logs (example 3.8)

Percentage Accurate: 1.5% → 100.0%

Time: 6.3s

Alternatives: 2

Speedup: 213.0×

Specification

\[n > 6.8 \cdot 10^{+15}\]

Math

\[\begin{array}{l} \\ \left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1 \end{array} \]

FPCore

(FPCore (n)
 :precision binary64
 (- (- (* (+ n 1.0) (log (+ n 1.0))) (* n (log n))) 1.0))

C

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

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    code = (((n + 1.0d0) * log((n + 1.0d0))) - (n * log(n))) - 1.0d0
end function

Java

public static double code(double n) {
	return (((n + 1.0) * Math.log((n + 1.0))) - (n * Math.log(n))) - 1.0;
}

Python

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

Julia

function code(n)
	return Float64(Float64(Float64(Float64(n + 1.0) * log(Float64(n + 1.0))) - Float64(n * log(n))) - 1.0)
end

MATLAB

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

Wolfram

code[n_] := N[(N[(N[(N[(n + 1.0), $MachinePrecision] * N[Log[N[(n + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(n * N[Log[n], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]

Initial Program: 1.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1 \end{array} \]

FPCore

(FPCore (n)
 :precision binary64
 (- (- (* (+ n 1.0) (log (+ n 1.0))) (* n (log n))) 1.0))

C

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

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    code = (((n + 1.0d0) * log((n + 1.0d0))) - (n * log(n))) - 1.0d0
end function

Java

public static double code(double n) {
	return (((n + 1.0) * Math.log((n + 1.0))) - (n * Math.log(n))) - 1.0;
}

Python

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

Julia

function code(n)
	return Float64(Float64(Float64(Float64(n + 1.0) * log(Float64(n + 1.0))) - Float64(n * log(n))) - 1.0)
end

MATLAB

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

Wolfram

code[n_] := N[(N[(N[(N[(n + 1.0), $MachinePrecision] * N[Log[N[(n + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(n * N[Log[n], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]

Alternative 1: 100.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \log n \end{array} \]

FPCore

(FPCore (n) :precision binary64 (log n))

C

double code(double n) {
	return log(n);
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    code = log(n)
end function

Java

public static double code(double n) {
	return Math.log(n);
}

Python

def code(n):
	return math.log(n)

Julia

function code(n)
	return log(n)
end

MATLAB

function tmp = code(n)
	tmp = log(n);
end

Wolfram

code[n_] := N[Log[n], $MachinePrecision]

Derivation

1. Initial program 1.5%

   \[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1 \]
2. Step-by-step derivation

   1. sub-neg 1.5%

      \[\leadsto \color{blue}{\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) + \left(-1\right)} \]

   2. +-commutative 1.5%

      \[\leadsto \color{blue}{\left(-1\right) + \left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right)} \]

   3. associate-+r- 3.1%

      \[\leadsto \color{blue}{\left(\left(-1\right) + \left(n + 1\right) \cdot \log \left(n + 1\right)\right) - n \cdot \log n} \]

   4. +-commutative 3.1%

      \[\leadsto \color{blue}{\left(\left(n + 1\right) \cdot \log \left(n + 1\right) + \left(-1\right)\right)} - n \cdot \log n \]

   5. fma-def 3.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(n + 1, \log \left(n + 1\right), -1\right)} - n \cdot \log n \]

   6. +-commutative 3.1%

      \[\leadsto \mathsf{fma}\left(n + 1, \log \color{blue}{\left(1 + n\right)}, -1\right) - n \cdot \log n \]

   7. log1p-def 3.1%

      \[\leadsto \mathsf{fma}\left(n + 1, \color{blue}{\mathsf{log1p}\left(n\right)}, -1\right) - n \cdot \log n \]

   8. metadata-eval 3.1%

      \[\leadsto \mathsf{fma}\left(n + 1, \mathsf{log1p}\left(n\right), \color{blue}{-1}\right) - n \cdot \log n \]

3. Simplified 3.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(n + 1, \mathsf{log1p}\left(n\right), -1\right) - n \cdot \log n} \]

4. Taylor expanded in n around inf 100.0%

   \[\leadsto \color{blue}{-1 \cdot \log \left(\frac{1}{n}\right)} \]
5. Step-by-step derivation

   1. log-rec 100.0%

      \[\leadsto -1 \cdot \color{blue}{\left(-\log n\right)} \]

   2. mul-1-neg 100.0%

      \[\leadsto \color{blue}{-\left(-\log n\right)} \]

   3. remove-double-neg 100.0%

      \[\leadsto \color{blue}{\log n} \]

6. Simplified 100.0%

   \[\leadsto \color{blue}{\log n} \]

7. Final simplification 100.0%

   \[\leadsto \log n \]

Alternative 2: 3.1% accurate, 213.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (n) :precision binary64 0.0)

C

double code(double n) {
	return 0.0;
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    code = 0.0d0
end function

Java

public static double code(double n) {
	return 0.0;
}

Python

def code(n):
	return 0.0

Julia

function code(n)
	return 0.0
end

MATLAB

function tmp = code(n)
	tmp = 0.0;
end

Wolfram

code[n_] := 0.0

Derivation

1. Initial program 1.5%

   \[\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) - 1 \]
2. Step-by-step derivation

   1. sub-neg 1.5%

      \[\leadsto \color{blue}{\left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right) + \left(-1\right)} \]

   2. +-commutative 1.5%

      \[\leadsto \color{blue}{\left(-1\right) + \left(\left(n + 1\right) \cdot \log \left(n + 1\right) - n \cdot \log n\right)} \]

   3. associate-+r- 3.1%

      \[\leadsto \color{blue}{\left(\left(-1\right) + \left(n + 1\right) \cdot \log \left(n + 1\right)\right) - n \cdot \log n} \]

   4. +-commutative 3.1%

      \[\leadsto \color{blue}{\left(\left(n + 1\right) \cdot \log \left(n + 1\right) + \left(-1\right)\right)} - n \cdot \log n \]

   5. fma-def 3.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(n + 1, \log \left(n + 1\right), -1\right)} - n \cdot \log n \]

   6. +-commutative 3.1%

      \[\leadsto \mathsf{fma}\left(n + 1, \log \color{blue}{\left(1 + n\right)}, -1\right) - n \cdot \log n \]

   7. log1p-def 3.1%

      \[\leadsto \mathsf{fma}\left(n + 1, \color{blue}{\mathsf{log1p}\left(n\right)}, -1\right) - n \cdot \log n \]

   8. metadata-eval 3.1%

      \[\leadsto \mathsf{fma}\left(n + 1, \mathsf{log1p}\left(n\right), \color{blue}{-1}\right) - n \cdot \log n \]

3. Simplified 3.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(n + 1, \mathsf{log1p}\left(n\right), -1\right) - n \cdot \log n} \]

4. Taylor expanded in n around -inf 0.0%

   \[\leadsto \color{blue}{-1 \cdot \left(n \cdot \left(-1 \cdot \left(\log -1 + -1 \cdot \log \left(\frac{-1}{n}\right)\right) - -1 \cdot \left(-1 \cdot \log \left(\frac{-1}{n}\right) + \log -1\right)\right)\right)} \]
5. Step-by-step derivation

   1. *-commutative 0.0%

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

   2. +-commutative 0.0%

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

   3. +-inverses 3.1%

      \[\leadsto -1 \cdot \left(\color{blue}{0} \cdot n\right) \]

   4. mul0-lft 3.1%

      \[\leadsto -1 \cdot \color{blue}{0} \]

   5. metadata-eval 3.1%

      \[\leadsto \color{blue}{0} \]

6. Simplified 3.1%

   \[\leadsto \color{blue}{0} \]

7. Final simplification 3.1%

   \[\leadsto 0 \]

Developer target: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \log \left(n + 1\right) - \left(\frac{1}{2 \cdot n} - \left(\frac{1}{3 \cdot \left(n \cdot n\right)} - \frac{4}{{n}^{3}}\right)\right) \end{array} \]

FPCore

(FPCore (n)
 :precision binary64
 (-
  (log (+ n 1.0))
  (- (/ 1.0 (* 2.0 n)) (- (/ 1.0 (* 3.0 (* n n))) (/ 4.0 (pow n 3.0))))))

C

double code(double n) {
	return log((n + 1.0)) - ((1.0 / (2.0 * n)) - ((1.0 / (3.0 * (n * n))) - (4.0 / pow(n, 3.0))));
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    code = log((n + 1.0d0)) - ((1.0d0 / (2.0d0 * n)) - ((1.0d0 / (3.0d0 * (n * n))) - (4.0d0 / (n ** 3.0d0))))
end function

Java

public static double code(double n) {
	return Math.log((n + 1.0)) - ((1.0 / (2.0 * n)) - ((1.0 / (3.0 * (n * n))) - (4.0 / Math.pow(n, 3.0))));
}

Python

def code(n):
	return math.log((n + 1.0)) - ((1.0 / (2.0 * n)) - ((1.0 / (3.0 * (n * n))) - (4.0 / math.pow(n, 3.0))))

Julia

function code(n)
	return Float64(log(Float64(n + 1.0)) - Float64(Float64(1.0 / Float64(2.0 * n)) - Float64(Float64(1.0 / Float64(3.0 * Float64(n * n))) - Float64(4.0 / (n ^ 3.0)))))
end

MATLAB

function tmp = code(n)
	tmp = log((n + 1.0)) - ((1.0 / (2.0 * n)) - ((1.0 / (3.0 * (n * n))) - (4.0 / (n ^ 3.0))));
end

Wolfram

code[n_] := N[(N[Log[N[(n + 1.0), $MachinePrecision]], $MachinePrecision] - N[(N[(1.0 / N[(2.0 * n), $MachinePrecision]), $MachinePrecision] - N[(N[(1.0 / N[(3.0 * N[(n * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(4.0 / N[Power[n, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2023187 
(FPCore (n)
  :name "logs (example 3.8)"
  :precision binary64
  :pre (> n 6.8e+15)

  :herbie-target
  (- (log (+ n 1.0)) (- (/ 1.0 (* 2.0 n)) (- (/ 1.0 (* 3.0 (* n n))) (/ 4.0 (pow n 3.0)))))

  (- (- (* (+ n 1.0) (log (+ n 1.0))) (* n (log n))) 1.0))