logs (example 3.8)

Percentage Accurate: 1.5% → 98.9%

Time: 14.4s

Alternatives: 5

Speedup: 1.0×

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: 98.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \mathsf{log1p}\left(n\right) + \left(n \cdot \mathsf{log1p}\left(n\right) - \mathsf{fma}\left(n, \log n, 1\right)\right) \end{array} \]

FPCore

(FPCore (n)
 :precision binary64
 (+ (log1p n) (- (* n (log1p n)) (fma n (log n) 1.0))))

C

double code(double n) {
	return log1p(n) + ((n * log1p(n)) - fma(n, log(n), 1.0));
}

Julia

function code(n)
	return Float64(log1p(n) + Float64(Float64(n * log1p(n)) - fma(n, log(n), 1.0)))
end

Wolfram

code[n_] := N[(N[Log[1 + n], $MachinePrecision] + N[(N[(n * N[Log[1 + n], $MachinePrecision]), $MachinePrecision] - N[(n * N[Log[n], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $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. Add Preprocessing
3. Step-by-step derivation

   1. associate--l- N/A

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

   2. *-commutative N/A

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

   3. +-commutative N/A

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

   4. distribute-lft-in N/A

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

   5. *-rgt-identity N/A

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

   6. associate--l+ N/A

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

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

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

   8. +-commutative N/A

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

   9. accelerator-lowering-log1p.f64 N/A

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

   10. --lowering--.f64 N/A

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

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

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

   12. +-commutative N/A

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

   13. accelerator-lowering-log1p.f64 N/A

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

   14. accelerator-lowering-fma.f64 N/A

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

   15. log-lowering-log.f64 98.4

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

4. Applied egg-rr 98.4%

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

5. Final simplification 98.4%

   \[\leadsto \mathsf{log1p}\left(n\right) + \left(n \cdot \mathsf{log1p}\left(n\right) - \mathsf{fma}\left(n, \log n, 1\right)\right) \]
6. Add Preprocessing

Alternative 2: 30.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{1}{-1 + \frac{1}{\frac{1}{\mathsf{fma}\left(n, \log \left(\frac{n + 1}{n}\right), \mathsf{log1p}\left(n\right)\right)}}}} \end{array} \]

FPCore

(FPCore (n)
 :precision binary64
 (/
  1.0
  (/ 1.0 (+ -1.0 (/ 1.0 (/ 1.0 (fma n (log (/ (+ n 1.0) n)) (log1p n))))))))

C

double code(double n) {
	return 1.0 / (1.0 / (-1.0 + (1.0 / (1.0 / fma(n, log(((n + 1.0) / n)), log1p(n))))));
}

Julia

function code(n)
	return Float64(1.0 / Float64(1.0 / Float64(-1.0 + Float64(1.0 / Float64(1.0 / fma(n, log(Float64(Float64(n + 1.0) / n)), log1p(n)))))))
end

Wolfram

code[n_] := N[(1.0 / N[(1.0 / N[(-1.0 + N[(1.0 / N[(1.0 / N[(n * N[Log[N[(N[(n + 1.0), $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] + N[Log[1 + n], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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. Add Preprocessing
3. Step-by-step derivation

   1. associate--l- N/A

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

   2. *-commutative N/A

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

   3. +-commutative N/A

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

   4. distribute-lft-in N/A

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

   5. *-rgt-identity N/A

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

   6. associate--l+ N/A

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

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

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

   8. +-commutative N/A

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

   9. accelerator-lowering-log1p.f64 N/A

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

   10. --lowering--.f64 N/A

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

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

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

   12. +-commutative N/A

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

   13. accelerator-lowering-log1p.f64 N/A

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

   14. accelerator-lowering-fma.f64 N/A

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

   15. log-lowering-log.f64 98.4

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

4. Applied egg-rr 98.4%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(n\right) + \left(\mathsf{log1p}\left(n\right) \cdot n - \mathsf{fma}\left(n, \log n, 1\right)\right)} \]
5. Step-by-step derivation

   1. flip-+ N/A

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

   2. div-inv N/A

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

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

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

6. Applied egg-rr 31.0%

   \[\leadsto \color{blue}{\left({\left(\mathsf{log1p}\left(n\right)\right)}^{2} - {\left(\mathsf{fma}\left(n, \log \left(\frac{1 + n}{n}\right), -1\right)\right)}^{2}\right) \cdot \frac{1}{\mathsf{log1p}\left(n\right) - \mathsf{fma}\left(n, \log \left(\frac{1 + n}{n}\right), -1\right)}} \]
7. Step-by-step derivation

   1. *-commutative N/A

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

   2. associate-/r/ N/A

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

   3. /-lowering-/.f64 N/A

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

   4. clear-num N/A

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

   5. unpow2 N/A

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

   6. unpow2 N/A

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

8. Applied egg-rr 31.0%

   \[\leadsto \color{blue}{\frac{1}{\frac{1}{-1 + \mathsf{fma}\left(n, \log \left(\frac{1 + n}{n}\right), \mathsf{log1p}\left(n\right)\right)}}} \]
9. Step-by-step derivation

   1. flip-+ N/A

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

   2. clear-num N/A

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

   3. /-lowering-/.f64 N/A

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

   4. clear-num N/A

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

   5. flip-+ N/A

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

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

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

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

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

10. Applied egg-rr 31.0%

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

11. Final simplification 31.0%

    \[\leadsto \frac{1}{\frac{1}{-1 + \frac{1}{\frac{1}{\mathsf{fma}\left(n, \log \left(\frac{n + 1}{n}\right), \mathsf{log1p}\left(n\right)\right)}}}} \]
12. Add Preprocessing

Alternative 3: 30.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{log1p}\left(n\right) + \mathsf{fma}\left(n, \log \left(\frac{n + 1}{n}\right), -1\right) \end{array} \]

FPCore

(FPCore (n)
 :precision binary64
 (+ (log1p n) (fma n (log (/ (+ n 1.0) n)) -1.0)))

C

double code(double n) {
	return log1p(n) + fma(n, log(((n + 1.0) / n)), -1.0);
}

Julia

function code(n)
	return Float64(log1p(n) + fma(n, log(Float64(Float64(n + 1.0) / n)), -1.0))
end

Wolfram

code[n_] := N[(N[Log[1 + n], $MachinePrecision] + N[(n * N[Log[N[(N[(n + 1.0), $MachinePrecision] / n), $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision]), $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. Add Preprocessing
3. Step-by-step derivation

   1. associate--l- N/A

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

   2. *-commutative N/A

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

   3. +-commutative N/A

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

   4. distribute-lft-in N/A

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

   5. *-rgt-identity N/A

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

   6. associate--l+ N/A

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

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

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

   8. +-commutative N/A

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

   9. accelerator-lowering-log1p.f64 N/A

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

   10. --lowering--.f64 N/A

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

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

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

   12. +-commutative N/A

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

   13. accelerator-lowering-log1p.f64 N/A

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

   14. accelerator-lowering-fma.f64 N/A

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

   15. log-lowering-log.f64 98.4

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

4. Applied egg-rr 98.4%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(n\right) + \left(\mathsf{log1p}\left(n\right) \cdot n - \mathsf{fma}\left(n, \log n, 1\right)\right)} \]
5. Step-by-step derivation

   1. +-commutative N/A

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

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

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

   3. associate--r+ N/A

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

   4. sub-neg N/A

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

   5. *-commutative N/A

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

   6. distribute-lft-out-- N/A

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

   7. metadata-eval N/A

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

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

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

   9. +-commutative N/A

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

   10. metadata-eval N/A

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

   11. sub-neg N/A

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

   12. diff-log N/A

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

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

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

   14. /-lowering-/.f64 N/A

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

   15. sub-neg N/A

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

   16. metadata-eval N/A

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

   17. +-commutative N/A

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

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

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

6. Applied egg-rr 31.0%

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

7. Final simplification 31.0%

   \[\leadsto \mathsf{log1p}\left(n\right) + \mathsf{fma}\left(n, \log \left(\frac{n + 1}{n}\right), -1\right) \]
8. Add Preprocessing

Alternative 4: 3.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ -1 + \mathsf{fma}\left(-\log n, n, \mathsf{log1p}\left(n\right) \cdot \left(n + 1\right)\right) \end{array} \]

FPCore

(FPCore (n)
 :precision binary64
 (+ -1.0 (fma (- (log n)) n (* (log1p n) (+ n 1.0)))))

C

double code(double n) {
	return -1.0 + fma(-log(n), n, (log1p(n) * (n + 1.0)));
}

Julia

function code(n)
	return Float64(-1.0 + fma(Float64(-log(n)), n, Float64(log1p(n) * Float64(n + 1.0))))
end

Wolfram

code[n_] := N[(-1.0 + N[((-N[Log[n], $MachinePrecision]) * n + N[(N[Log[1 + n], $MachinePrecision] * N[(n + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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. Add Preprocessing
3. Step-by-step derivation

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. distribute-lft-neg-in N/A

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

   5. accelerator-lowering-fma.f64 N/A

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

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

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

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

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

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

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

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

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

   10. +-commutative N/A

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

   11. accelerator-lowering-log1p.f64 3.2

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

4. Applied egg-rr 3.2%

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

5. Final simplification 3.2%

   \[\leadsto -1 + \mathsf{fma}\left(-\log n, n, \mathsf{log1p}\left(n\right) \cdot \left(n + 1\right)\right) \]
6. Add Preprocessing

Alternative 5: 3.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(-\log n, n, \mathsf{fma}\left(n + 1, \mathsf{log1p}\left(n\right), -1\right)\right) \end{array} \]

FPCore

(FPCore (n)
 :precision binary64
 (fma (- (log n)) n (fma (+ n 1.0) (log1p n) -1.0)))

C

double code(double n) {
	return fma(-log(n), n, fma((n + 1.0), log1p(n), -1.0));
}

Julia

function code(n)
	return fma(Float64(-log(n)), n, fma(Float64(n + 1.0), log1p(n), -1.0))
end

Wolfram

code[n_] := N[((-N[Log[n], $MachinePrecision]) * n + N[(N[(n + 1.0), $MachinePrecision] * N[Log[1 + n], $MachinePrecision] + -1.0), $MachinePrecision]), $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. Add Preprocessing
3. Step-by-step derivation

   1. sub-neg N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. associate-+l+ N/A

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

   5. *-commutative N/A

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

   6. distribute-lft-neg-in N/A

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

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

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

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

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

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

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

   10. accelerator-lowering-fma.f64 N/A

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

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

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

   12. +-commutative N/A

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

   13. accelerator-lowering-log1p.f64 N/A

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

   14. metadata-eval 3.2

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

4. Applied egg-rr 3.2%

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

Developer Target 1: 100.0% accurate, 0.9× 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 2024207 
(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))