logs (example 3.8)

Percentage Accurate: 1.6% → 100.0%

Time: 8.9s

Alternatives: 4

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: 1.6% 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 1.6%

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. sub-neg N/A

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

   4. +-commutative N/A

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

   5. associate-+r+ N/A

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

   6. unsub-neg N/A

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

   7. associate-+l- N/A

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

   8. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(\mathsf{neg}\left(1\right)\right), \color{blue}{\left(n \cdot \log n - \left(n + 1\right) \cdot \log \left(n + 1\right)\right)}\right) \]

   9. metadata-eval N/A

      \[\leadsto \mathsf{\_.f64}\left(-1, \left(\color{blue}{n \cdot \log n} - \left(n + 1\right) \cdot \log \left(n + 1\right)\right)\right) \]

   10. sub-neg N/A

       \[\leadsto \mathsf{\_.f64}\left(-1, \left(n \cdot \log n + \color{blue}{\left(\mathsf{neg}\left(\left(n + 1\right) \cdot \log \left(n + 1\right)\right)\right)}\right)\right) \]

   11. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(-1, \mathsf{+.f64}\left(\left(n \cdot \log n\right), \color{blue}{\left(\mathsf{neg}\left(\left(n + 1\right) \cdot \log \left(n + 1\right)\right)\right)}\right)\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(-1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, \log n\right), \left(\mathsf{neg}\left(\color{blue}{\left(n + 1\right) \cdot \log \left(n + 1\right)}\right)\right)\right)\right) \]

   13. log-lowering-log.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(-1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, \mathsf{log.f64}\left(n\right)\right), \left(\mathsf{neg}\left(\left(n + 1\right) \cdot \color{blue}{\log \left(n + 1\right)}\right)\right)\right)\right) \]

   14. *-commutative N/A

       \[\leadsto \mathsf{\_.f64}\left(-1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, \mathsf{log.f64}\left(n\right)\right), \left(\mathsf{neg}\left(\log \left(n + 1\right) \cdot \left(n + 1\right)\right)\right)\right)\right) \]

   15. distribute-rgt-neg-in N/A

       \[\leadsto \mathsf{\_.f64}\left(-1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, \mathsf{log.f64}\left(n\right)\right), \left(\log \left(n + 1\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\left(n + 1\right)\right)\right)}\right)\right)\right) \]

   16. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(-1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, \mathsf{log.f64}\left(n\right)\right), \mathsf{*.f64}\left(\log \left(n + 1\right), \color{blue}{\left(\mathsf{neg}\left(\left(n + 1\right)\right)\right)}\right)\right)\right) \]

3. Simplified 1.6%

   \[\leadsto \color{blue}{-1 - \left(n \cdot \log n + \mathsf{log1p}\left(n\right) \cdot \left(-1 - n\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in n around inf

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

7. Simplified 99.9%

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

8. Taylor expanded in n around 0

   \[\leadsto \color{blue}{\log n} \]
9. Step-by-step derivation

   1. log-lowering-log.f64 100.0%

      \[\leadsto \mathsf{log.f64}\left(n\right) \]

10. Simplified 100.0%

    \[\leadsto \color{blue}{\log n} \]
11. Add Preprocessing

Alternative 2: 5.0% accurate, 19.4× speedup

Math

\[\begin{array}{l} \\ \frac{0.25}{n} \cdot \left(\frac{1}{n} \cdot \frac{n}{0.5}\right) \end{array} \]

FPCore

(FPCore (n) :precision binary64 (* (/ 0.25 n) (* (/ 1.0 n) (/ n 0.5))))

C

double code(double n) {
	return (0.25 / n) * ((1.0 / n) * (n / 0.5));
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    code = (0.25d0 / n) * ((1.0d0 / n) * (n / 0.5d0))
end function

Java

public static double code(double n) {
	return (0.25 / n) * ((1.0 / n) * (n / 0.5));
}

Python

def code(n):
	return (0.25 / n) * ((1.0 / n) * (n / 0.5))

Julia

function code(n)
	return Float64(Float64(0.25 / n) * Float64(Float64(1.0 / n) * Float64(n / 0.5)))
end

MATLAB

function tmp = code(n)
	tmp = (0.25 / n) * ((1.0 / n) * (n / 0.5));
end

Wolfram

code[n_] := N[(N[(0.25 / n), $MachinePrecision] * N[(N[(1.0 / n), $MachinePrecision] * N[(n / 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 1.6%

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. sub-neg N/A

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

   4. +-commutative N/A

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

   5. associate-+r+ N/A

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

   6. unsub-neg N/A

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

   7. associate-+l- N/A

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

   8. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(\mathsf{neg}\left(1\right)\right), \color{blue}{\left(n \cdot \log n - \left(n + 1\right) \cdot \log \left(n + 1\right)\right)}\right) \]

   9. metadata-eval N/A

      \[\leadsto \mathsf{\_.f64}\left(-1, \left(\color{blue}{n \cdot \log n} - \left(n + 1\right) \cdot \log \left(n + 1\right)\right)\right) \]

   10. sub-neg N/A

       \[\leadsto \mathsf{\_.f64}\left(-1, \left(n \cdot \log n + \color{blue}{\left(\mathsf{neg}\left(\left(n + 1\right) \cdot \log \left(n + 1\right)\right)\right)}\right)\right) \]

   11. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(-1, \mathsf{+.f64}\left(\left(n \cdot \log n\right), \color{blue}{\left(\mathsf{neg}\left(\left(n + 1\right) \cdot \log \left(n + 1\right)\right)\right)}\right)\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(-1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, \log n\right), \left(\mathsf{neg}\left(\color{blue}{\left(n + 1\right) \cdot \log \left(n + 1\right)}\right)\right)\right)\right) \]

   13. log-lowering-log.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(-1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, \mathsf{log.f64}\left(n\right)\right), \left(\mathsf{neg}\left(\left(n + 1\right) \cdot \color{blue}{\log \left(n + 1\right)}\right)\right)\right)\right) \]

   14. *-commutative N/A

       \[\leadsto \mathsf{\_.f64}\left(-1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, \mathsf{log.f64}\left(n\right)\right), \left(\mathsf{neg}\left(\log \left(n + 1\right) \cdot \left(n + 1\right)\right)\right)\right)\right) \]

   15. distribute-rgt-neg-in N/A

       \[\leadsto \mathsf{\_.f64}\left(-1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, \mathsf{log.f64}\left(n\right)\right), \left(\log \left(n + 1\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\left(n + 1\right)\right)\right)}\right)\right)\right) \]

   16. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(-1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, \mathsf{log.f64}\left(n\right)\right), \mathsf{*.f64}\left(\log \left(n + 1\right), \color{blue}{\left(\mathsf{neg}\left(\left(n + 1\right)\right)\right)}\right)\right)\right) \]

3. Simplified 1.6%

   \[\leadsto \color{blue}{-1 - \left(n \cdot \log n + \mathsf{log1p}\left(n\right) \cdot \left(-1 - n\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in n around inf

   \[\leadsto \mathsf{\_.f64}\left(-1, \color{blue}{\left(n \cdot \left(\left(\log \left(\frac{1}{n}\right) + \left(-1 \cdot \log \left(\frac{1}{n}\right) + -1 \cdot \frac{1 + -1 \cdot \log \left(\frac{1}{n}\right)}{n}\right)\right) - \frac{\frac{1}{2}}{{n}^{2}}\right)\right)}\right) \]
6. Step-by-step derivation

   1. associate-+r+ N/A

      \[\leadsto \mathsf{\_.f64}\left(-1, \left(n \cdot \left(\left(\left(\log \left(\frac{1}{n}\right) + -1 \cdot \log \left(\frac{1}{n}\right)\right) + -1 \cdot \frac{1 + -1 \cdot \log \left(\frac{1}{n}\right)}{n}\right) - \frac{\color{blue}{\frac{1}{2}}}{{n}^{2}}\right)\right)\right) \]

   2. distribute-rgt1-in N/A

      \[\leadsto \mathsf{\_.f64}\left(-1, \left(n \cdot \left(\left(\left(-1 + 1\right) \cdot \log \left(\frac{1}{n}\right) + -1 \cdot \frac{1 + -1 \cdot \log \left(\frac{1}{n}\right)}{n}\right) - \frac{\frac{1}{2}}{{n}^{2}}\right)\right)\right) \]

   3. metadata-eval N/A

      \[\leadsto \mathsf{\_.f64}\left(-1, \left(n \cdot \left(\left(0 \cdot \log \left(\frac{1}{n}\right) + -1 \cdot \frac{1 + -1 \cdot \log \left(\frac{1}{n}\right)}{n}\right) - \frac{\frac{1}{2}}{{n}^{2}}\right)\right)\right) \]

   4. mul0-lft N/A

      \[\leadsto \mathsf{\_.f64}\left(-1, \left(n \cdot \left(\left(0 + -1 \cdot \frac{1 + -1 \cdot \log \left(\frac{1}{n}\right)}{n}\right) - \frac{\frac{1}{2}}{{n}^{2}}\right)\right)\right) \]

   5. +-lft-identity N/A

      \[\leadsto \mathsf{\_.f64}\left(-1, \left(n \cdot \left(-1 \cdot \frac{1 + -1 \cdot \log \left(\frac{1}{n}\right)}{n} - \frac{\color{blue}{\frac{1}{2}}}{{n}^{2}}\right)\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(-1, \mathsf{*.f64}\left(n, \color{blue}{\left(-1 \cdot \frac{1 + -1 \cdot \log \left(\frac{1}{n}\right)}{n} - \frac{\frac{1}{2}}{{n}^{2}}\right)}\right)\right) \]

   7. sub-neg N/A

      \[\leadsto \mathsf{\_.f64}\left(-1, \mathsf{*.f64}\left(n, \left(-1 \cdot \frac{1 + -1 \cdot \log \left(\frac{1}{n}\right)}{n} + \color{blue}{\left(\mathsf{neg}\left(\frac{\frac{1}{2}}{{n}^{2}}\right)\right)}\right)\right)\right) \]

   8. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(-1, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(\left(-1 \cdot \frac{1 + -1 \cdot \log \left(\frac{1}{n}\right)}{n}\right), \color{blue}{\left(\mathsf{neg}\left(\frac{\frac{1}{2}}{{n}^{2}}\right)\right)}\right)\right)\right) \]

7. Simplified 99.9%

   \[\leadsto -1 - \color{blue}{n \cdot \left(\frac{-1 - \log n}{n} + \frac{-0.5}{n \cdot n}\right)} \]

8. Taylor expanded in n around 0

   \[\leadsto \color{blue}{\frac{\frac{1}{2}}{n}} \]
9. Step-by-step derivation

   1. /-lowering-/.f64 4.9%

      \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{n}\right) \]

10. Simplified 4.9%

    \[\leadsto \color{blue}{\frac{0.5}{n}} \]
11. Step-by-step derivation

    1. frac-2neg N/A

       \[\leadsto \frac{\mathsf{neg}\left(\frac{1}{2}\right)}{\color{blue}{\mathsf{neg}\left(n\right)}} \]

    2. neg-sub0 N/A

       \[\leadsto \frac{\mathsf{neg}\left(\frac{1}{2}\right)}{0 - \color{blue}{n}} \]

    3. flip-- N/A

       \[\leadsto \frac{\mathsf{neg}\left(\frac{1}{2}\right)}{\frac{0 \cdot 0 - n \cdot n}{\color{blue}{0 + n}}} \]

    4. +-lft-identity N/A

       \[\leadsto \frac{\mathsf{neg}\left(\frac{1}{2}\right)}{\frac{0 \cdot 0 - n \cdot n}{n}} \]

    5. associate-/r/ N/A

       \[\leadsto \frac{\mathsf{neg}\left(\frac{1}{2}\right)}{0 \cdot 0 - n \cdot n} \cdot \color{blue}{n} \]

    6. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\left(\frac{\mathsf{neg}\left(\frac{1}{2}\right)}{0 \cdot 0 - n \cdot n}\right), \color{blue}{n}\right) \]

    7. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right), \left(0 \cdot 0 - n \cdot n\right)\right), n\right) \]

    8. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, \left(0 \cdot 0 - n \cdot n\right)\right), n\right) \]

    9. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, \left(0 - n \cdot n\right)\right), n\right) \]

    10. +-lft-identity N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, \left(0 - n \cdot \left(0 + n\right)\right)\right), n\right) \]

    11. +-commutative N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, \left(0 - n \cdot \left(n + 0\right)\right)\right), n\right) \]

    12. distribute-rgt-out N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, \left(0 - \left(n \cdot n + 0 \cdot n\right)\right)\right), n\right) \]

    13. +-lft-identity N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, \left(0 - \left(0 + \left(n \cdot n + 0 \cdot n\right)\right)\right)\right), n\right) \]

    14. metadata-eval N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, \left(0 - \left(0 \cdot 0 + \left(n \cdot n + 0 \cdot n\right)\right)\right)\right), n\right) \]

    15. --lowering--.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{\_.f64}\left(0, \left(0 \cdot 0 + \left(n \cdot n + 0 \cdot n\right)\right)\right)\right), n\right) \]

    16. metadata-eval N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{\_.f64}\left(0, \left(0 + \left(n \cdot n + 0 \cdot n\right)\right)\right)\right), n\right) \]

    17. +-lft-identity N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{\_.f64}\left(0, \left(n \cdot n + 0 \cdot n\right)\right)\right), n\right) \]

    18. distribute-rgt-out N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{\_.f64}\left(0, \left(n \cdot \left(n + 0\right)\right)\right)\right), n\right) \]

    19. +-commutative N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{\_.f64}\left(0, \left(n \cdot \left(0 + n\right)\right)\right)\right), n\right) \]

    20. +-lft-identity N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{\_.f64}\left(0, \left(n \cdot n\right)\right)\right), n\right) \]

    21. *-lowering-*.f64 4.5%

        \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(n, n\right)\right)\right), n\right) \]

12. Applied egg-rr 4.5%

    \[\leadsto \color{blue}{\frac{-0.5}{0 - n \cdot n} \cdot n} \]

14. Applied egg-rr 4.9%

    \[\leadsto \color{blue}{\frac{0.25}{n} \cdot \left(\frac{1}{n} \cdot \frac{n}{0.5}\right)} \]
15. Add Preprocessing

Alternative 3: 5.0% accurate, 71.0× speedup

Math

\[\begin{array}{l} \\ \frac{0.5}{n} \end{array} \]

FPCore

(FPCore (n) :precision binary64 (/ 0.5 n))

C

double code(double n) {
	return 0.5 / n;
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    code = 0.5d0 / n
end function

Java

public static double code(double n) {
	return 0.5 / n;
}

Python

def code(n):
	return 0.5 / n

Julia

function code(n)
	return Float64(0.5 / n)
end

MATLAB

function tmp = code(n)
	tmp = 0.5 / n;
end

Wolfram

code[n_] := N[(0.5 / n), $MachinePrecision]

Derivation

1. Initial program 1.6%

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. sub-neg N/A

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

   4. +-commutative N/A

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

   5. associate-+r+ N/A

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

   6. unsub-neg N/A

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

   7. associate-+l- N/A

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

   8. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(\mathsf{neg}\left(1\right)\right), \color{blue}{\left(n \cdot \log n - \left(n + 1\right) \cdot \log \left(n + 1\right)\right)}\right) \]

   9. metadata-eval N/A

      \[\leadsto \mathsf{\_.f64}\left(-1, \left(\color{blue}{n \cdot \log n} - \left(n + 1\right) \cdot \log \left(n + 1\right)\right)\right) \]

   10. sub-neg N/A

       \[\leadsto \mathsf{\_.f64}\left(-1, \left(n \cdot \log n + \color{blue}{\left(\mathsf{neg}\left(\left(n + 1\right) \cdot \log \left(n + 1\right)\right)\right)}\right)\right) \]

   11. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(-1, \mathsf{+.f64}\left(\left(n \cdot \log n\right), \color{blue}{\left(\mathsf{neg}\left(\left(n + 1\right) \cdot \log \left(n + 1\right)\right)\right)}\right)\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(-1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, \log n\right), \left(\mathsf{neg}\left(\color{blue}{\left(n + 1\right) \cdot \log \left(n + 1\right)}\right)\right)\right)\right) \]

   13. log-lowering-log.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(-1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, \mathsf{log.f64}\left(n\right)\right), \left(\mathsf{neg}\left(\left(n + 1\right) \cdot \color{blue}{\log \left(n + 1\right)}\right)\right)\right)\right) \]

   14. *-commutative N/A

       \[\leadsto \mathsf{\_.f64}\left(-1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, \mathsf{log.f64}\left(n\right)\right), \left(\mathsf{neg}\left(\log \left(n + 1\right) \cdot \left(n + 1\right)\right)\right)\right)\right) \]

   15. distribute-rgt-neg-in N/A

       \[\leadsto \mathsf{\_.f64}\left(-1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, \mathsf{log.f64}\left(n\right)\right), \left(\log \left(n + 1\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\left(n + 1\right)\right)\right)}\right)\right)\right) \]

   16. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(-1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, \mathsf{log.f64}\left(n\right)\right), \mathsf{*.f64}\left(\log \left(n + 1\right), \color{blue}{\left(\mathsf{neg}\left(\left(n + 1\right)\right)\right)}\right)\right)\right) \]

3. Simplified 1.6%

   \[\leadsto \color{blue}{-1 - \left(n \cdot \log n + \mathsf{log1p}\left(n\right) \cdot \left(-1 - n\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in n around inf

   \[\leadsto \mathsf{\_.f64}\left(-1, \color{blue}{\left(n \cdot \left(\left(\log \left(\frac{1}{n}\right) + \left(-1 \cdot \log \left(\frac{1}{n}\right) + -1 \cdot \frac{1 + -1 \cdot \log \left(\frac{1}{n}\right)}{n}\right)\right) - \frac{\frac{1}{2}}{{n}^{2}}\right)\right)}\right) \]
6. Step-by-step derivation

   1. associate-+r+ N/A

      \[\leadsto \mathsf{\_.f64}\left(-1, \left(n \cdot \left(\left(\left(\log \left(\frac{1}{n}\right) + -1 \cdot \log \left(\frac{1}{n}\right)\right) + -1 \cdot \frac{1 + -1 \cdot \log \left(\frac{1}{n}\right)}{n}\right) - \frac{\color{blue}{\frac{1}{2}}}{{n}^{2}}\right)\right)\right) \]

   2. distribute-rgt1-in N/A

      \[\leadsto \mathsf{\_.f64}\left(-1, \left(n \cdot \left(\left(\left(-1 + 1\right) \cdot \log \left(\frac{1}{n}\right) + -1 \cdot \frac{1 + -1 \cdot \log \left(\frac{1}{n}\right)}{n}\right) - \frac{\frac{1}{2}}{{n}^{2}}\right)\right)\right) \]

   3. metadata-eval N/A

      \[\leadsto \mathsf{\_.f64}\left(-1, \left(n \cdot \left(\left(0 \cdot \log \left(\frac{1}{n}\right) + -1 \cdot \frac{1 + -1 \cdot \log \left(\frac{1}{n}\right)}{n}\right) - \frac{\frac{1}{2}}{{n}^{2}}\right)\right)\right) \]

   4. mul0-lft N/A

      \[\leadsto \mathsf{\_.f64}\left(-1, \left(n \cdot \left(\left(0 + -1 \cdot \frac{1 + -1 \cdot \log \left(\frac{1}{n}\right)}{n}\right) - \frac{\frac{1}{2}}{{n}^{2}}\right)\right)\right) \]

   5. +-lft-identity N/A

      \[\leadsto \mathsf{\_.f64}\left(-1, \left(n \cdot \left(-1 \cdot \frac{1 + -1 \cdot \log \left(\frac{1}{n}\right)}{n} - \frac{\color{blue}{\frac{1}{2}}}{{n}^{2}}\right)\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(-1, \mathsf{*.f64}\left(n, \color{blue}{\left(-1 \cdot \frac{1 + -1 \cdot \log \left(\frac{1}{n}\right)}{n} - \frac{\frac{1}{2}}{{n}^{2}}\right)}\right)\right) \]

   7. sub-neg N/A

      \[\leadsto \mathsf{\_.f64}\left(-1, \mathsf{*.f64}\left(n, \left(-1 \cdot \frac{1 + -1 \cdot \log \left(\frac{1}{n}\right)}{n} + \color{blue}{\left(\mathsf{neg}\left(\frac{\frac{1}{2}}{{n}^{2}}\right)\right)}\right)\right)\right) \]

   8. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(-1, \mathsf{*.f64}\left(n, \mathsf{+.f64}\left(\left(-1 \cdot \frac{1 + -1 \cdot \log \left(\frac{1}{n}\right)}{n}\right), \color{blue}{\left(\mathsf{neg}\left(\frac{\frac{1}{2}}{{n}^{2}}\right)\right)}\right)\right)\right) \]

7. Simplified 99.9%

   \[\leadsto -1 - \color{blue}{n \cdot \left(\frac{-1 - \log n}{n} + \frac{-0.5}{n \cdot n}\right)} \]

8. Taylor expanded in n around 0

   \[\leadsto \color{blue}{\frac{\frac{1}{2}}{n}} \]
9. Step-by-step derivation

   1. /-lowering-/.f64 4.9%

      \[\leadsto \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{n}\right) \]

10. Simplified 4.9%

    \[\leadsto \color{blue}{\frac{0.5}{n}} \]
11. Add Preprocessing

Alternative 4: 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 1.6%

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. sub-neg N/A

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

   4. +-commutative N/A

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

   5. associate-+r+ N/A

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

   6. unsub-neg N/A

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

   7. associate-+l- N/A

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

   8. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(\mathsf{neg}\left(1\right)\right), \color{blue}{\left(n \cdot \log n - \left(n + 1\right) \cdot \log \left(n + 1\right)\right)}\right) \]

   9. metadata-eval N/A

      \[\leadsto \mathsf{\_.f64}\left(-1, \left(\color{blue}{n \cdot \log n} - \left(n + 1\right) \cdot \log \left(n + 1\right)\right)\right) \]

   10. sub-neg N/A

       \[\leadsto \mathsf{\_.f64}\left(-1, \left(n \cdot \log n + \color{blue}{\left(\mathsf{neg}\left(\left(n + 1\right) \cdot \log \left(n + 1\right)\right)\right)}\right)\right) \]

   11. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(-1, \mathsf{+.f64}\left(\left(n \cdot \log n\right), \color{blue}{\left(\mathsf{neg}\left(\left(n + 1\right) \cdot \log \left(n + 1\right)\right)\right)}\right)\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(-1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, \log n\right), \left(\mathsf{neg}\left(\color{blue}{\left(n + 1\right) \cdot \log \left(n + 1\right)}\right)\right)\right)\right) \]

   13. log-lowering-log.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(-1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, \mathsf{log.f64}\left(n\right)\right), \left(\mathsf{neg}\left(\left(n + 1\right) \cdot \color{blue}{\log \left(n + 1\right)}\right)\right)\right)\right) \]

   14. *-commutative N/A

       \[\leadsto \mathsf{\_.f64}\left(-1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, \mathsf{log.f64}\left(n\right)\right), \left(\mathsf{neg}\left(\log \left(n + 1\right) \cdot \left(n + 1\right)\right)\right)\right)\right) \]

   15. distribute-rgt-neg-in N/A

       \[\leadsto \mathsf{\_.f64}\left(-1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, \mathsf{log.f64}\left(n\right)\right), \left(\log \left(n + 1\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\left(n + 1\right)\right)\right)}\right)\right)\right) \]

   16. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(-1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(n, \mathsf{log.f64}\left(n\right)\right), \mathsf{*.f64}\left(\log \left(n + 1\right), \color{blue}{\left(\mathsf{neg}\left(\left(n + 1\right)\right)\right)}\right)\right)\right) \]

3. Simplified 1.6%

   \[\leadsto \color{blue}{-1 - \left(n \cdot \log n + \mathsf{log1p}\left(n\right) \cdot \left(-1 - n\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in n around inf

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

   1. associate-*r* N/A

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

   2. distribute-rgt1-in N/A

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

   3. metadata-eval N/A

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

   4. mul0-lft N/A

      \[\leadsto \left(-1 \cdot n\right) \cdot 0 \]

   5. mul0-rgt 3.1%

      \[\leadsto 0 \]

7. Simplified 3.1%

   \[\leadsto \color{blue}{0} \]
8. Add Preprocessing

Developer Target 1: 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 2024161 
(FPCore (n)
  :name "logs (example 3.8)"
  :precision binary64
  :pre (> n 6.8e+15)

  :alt
  (! :herbie-platform default (- (log (+ n 1)) (- (/ 1 (* 2 n)) (- (/ 1 (* 3 (* n n))) (/ 4 (pow n 3))))))

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