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

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)) <= 0.0002) {
		tmp = (1.0 / N) + (((0.3333333333333333 / pow(N, 3.0)) + (-0.5 / (N * N))) + (-0.25 / pow(N, 4.0)));
	} else {
		tmp = -log((N / (N + 1.0)));
	}
	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) :: tmp
    if ((log((n + 1.0d0)) - log(n)) <= 0.0002d0) then
        tmp = (1.0d0 / n) + (((0.3333333333333333d0 / (n ** 3.0d0)) + ((-0.5d0) / (n * n))) + ((-0.25d0) / (n ** 4.0d0)))
    else
        tmp = -log((n / (n + 1.0d0)))
    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 tmp;
	if ((Math.log((N + 1.0)) - Math.log(N)) <= 0.0002) {
		tmp = (1.0 / N) + (((0.3333333333333333 / Math.pow(N, 3.0)) + (-0.5 / (N * N))) + (-0.25 / Math.pow(N, 4.0)));
	} else {
		tmp = -Math.log((N / (N + 1.0)));
	}
	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)) <= 0.0002:
		tmp = (1.0 / N) + (((0.3333333333333333 / math.pow(N, 3.0)) + (-0.5 / (N * N))) + (-0.25 / math.pow(N, 4.0)))
	else:
		tmp = -math.log((N / (N + 1.0)))
	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)) <= 0.0002)
		tmp = Float64(Float64(1.0 / N) + Float64(Float64(Float64(0.3333333333333333 / (N ^ 3.0)) + Float64(-0.5 / Float64(N * N))) + Float64(-0.25 / (N ^ 4.0))));
	else
		tmp = Float64(-log(Float64(N / Float64(N + 1.0))));
	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)) <= 0.0002)
		tmp = (1.0 / N) + (((0.3333333333333333 / (N ^ 3.0)) + (-0.5 / (N * N))) + (-0.25 / (N ^ 4.0)));
	else
		tmp = -log((N / (N + 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

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

↓

code[N_] := 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[(N[(0.3333333333333333 / N[Power[N, 3.0], $MachinePrecision]), $MachinePrecision] + N[(-0.5 / N[(N * N), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.25 / N[Power[N, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-N[Log[N[(N / N[(N + 1.0), $MachinePrecision]), $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

      \[\leadsto \color{blue}{\mathsf{log1p}\left(N\right) - \log N} \]
      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 59.5

      \[\leadsto \color{blue}{\sqrt[3]{{\left(\mathsf{log1p}\left(N\right) - \log N\right)}^{3}}} \]

   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

      \[\leadsto \color{blue}{\frac{1}{N} + \left(\left(\frac{0.3333333333333333}{{N}^{3}} - \frac{0.5}{N \cdot N}\right) + \frac{-0.25}{{N}^{4}}\right)} \]
      Proof
      (+.f64 (/.f64 1 N) (+.f64 (-.f64 (/.f64 1/3 (pow.f64 N 3)) (/.f64 1/2 (*.f64 N N))) (/.f64 -1/4 (pow.f64 N 4)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 1 N) (+.f64 (-.f64 (/.f64 (Rewrite<= metadata-eval (*.f64 1/3 1)) (pow.f64 N 3)) (/.f64 1/2 (*.f64 N N))) (/.f64 -1/4 (pow.f64 N 4)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 1 N) (+.f64 (-.f64 (Rewrite<= associate-*r/_binary64 (*.f64 1/3 (/.f64 1 (pow.f64 N 3)))) (/.f64 1/2 (*.f64 N N))) (/.f64 -1/4 (pow.f64 N 4)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 1 N) (+.f64 (-.f64 (*.f64 1/3 (/.f64 1 (pow.f64 N 3))) (/.f64 (Rewrite<= metadata-eval (*.f64 1/2 1)) (*.f64 N N))) (/.f64 -1/4 (pow.f64 N 4)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 1 N) (+.f64 (-.f64 (*.f64 1/3 (/.f64 1 (pow.f64 N 3))) (/.f64 (*.f64 1/2 1) (Rewrite<= unpow2_binary64 (pow.f64 N 2)))) (/.f64 -1/4 (pow.f64 N 4)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 1 N) (+.f64 (-.f64 (*.f64 1/3 (/.f64 1 (pow.f64 N 3))) (Rewrite<= associate-*r/_binary64 (*.f64 1/2 (/.f64 1 (pow.f64 N 2))))) (/.f64 -1/4 (pow.f64 N 4)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 1 N) (+.f64 (-.f64 (*.f64 1/3 (/.f64 1 (pow.f64 N 3))) (*.f64 1/2 (/.f64 1 (pow.f64 N 2)))) (/.f64 (Rewrite<= metadata-eval (neg.f64 1/4)) (pow.f64 N 4)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 1 N) (+.f64 (-.f64 (*.f64 1/3 (/.f64 1 (pow.f64 N 3))) (*.f64 1/2 (/.f64 1 (pow.f64 N 2)))) (Rewrite<= distribute-neg-frac_binary64 (neg.f64 (/.f64 1/4 (pow.f64 N 4)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 1 N) (+.f64 (-.f64 (*.f64 1/3 (/.f64 1 (pow.f64 N 3))) (*.f64 1/2 (/.f64 1 (pow.f64 N 2)))) (neg.f64 (/.f64 (Rewrite<= metadata-eval (*.f64 1/4 1)) (pow.f64 N 4))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 1 N) (+.f64 (-.f64 (*.f64 1/3 (/.f64 1 (pow.f64 N 3))) (*.f64 1/2 (/.f64 1 (pow.f64 N 2)))) (neg.f64 (Rewrite<= associate-*r/_binary64 (*.f64 1/4 (/.f64 1 (pow.f64 N 4))))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 1 N) (Rewrite<= sub-neg_binary64 (-.f64 (-.f64 (*.f64 1/3 (/.f64 1 (pow.f64 N 3))) (*.f64 1/2 (/.f64 1 (pow.f64 N 2)))) (*.f64 1/4 (/.f64 1 (pow.f64 N 4)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 1 N) (Rewrite<= associate--r+_binary64 (-.f64 (*.f64 1/3 (/.f64 1 (pow.f64 N 3))) (+.f64 (*.f64 1/2 (/.f64 1 (pow.f64 N 2))) (*.f64 1/4 (/.f64 1 (pow.f64 N 4))))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 1 N) (-.f64 (*.f64 1/3 (/.f64 1 (pow.f64 N 3))) (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 1/4 (/.f64 1 (pow.f64 N 4))) (*.f64 1/2 (/.f64 1 (pow.f64 N 2))))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate--l+_binary64 (-.f64 (+.f64 (/.f64 1 N) (*.f64 1/3 (/.f64 1 (pow.f64 N 3)))) (+.f64 (*.f64 1/4 (/.f64 1 (pow.f64 N 4))) (*.f64 1/2 (/.f64 1 (pow.f64 N 2)))))): 2 points increase in error, 0 points decrease in error

   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

      \[\leadsto \color{blue}{\mathsf{log1p}\left(N\right) - \log N} \]
      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 0.5

      \[\leadsto \color{blue}{\sqrt[3]{{\left(\mathsf{log1p}\left(N\right) - \log N\right)}^{3}}} \]

   4. Applied egg-rr 63.4

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

   5. Applied egg-rr 63.4

      \[\leadsto \color{blue}{\log \left(\frac{N + 1}{N}\right)} + \mathsf{fma}\left(-\sqrt{\log N}, \sqrt{\log N}, \log N\right) \]

   6. Applied egg-rr 0.0

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

4. Final simplification 0.0

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

Alternatives

Alternative 1
Error	0.0
Cost	20868

\[\begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 0.0002:\\ \;\;\;\;\left(\frac{1}{N} + \frac{-0.5}{N \cdot N}\right) + \frac{1}{{N}^{3}} \cdot \left(0.3333333333333333 - \frac{0.25}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;-\log \left(\frac{N}{N + 1}\right)\\ \end{array} \]

Alternative 2
Error	0.0
Cost	6916

\[\begin{array}{l} \mathbf{if}\;N \leq 6550.341578586145:\\ \;\;\;\;-\log \left(\frac{N}{N + 1}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N} + \frac{-0.5 + \frac{0.3333333333333333}{N}}{N \cdot N}\\ \end{array} \]

Alternative 3
Error	0.1
Cost	6852

\[\begin{array}{l} \mathbf{if}\;N \leq 6550.341578586145:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N} + \frac{-0.5 + \frac{0.3333333333333333}{N}}{N \cdot N}\\ \end{array} \]

Alternative 4
Error	0.5
Cost	6724

\[\begin{array}{l} \mathbf{if}\;N \leq 0.048334303283473976:\\ \;\;\;\;N - \log N\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N} + \frac{-0.5 + \frac{0.3333333333333333}{N}}{N \cdot N}\\ \end{array} \]

Alternative 5
Error	0.8
Cost	6660

\[\begin{array}{l} \mathbf{if}\;N \leq 0.048334303283473976:\\ \;\;\;\;-\log N\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N} + \frac{-0.5 + \frac{0.3333333333333333}{N}}{N \cdot N}\\ \end{array} \]

Alternative 6
Error	30.7
Cost	192

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

Alternative 7
Error	61.3
Cost	64

\[0 \]

Alternative 8
Error	61.1
Cost	64

\[N \]

Reproduce

herbie shell --seed 2022311 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1.0)) (log N)))