neg log

Percentage Accurate: 100.0% → 100.0%

Time: 9.3s

Alternatives: 9

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ -\log \left(\frac{1}{x} - 1\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- (log (- (/ 1.0 x) 1.0))))

C

double code(double x) {
	return -log(((1.0 / x) - 1.0));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = -log(((1.0d0 / x) - 1.0d0))
end function

Java

public static double code(double x) {
	return -Math.log(((1.0 / x) - 1.0));
}

Python

def code(x):
	return -math.log(((1.0 / x) - 1.0))

Julia

function code(x)
	return Float64(-log(Float64(Float64(1.0 / x) - 1.0)))
end

MATLAB

function tmp = code(x)
	tmp = -log(((1.0 / x) - 1.0));
end

Wolfram

code[x_] := (-N[Log[N[(N[(1.0 / x), $MachinePrecision] - 1.0), $MachinePrecision]], $MachinePrecision])

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ -\log \left(\frac{1}{x} - 1\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- (log (- (/ 1.0 x) 1.0))))

C

double code(double x) {
	return -log(((1.0 / x) - 1.0));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = -log(((1.0d0 / x) - 1.0d0))
end function

Java

public static double code(double x) {
	return -Math.log(((1.0 / x) - 1.0));
}

Python

def code(x):
	return -math.log(((1.0 / x) - 1.0))

Julia

function code(x)
	return Float64(-log(Float64(Float64(1.0 / x) - 1.0)))
end

MATLAB

function tmp = code(x)
	tmp = -log(((1.0 / x) - 1.0));
end

Wolfram

code[x_] := (-N[Log[N[(N[(1.0 / x), $MachinePrecision] - 1.0), $MachinePrecision]], $MachinePrecision])

Alternative 1: 100.0% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (log (/ (* (+ 1.0 x) (fma x x 1.0)) (- (/ 1.0 x) (* x (* x x))))))

C

double code(double x) {
	return log((((1.0 + x) * fma(x, x, 1.0)) / ((1.0 / x) - (x * (x * x)))));
}

Julia

function code(x)
	return log(Float64(Float64(Float64(1.0 + x) * fma(x, x, 1.0)) / Float64(Float64(1.0 / x) - Float64(x * Float64(x * x)))))
end

Wolfram

code[x_] := N[Log[N[(N[(N[(1.0 + x), $MachinePrecision] * N[(x * x + 1.0), $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 / x), $MachinePrecision] - N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[-\log \left(\frac{1}{x} - 1\right) \]
2. Add Preprocessing

4. Applied egg-rr 26.5%

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

5. Taylor expanded in x around 0

   \[\leadsto \mathsf{neg}\left(\log \left(\frac{\frac{1}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)} + -1}{\color{blue}{\frac{1 + x \cdot \left(1 + x \cdot \left(1 + x\right)\right)}{{x}^{3}}}}\right)\right) \]
6. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{neg}\left(\log \left(\frac{\frac{1}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)} + -1}{\color{blue}{\frac{1 + x \cdot \left(1 + x \cdot \left(1 + x\right)\right)}{{x}^{3}}}}\right)\right) \]

   2. distribute-rgt-in N/A

      \[\leadsto \mathsf{neg}\left(\log \left(\frac{\frac{1}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)} + -1}{\frac{1 + \color{blue}{\left(1 \cdot x + \left(x \cdot \left(1 + x\right)\right) \cdot x\right)}}{{x}^{3}}}\right)\right) \]

   3. *-lft-identity N/A

      \[\leadsto \mathsf{neg}\left(\log \left(\frac{\frac{1}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)} + -1}{\frac{1 + \left(\color{blue}{x} + \left(x \cdot \left(1 + x\right)\right) \cdot x\right)}{{x}^{3}}}\right)\right) \]

   4. associate-+r+ N/A

      \[\leadsto \mathsf{neg}\left(\log \left(\frac{\frac{1}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)} + -1}{\frac{\color{blue}{\left(1 + x\right) + \left(x \cdot \left(1 + x\right)\right) \cdot x}}{{x}^{3}}}\right)\right) \]

   5. *-rgt-identity N/A

      \[\leadsto \mathsf{neg}\left(\log \left(\frac{\frac{1}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)} + -1}{\frac{\color{blue}{\left(1 + x\right) \cdot 1} + \left(x \cdot \left(1 + x\right)\right) \cdot x}{{x}^{3}}}\right)\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{neg}\left(\log \left(\frac{\frac{1}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)} + -1}{\frac{\left(1 + x\right) \cdot 1 + \color{blue}{\left(\left(1 + x\right) \cdot x\right)} \cdot x}{{x}^{3}}}\right)\right) \]

   7. associate-*l* N/A

      \[\leadsto \mathsf{neg}\left(\log \left(\frac{\frac{1}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)} + -1}{\frac{\left(1 + x\right) \cdot 1 + \color{blue}{\left(1 + x\right) \cdot \left(x \cdot x\right)}}{{x}^{3}}}\right)\right) \]

   8. unpow2 N/A

      \[\leadsto \mathsf{neg}\left(\log \left(\frac{\frac{1}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)} + -1}{\frac{\left(1 + x\right) \cdot 1 + \left(1 + x\right) \cdot \color{blue}{{x}^{2}}}{{x}^{3}}}\right)\right) \]

   9. distribute-lft-out N/A

      \[\leadsto \mathsf{neg}\left(\log \left(\frac{\frac{1}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)} + -1}{\frac{\color{blue}{\left(1 + x\right) \cdot \left(1 + {x}^{2}\right)}}{{x}^{3}}}\right)\right) \]

   10. +-commutative N/A

       \[\leadsto \mathsf{neg}\left(\log \left(\frac{\frac{1}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)} + -1}{\frac{\left(1 + x\right) \cdot \color{blue}{\left({x}^{2} + 1\right)}}{{x}^{3}}}\right)\right) \]

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

       \[\leadsto \mathsf{neg}\left(\log \left(\frac{\frac{1}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)} + -1}{\frac{\color{blue}{\left(1 + x\right) \cdot \left({x}^{2} + 1\right)}}{{x}^{3}}}\right)\right) \]

   12. +-commutative N/A

       \[\leadsto \mathsf{neg}\left(\log \left(\frac{\frac{1}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)} + -1}{\frac{\color{blue}{\left(x + 1\right)} \cdot \left({x}^{2} + 1\right)}{{x}^{3}}}\right)\right) \]

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

       \[\leadsto \mathsf{neg}\left(\log \left(\frac{\frac{1}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)} + -1}{\frac{\color{blue}{\left(x + 1\right)} \cdot \left({x}^{2} + 1\right)}{{x}^{3}}}\right)\right) \]

   14. unpow2 N/A

       \[\leadsto \mathsf{neg}\left(\log \left(\frac{\frac{1}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)} + -1}{\frac{\left(x + 1\right) \cdot \left(\color{blue}{x \cdot x} + 1\right)}{{x}^{3}}}\right)\right) \]

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

       \[\leadsto \mathsf{neg}\left(\log \left(\frac{\frac{1}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)} + -1}{\frac{\left(x + 1\right) \cdot \color{blue}{\mathsf{fma}\left(x, x, 1\right)}}{{x}^{3}}}\right)\right) \]

   16. cube-mult N/A

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

   17. unpow2 N/A

       \[\leadsto \mathsf{neg}\left(\log \left(\frac{\frac{1}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)} + -1}{\frac{\left(x + 1\right) \cdot \mathsf{fma}\left(x, x, 1\right)}{x \cdot \color{blue}{{x}^{2}}}}\right)\right) \]

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

       \[\leadsto \mathsf{neg}\left(\log \left(\frac{\frac{1}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)} + -1}{\frac{\left(x + 1\right) \cdot \mathsf{fma}\left(x, x, 1\right)}{\color{blue}{x \cdot {x}^{2}}}}\right)\right) \]

   19. unpow2 N/A

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

   20. *-lowering-*.f64 26.5

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

7. Simplified 26.5%

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

   1. neg-log N/A

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

   2. clear-num N/A

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

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

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

   4. associate-/l/ N/A

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

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

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

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

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

   7. +-commutative N/A

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

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

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

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

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

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

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

9. Applied egg-rr 26.5%

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

10. Taylor expanded in x around 0

    \[\leadsto \log \left(\frac{\left(1 + x\right) \cdot \mathsf{fma}\left(x, x, 1\right)}{\color{blue}{\frac{1 + -1 \cdot {x}^{4}}{x}}}\right) \]
11. Step-by-step derivation

    1. mul-1-neg N/A

       \[\leadsto \log \left(\frac{\left(1 + x\right) \cdot \mathsf{fma}\left(x, x, 1\right)}{\frac{1 + \color{blue}{\left(\mathsf{neg}\left({x}^{4}\right)\right)}}{x}}\right) \]

    2. unsub-neg N/A

       \[\leadsto \log \left(\frac{\left(1 + x\right) \cdot \mathsf{fma}\left(x, x, 1\right)}{\frac{\color{blue}{1 - {x}^{4}}}{x}}\right) \]

    3. div-sub N/A

       \[\leadsto \log \left(\frac{\left(1 + x\right) \cdot \mathsf{fma}\left(x, x, 1\right)}{\color{blue}{\frac{1}{x} - \frac{{x}^{4}}{x}}}\right) \]

    4. metadata-eval N/A

       \[\leadsto \log \left(\frac{\left(1 + x\right) \cdot \mathsf{fma}\left(x, x, 1\right)}{\frac{1}{x} - \frac{{x}^{\color{blue}{\left(3 + 1\right)}}}{x}}\right) \]

    5. pow-plus N/A

       \[\leadsto \log \left(\frac{\left(1 + x\right) \cdot \mathsf{fma}\left(x, x, 1\right)}{\frac{1}{x} - \frac{\color{blue}{{x}^{3} \cdot x}}{x}}\right) \]

    6. *-lft-identity N/A

       \[\leadsto \log \left(\frac{\left(1 + x\right) \cdot \mathsf{fma}\left(x, x, 1\right)}{\frac{1}{x} - \frac{{x}^{3} \cdot \color{blue}{\left(1 \cdot x\right)}}{x}}\right) \]

    7. lft-mult-inverse N/A

       \[\leadsto \log \left(\frac{\left(1 + x\right) \cdot \mathsf{fma}\left(x, x, 1\right)}{\frac{1}{x} - \frac{{x}^{3} \cdot \left(\color{blue}{\left(\frac{1}{{x}^{2}} \cdot {x}^{2}\right)} \cdot x\right)}{x}}\right) \]

    8. associate-*r* N/A

       \[\leadsto \log \left(\frac{\left(1 + x\right) \cdot \mathsf{fma}\left(x, x, 1\right)}{\frac{1}{x} - \frac{{x}^{3} \cdot \color{blue}{\left(\frac{1}{{x}^{2}} \cdot \left({x}^{2} \cdot x\right)\right)}}{x}}\right) \]

    9. unpow2 N/A

       \[\leadsto \log \left(\frac{\left(1 + x\right) \cdot \mathsf{fma}\left(x, x, 1\right)}{\frac{1}{x} - \frac{{x}^{3} \cdot \left(\frac{1}{{x}^{2}} \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot x\right)\right)}{x}}\right) \]

    10. unpow3 N/A

        \[\leadsto \log \left(\frac{\left(1 + x\right) \cdot \mathsf{fma}\left(x, x, 1\right)}{\frac{1}{x} - \frac{{x}^{3} \cdot \left(\frac{1}{{x}^{2}} \cdot \color{blue}{{x}^{3}}\right)}{x}}\right) \]

    11. associate-*l/ N/A

        \[\leadsto \log \left(\frac{\left(1 + x\right) \cdot \mathsf{fma}\left(x, x, 1\right)}{\frac{1}{x} - \frac{{x}^{3} \cdot \color{blue}{\frac{1 \cdot {x}^{3}}{{x}^{2}}}}{x}}\right) \]

    12. *-lft-identity N/A

        \[\leadsto \log \left(\frac{\left(1 + x\right) \cdot \mathsf{fma}\left(x, x, 1\right)}{\frac{1}{x} - \frac{{x}^{3} \cdot \frac{\color{blue}{{x}^{3}}}{{x}^{2}}}{x}}\right) \]

    13. associate-/l* N/A

        \[\leadsto \log \left(\frac{\left(1 + x\right) \cdot \mathsf{fma}\left(x, x, 1\right)}{\frac{1}{x} - \frac{\color{blue}{\frac{{x}^{3} \cdot {x}^{3}}{{x}^{2}}}}{x}}\right) \]

    14. pow-sqr N/A

        \[\leadsto \log \left(\frac{\left(1 + x\right) \cdot \mathsf{fma}\left(x, x, 1\right)}{\frac{1}{x} - \frac{\frac{\color{blue}{{x}^{\left(2 \cdot 3\right)}}}{{x}^{2}}}{x}}\right) \]

    15. metadata-eval N/A

        \[\leadsto \log \left(\frac{\left(1 + x\right) \cdot \mathsf{fma}\left(x, x, 1\right)}{\frac{1}{x} - \frac{\frac{{x}^{\color{blue}{6}}}{{x}^{2}}}{x}}\right) \]

    16. associate-/l/ N/A

        \[\leadsto \log \left(\frac{\left(1 + x\right) \cdot \mathsf{fma}\left(x, x, 1\right)}{\frac{1}{x} - \color{blue}{\frac{{x}^{6}}{x \cdot {x}^{2}}}}\right) \]

    17. metadata-eval N/A

        \[\leadsto \log \left(\frac{\left(1 + x\right) \cdot \mathsf{fma}\left(x, x, 1\right)}{\frac{1}{x} - \frac{{x}^{\color{blue}{\left(2 \cdot 3\right)}}}{x \cdot {x}^{2}}}\right) \]

    18. pow-sqr N/A

        \[\leadsto \log \left(\frac{\left(1 + x\right) \cdot \mathsf{fma}\left(x, x, 1\right)}{\frac{1}{x} - \frac{\color{blue}{{x}^{3} \cdot {x}^{3}}}{x \cdot {x}^{2}}}\right) \]

    19. unpow2 N/A

        \[\leadsto \log \left(\frac{\left(1 + x\right) \cdot \mathsf{fma}\left(x, x, 1\right)}{\frac{1}{x} - \frac{{x}^{3} \cdot {x}^{3}}{x \cdot \color{blue}{\left(x \cdot x\right)}}}\right) \]

    20. cube-mult N/A

        \[\leadsto \log \left(\frac{\left(1 + x\right) \cdot \mathsf{fma}\left(x, x, 1\right)}{\frac{1}{x} - \frac{{x}^{3} \cdot {x}^{3}}{\color{blue}{{x}^{3}}}}\right) \]

    21. associate-/l* N/A

        \[\leadsto \log \left(\frac{\left(1 + x\right) \cdot \mathsf{fma}\left(x, x, 1\right)}{\frac{1}{x} - \color{blue}{{x}^{3} \cdot \frac{{x}^{3}}{{x}^{3}}}}\right) \]

12. Simplified 100.0%

    \[\leadsto \log \left(\frac{\left(1 + x\right) \cdot \mathsf{fma}\left(x, x, 1\right)}{\color{blue}{\frac{1}{x} - x \cdot \left(x \cdot x\right)}}\right) \]
13. Add Preprocessing

Alternative 2: 7.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot x\right)\\ \mathbf{if}\;-\log \left(\frac{1}{x} + -1\right) \leq -250:\\ \;\;\;\;x \cdot \left(x \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, x \cdot x, t\_0 \cdot \left(t\_0 \cdot 0.125\right)\right)}{\mathsf{fma}\left(x, \left(x \cdot x\right) \cdot -0.5, \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 0.25\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* x (* x x))))
   (if (<= (- (log (+ (/ 1.0 x) -1.0))) -250.0)
     (* x (* x 0.5))
     (/
      (fma x (* x x) (* t_0 (* t_0 0.125)))
      (fma x (* (* x x) -0.5) (* (* x x) (* (* x x) 0.25)))))))

C

double code(double x) {
	double t_0 = x * (x * x);
	double tmp;
	if (-log(((1.0 / x) + -1.0)) <= -250.0) {
		tmp = x * (x * 0.5);
	} else {
		tmp = fma(x, (x * x), (t_0 * (t_0 * 0.125))) / fma(x, ((x * x) * -0.5), ((x * x) * ((x * x) * 0.25)));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(x * Float64(x * x))
	tmp = 0.0
	if (Float64(-log(Float64(Float64(1.0 / x) + -1.0))) <= -250.0)
		tmp = Float64(x * Float64(x * 0.5));
	else
		tmp = Float64(fma(x, Float64(x * x), Float64(t_0 * Float64(t_0 * 0.125))) / fma(x, Float64(Float64(x * x) * -0.5), Float64(Float64(x * x) * Float64(Float64(x * x) * 0.25))));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[(-N[Log[N[(N[(1.0 / x), $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision]), -250.0], N[(x * N[(x * 0.5), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(x * x), $MachinePrecision] + N[(t$95$0 * N[(t$95$0 * 0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * N[(N[(x * x), $MachinePrecision] * -0.5), $MachinePrecision] + N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (neg.f64 (log.f64 (-.f64 (/.f64 #s(literal 1 binary64) x) #s(literal 1 binary64)))) < -250

   1. Initial program 100.0%

      \[-\log \left(\frac{1}{x} - 1\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + \frac{1}{2} \cdot x\right) - -1 \cdot \log x} \]
   4. Step-by-step derivation

      1. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, 1 + \frac{1}{2} \cdot x, \log x\right)} \]

      5. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\frac{1}{2} \cdot x + 1}, \log x\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{1}{2}} + 1, \log x\right) \]

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

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, \frac{1}{2}, 1\right)}, \log x\right) \]

      8. log-lowering-log.f64 100.0

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, 1\right), \color{blue}{\log x}\right) \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, 1\right), \log x\right)} \]

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot {x}^{2}} \]
   7. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(x \cdot x\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot x\right) \cdot x} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{x \cdot \left(\frac{1}{2} \cdot x\right)} \]

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

         \[\leadsto \color{blue}{x \cdot \left(\frac{1}{2} \cdot x\right)} \]

      5. *-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(x \cdot \frac{1}{2}\right)} \]

      6. *-lowering-*.f64 3.0

         \[\leadsto x \cdot \color{blue}{\left(x \cdot 0.5\right)} \]

   8. Simplified 3.0%

      \[\leadsto \color{blue}{x \cdot \left(x \cdot 0.5\right)} \]

   if -250 < (neg.f64 (log.f64 (-.f64 (/.f64 #s(literal 1 binary64) x) #s(literal 1 binary64))))

   1. Initial program 100.0%

      \[-\log \left(\frac{1}{x} - 1\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + \frac{1}{2} \cdot x\right) - -1 \cdot \log x} \]
   4. Step-by-step derivation

      1. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, 1 + \frac{1}{2} \cdot x, \log x\right)} \]

      5. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\frac{1}{2} \cdot x + 1}, \log x\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{1}{2}} + 1, \log x\right) \]

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

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, \frac{1}{2}, 1\right)}, \log x\right) \]

      8. log-lowering-log.f64 97.8

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, 1\right), \color{blue}{\log x}\right) \]

   5. Simplified 97.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, 1\right), \log x\right)} \]

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{x}\right)} \]
   7. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto \color{blue}{{x}^{2} \cdot \frac{1}{2} + {x}^{2} \cdot \frac{1}{x}} \]

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. unpow2 N/A

         \[\leadsto x \cdot \left(\frac{1}{2} \cdot x\right) + \color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{x} \]

      6. associate-*l* N/A

         \[\leadsto x \cdot \left(\frac{1}{2} \cdot x\right) + \color{blue}{x \cdot \left(x \cdot \frac{1}{x}\right)} \]

      7. rgt-mult-inverse N/A

         \[\leadsto x \cdot \left(\frac{1}{2} \cdot x\right) + x \cdot \color{blue}{1} \]

      8. *-rgt-identity N/A

         \[\leadsto x \cdot \left(\frac{1}{2} \cdot x\right) + \color{blue}{x} \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{1}{2} \cdot x, x\right)} \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{1}{2}}, x\right) \]

      11. *-lowering-*.f64 1.9

          \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot 0.5}, x\right) \]

   8. Simplified 1.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot 0.5, x\right)} \]
   9. Step-by-step derivation

      1. flip3-+ N/A

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

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

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

   10. Applied egg-rr 1.9%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, x \cdot x, \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot 0.125\right)\right)}{\mathsf{fma}\left(x, x - x \cdot \left(x \cdot 0.5\right), \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 0.25\right)\right)}} \]

   11. Taylor expanded in x around inf

       \[\leadsto \frac{\mathsf{fma}\left(x, x \cdot x, \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{8}\right)\right)}{\mathsf{fma}\left(x, \color{blue}{\frac{-1}{2} \cdot {x}^{2}}, \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{4}\right)\right)} \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(x, x \cdot x, \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{8}\right)\right)}{\mathsf{fma}\left(x, \color{blue}{{x}^{2} \cdot \frac{-1}{2}}, \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{4}\right)\right)} \]

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

          \[\leadsto \frac{\mathsf{fma}\left(x, x \cdot x, \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{8}\right)\right)}{\mathsf{fma}\left(x, \color{blue}{{x}^{2} \cdot \frac{-1}{2}}, \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{4}\right)\right)} \]

       3. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(x, x \cdot x, \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{8}\right)\right)}{\mathsf{fma}\left(x, \color{blue}{\left(x \cdot x\right)} \cdot \frac{-1}{2}, \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{4}\right)\right)} \]

       4. *-lowering-*.f64 14.9

          \[\leadsto \frac{\mathsf{fma}\left(x, x \cdot x, \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot 0.125\right)\right)}{\mathsf{fma}\left(x, \color{blue}{\left(x \cdot x\right)} \cdot -0.5, \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 0.25\right)\right)} \]

   13. Simplified 14.9%

       \[\leadsto \frac{\mathsf{fma}\left(x, x \cdot x, \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot 0.125\right)\right)}{\mathsf{fma}\left(x, \color{blue}{\left(x \cdot x\right) \cdot -0.5}, \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 0.25\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 7.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-\log \left(\frac{1}{x} + -1\right) \leq -250:\\ \;\;\;\;x \cdot \left(x \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, x \cdot x, \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot 0.125\right)\right)}{\mathsf{fma}\left(x, \left(x \cdot x\right) \cdot -0.5, \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 0.25\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 6.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot x\right)\\ \mathbf{if}\;-\log \left(\frac{1}{x} + -1\right) \leq -250:\\ \;\;\;\;x \cdot \left(x \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, x \cdot x, t\_0 \cdot \left(t\_0 \cdot 0.125\right)\right)}{\left(x \cdot t\_0\right) \cdot \left(0.25 + \frac{-0.5}{x}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* x (* x x))))
   (if (<= (- (log (+ (/ 1.0 x) -1.0))) -250.0)
     (* x (* x 0.5))
     (/
      (fma x (* x x) (* t_0 (* t_0 0.125)))
      (* (* x t_0) (+ 0.25 (/ -0.5 x)))))))

C

double code(double x) {
	double t_0 = x * (x * x);
	double tmp;
	if (-log(((1.0 / x) + -1.0)) <= -250.0) {
		tmp = x * (x * 0.5);
	} else {
		tmp = fma(x, (x * x), (t_0 * (t_0 * 0.125))) / ((x * t_0) * (0.25 + (-0.5 / x)));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(x * Float64(x * x))
	tmp = 0.0
	if (Float64(-log(Float64(Float64(1.0 / x) + -1.0))) <= -250.0)
		tmp = Float64(x * Float64(x * 0.5));
	else
		tmp = Float64(fma(x, Float64(x * x), Float64(t_0 * Float64(t_0 * 0.125))) / Float64(Float64(x * t_0) * Float64(0.25 + Float64(-0.5 / x))));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[(-N[Log[N[(N[(1.0 / x), $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision]), -250.0], N[(x * N[(x * 0.5), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(x * x), $MachinePrecision] + N[(t$95$0 * N[(t$95$0 * 0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(x * t$95$0), $MachinePrecision] * N[(0.25 + N[(-0.5 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (neg.f64 (log.f64 (-.f64 (/.f64 #s(literal 1 binary64) x) #s(literal 1 binary64)))) < -250

   1. Initial program 100.0%

      \[-\log \left(\frac{1}{x} - 1\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + \frac{1}{2} \cdot x\right) - -1 \cdot \log x} \]
   4. Step-by-step derivation

      1. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, 1 + \frac{1}{2} \cdot x, \log x\right)} \]

      5. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\frac{1}{2} \cdot x + 1}, \log x\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{1}{2}} + 1, \log x\right) \]

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

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, \frac{1}{2}, 1\right)}, \log x\right) \]

      8. log-lowering-log.f64 100.0

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, 1\right), \color{blue}{\log x}\right) \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, 1\right), \log x\right)} \]

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot {x}^{2}} \]
   7. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(x \cdot x\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot x\right) \cdot x} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{x \cdot \left(\frac{1}{2} \cdot x\right)} \]

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

         \[\leadsto \color{blue}{x \cdot \left(\frac{1}{2} \cdot x\right)} \]

      5. *-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(x \cdot \frac{1}{2}\right)} \]

      6. *-lowering-*.f64 3.0

         \[\leadsto x \cdot \color{blue}{\left(x \cdot 0.5\right)} \]

   8. Simplified 3.0%

      \[\leadsto \color{blue}{x \cdot \left(x \cdot 0.5\right)} \]

   if -250 < (neg.f64 (log.f64 (-.f64 (/.f64 #s(literal 1 binary64) x) #s(literal 1 binary64))))

   1. Initial program 100.0%

      \[-\log \left(\frac{1}{x} - 1\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + \frac{1}{2} \cdot x\right) - -1 \cdot \log x} \]
   4. Step-by-step derivation

      1. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, 1 + \frac{1}{2} \cdot x, \log x\right)} \]

      5. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\frac{1}{2} \cdot x + 1}, \log x\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{1}{2}} + 1, \log x\right) \]

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

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, \frac{1}{2}, 1\right)}, \log x\right) \]

      8. log-lowering-log.f64 97.8

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, 1\right), \color{blue}{\log x}\right) \]

   5. Simplified 97.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, 1\right), \log x\right)} \]

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{x}\right)} \]
   7. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto \color{blue}{{x}^{2} \cdot \frac{1}{2} + {x}^{2} \cdot \frac{1}{x}} \]

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. unpow2 N/A

         \[\leadsto x \cdot \left(\frac{1}{2} \cdot x\right) + \color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{x} \]

      6. associate-*l* N/A

         \[\leadsto x \cdot \left(\frac{1}{2} \cdot x\right) + \color{blue}{x \cdot \left(x \cdot \frac{1}{x}\right)} \]

      7. rgt-mult-inverse N/A

         \[\leadsto x \cdot \left(\frac{1}{2} \cdot x\right) + x \cdot \color{blue}{1} \]

      8. *-rgt-identity N/A

         \[\leadsto x \cdot \left(\frac{1}{2} \cdot x\right) + \color{blue}{x} \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{1}{2} \cdot x, x\right)} \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{1}{2}}, x\right) \]

      11. *-lowering-*.f64 1.9

          \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot 0.5}, x\right) \]

   8. Simplified 1.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot 0.5, x\right)} \]
   9. Step-by-step derivation

      1. flip3-+ N/A

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

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

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

   10. Applied egg-rr 1.9%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, x \cdot x, \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot 0.125\right)\right)}{\mathsf{fma}\left(x, x - x \cdot \left(x \cdot 0.5\right), \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 0.25\right)\right)}} \]

   11. Taylor expanded in x around inf

       \[\leadsto \frac{\mathsf{fma}\left(x, x \cdot x, \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{8}\right)\right)}{\color{blue}{{x}^{4} \cdot \left(\frac{1}{4} - \frac{1}{2} \cdot \frac{1}{x}\right)}} \]

   13. Simplified 11.9%

       \[\leadsto \frac{\mathsf{fma}\left(x, x \cdot x, \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot 0.125\right)\right)}{\color{blue}{\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(0.25 + \frac{-0.5}{x}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 6.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-\log \left(\frac{1}{x} + -1\right) \leq -250:\\ \;\;\;\;x \cdot \left(x \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, x \cdot x, \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot 0.125\right)\right)}{\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(0.25 + \frac{-0.5}{x}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 100.0% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (- (log (+ (/ 1.0 x) -1.0))))

C

double code(double x) {
	return -log(((1.0 / x) + -1.0));
}

Fortran

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

Java

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

Python

def code(x):
	return -math.log(((1.0 / x) + -1.0))

Julia

function code(x)
	return Float64(-log(Float64(Float64(1.0 / x) + -1.0)))
end

MATLAB

function tmp = code(x)
	tmp = -log(((1.0 / x) + -1.0));
end

Wolfram

code[x_] := (-N[Log[N[(N[(1.0 / x), $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision])

Derivation

1. Initial program 100.0%

   \[-\log \left(\frac{1}{x} - 1\right) \]
2. Add Preprocessing

3. Final simplification 100.0%

   \[\leadsto -\log \left(\frac{1}{x} + -1\right) \]
4. Add Preprocessing

Alternative 5: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, 1\right), \log x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (fma x (fma x 0.5 1.0) (log x)))

C

double code(double x) {
	return fma(x, fma(x, 0.5, 1.0), log(x));
}

Julia

function code(x)
	return fma(x, fma(x, 0.5, 1.0), log(x))
end

Wolfram

code[x_] := N[(x * N[(x * 0.5 + 1.0), $MachinePrecision] + N[Log[x], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[-\log \left(\frac{1}{x} - 1\right) \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(1 + \frac{1}{2} \cdot x\right) - -1 \cdot \log x} \]
4. Step-by-step derivation

   1. cancel-sign-sub-inv N/A

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

   2. metadata-eval N/A

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

   3. *-lft-identity N/A

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

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

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, 1 + \frac{1}{2} \cdot x, \log x\right)} \]

   5. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{\frac{1}{2} \cdot x + 1}, \log x\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{1}{2}} + 1, \log x\right) \]

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

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, \frac{1}{2}, 1\right)}, \log x\right) \]

   8. log-lowering-log.f64 99.2

      \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, 1\right), \color{blue}{\log x}\right) \]

5. Simplified 99.2%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, 1\right), \log x\right)} \]
6. Add Preprocessing

Alternative 6: 99.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ x + \log x \end{array} \]

FPCore

(FPCore (x) :precision binary64 (+ x (log x)))

C

double code(double x) {
	return x + log(x);
}

Fortran

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

Java

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

Python

def code(x):
	return x + math.log(x)

Julia

function code(x)
	return Float64(x + log(x))
end

MATLAB

function tmp = code(x)
	tmp = x + log(x);
end

Wolfram

code[x_] := N[(x + N[Log[x], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[-\log \left(\frac{1}{x} - 1\right) \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x - -1 \cdot \log x} \]
4. Step-by-step derivation

   1. sub-neg N/A

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

   2. mul-1-neg N/A

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

   3. remove-double-neg N/A

      \[\leadsto x + \color{blue}{\log x} \]

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

      \[\leadsto \color{blue}{x + \log x} \]

   5. log-lowering-log.f64 98.9

      \[\leadsto x + \color{blue}{\log x} \]

5. Simplified 98.9%

   \[\leadsto \color{blue}{x + \log x} \]
6. Add Preprocessing

Alternative 7: 98.3% accurate, 1.2× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (log x))

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x)
	return log(x)
end

MATLAB

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

Wolfram

code[x_] := N[Log[x], $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[-\log \left(\frac{1}{x} - 1\right) \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\log x} \]
4. Step-by-step derivation

   1. log-lowering-log.f64 97.5

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

5. Simplified 97.5%

   \[\leadsto \color{blue}{\log x} \]
6. Add Preprocessing

Alternative 8: 2.7% accurate, 10.6× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(x \cdot 0.5\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* x (* x 0.5)))

C

double code(double x) {
	return x * (x * 0.5);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * (x * 0.5d0)
end function

Java

public static double code(double x) {
	return x * (x * 0.5);
}

Python

def code(x):
	return x * (x * 0.5)

Julia

function code(x)
	return Float64(x * Float64(x * 0.5))
end

MATLAB

function tmp = code(x)
	tmp = x * (x * 0.5);
end

Wolfram

code[x_] := N[(x * N[(x * 0.5), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[-\log \left(\frac{1}{x} - 1\right) \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(1 + \frac{1}{2} \cdot x\right) - -1 \cdot \log x} \]
4. Step-by-step derivation

   1. cancel-sign-sub-inv N/A

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

   2. metadata-eval N/A

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

   3. *-lft-identity N/A

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

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

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, 1 + \frac{1}{2} \cdot x, \log x\right)} \]

   5. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{\frac{1}{2} \cdot x + 1}, \log x\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{1}{2}} + 1, \log x\right) \]

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

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, \frac{1}{2}, 1\right)}, \log x\right) \]

   8. log-lowering-log.f64 99.2

      \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, 1\right), \color{blue}{\log x}\right) \]

5. Simplified 99.2%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, 1\right), \log x\right)} \]

6. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{1}{2} \cdot {x}^{2}} \]
7. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(x \cdot x\right)} \]

   2. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot x\right) \cdot x} \]

   3. *-commutative N/A

      \[\leadsto \color{blue}{x \cdot \left(\frac{1}{2} \cdot x\right)} \]

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

      \[\leadsto \color{blue}{x \cdot \left(\frac{1}{2} \cdot x\right)} \]

   5. *-commutative N/A

      \[\leadsto x \cdot \color{blue}{\left(x \cdot \frac{1}{2}\right)} \]

   6. *-lowering-*.f64 2.7

      \[\leadsto x \cdot \color{blue}{\left(x \cdot 0.5\right)} \]

8. Simplified 2.7%

   \[\leadsto \color{blue}{x \cdot \left(x \cdot 0.5\right)} \]
9. Add Preprocessing

Alternative 9: 2.3% accurate, 117.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 x)

C

double code(double x) {
	return x;
}

Fortran

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

Java

public static double code(double x) {
	return x;
}

Python

def code(x):
	return x

Julia

function code(x)
	return x
end

MATLAB

function tmp = code(x)
	tmp = x;
end

Wolfram

code[x_] := x

Derivation

1. Initial program 100.0%

   \[-\log \left(\frac{1}{x} - 1\right) \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x - -1 \cdot \log x} \]
4. Step-by-step derivation

   1. sub-neg N/A

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

   2. mul-1-neg N/A

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

   3. remove-double-neg N/A

      \[\leadsto x + \color{blue}{\log x} \]

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

      \[\leadsto \color{blue}{x + \log x} \]

   5. log-lowering-log.f64 98.9

      \[\leadsto x + \color{blue}{\log x} \]

5. Simplified 98.9%

   \[\leadsto \color{blue}{x + \log x} \]

6. Taylor expanded in x around inf

   \[\leadsto \color{blue}{x} \]
7. Step-by-step derivation

8. Simplified 2.3%

   \[\leadsto \color{blue}{x} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024205 
(FPCore (x)
  :name "neg log"
  :precision binary64
  (- (log (- (/ 1.0 x) 1.0))))