bug500 (missed optimization)

Percentage Accurate: 69.2% → 99.0%

Time: 9.7s

Alternatives: 6

Speedup: 14.7×

Specification

\[-1000 < x \land x < 1000\]

Math

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

FPCore

(FPCore (x) :precision binary64 (- (sin x) x))

C

double code(double x) {
	return sin(x) - x;
}

Fortran

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

Java

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

Python

def code(x):
	return math.sin(x) - x

Julia

function code(x)
	return Float64(sin(x) - x)
end

MATLAB

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

Wolfram

code[x_] := N[(N[Sin[x], $MachinePrecision] - x), $MachinePrecision]

Initial Program: 69.2% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (- (sin x) x))

C

double code(double x) {
	return sin(x) - x;
}

Fortran

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

Java

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

Python

def code(x):
	return math.sin(x) - x

Julia

function code(x)
	return Float64(sin(x) - x)
end

MATLAB

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

Wolfram

code[x_] := N[(N[Sin[x], $MachinePrecision] - x), $MachinePrecision]

Alternative 1: 99.0% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ x \cdot \frac{x}{\frac{-6}{x} + x \cdot \left(-0.3 + x \cdot \left(x \cdot \left(-0.007857142857142858 + \left(x \cdot x\right) \cdot -0.0001349206349206349\right)\right)\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  x
  (/
   x
   (+
    (/ -6.0 x)
    (*
     x
     (+
      -0.3
      (*
       x
       (*
        x
        (+ -0.007857142857142858 (* (* x x) -0.0001349206349206349))))))))))

C

double code(double x) {
	return x * (x / ((-6.0 / x) + (x * (-0.3 + (x * (x * (-0.007857142857142858 + ((x * x) * -0.0001349206349206349))))))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * (x / (((-6.0d0) / x) + (x * ((-0.3d0) + (x * (x * ((-0.007857142857142858d0) + ((x * x) * (-0.0001349206349206349d0)))))))))
end function

Java

public static double code(double x) {
	return x * (x / ((-6.0 / x) + (x * (-0.3 + (x * (x * (-0.007857142857142858 + ((x * x) * -0.0001349206349206349))))))));
}

Python

def code(x):
	return x * (x / ((-6.0 / x) + (x * (-0.3 + (x * (x * (-0.007857142857142858 + ((x * x) * -0.0001349206349206349))))))))

Julia

function code(x)
	return Float64(x * Float64(x / Float64(Float64(-6.0 / x) + Float64(x * Float64(-0.3 + Float64(x * Float64(x * Float64(-0.007857142857142858 + Float64(Float64(x * x) * -0.0001349206349206349)))))))))
end

MATLAB

function tmp = code(x)
	tmp = x * (x / ((-6.0 / x) + (x * (-0.3 + (x * (x * (-0.007857142857142858 + ((x * x) * -0.0001349206349206349))))))));
end

Wolfram

code[x_] := N[(x * N[(x / N[(N[(-6.0 / x), $MachinePrecision] + N[(x * N[(-0.3 + N[(x * N[(x * N[(-0.007857142857142858 + N[(N[(x * x), $MachinePrecision] * -0.0001349206349206349), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 67.7%

   \[\sin x - x \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{{x}^{3} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + {x}^{2} \cdot \left(\frac{1}{362880} \cdot {x}^{2} - \frac{1}{5040}\right)\right) - \frac{1}{6}\right)} \]
4. Step-by-step derivation

   1. cube-mult N/A

      \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\color{blue}{{x}^{2} \cdot \left(\frac{1}{120} + {x}^{2} \cdot \left(\frac{1}{362880} \cdot {x}^{2} - \frac{1}{5040}\right)\right)} - \frac{1}{6}\right) \]

   2. unpow2 N/A

      \[\leadsto \left(x \cdot {x}^{2}\right) \cdot \left({x}^{2} \cdot \color{blue}{\left(\frac{1}{120} + {x}^{2} \cdot \left(\frac{1}{362880} \cdot {x}^{2} - \frac{1}{5040}\right)\right)} - \frac{1}{6}\right) \]

   3. associate-*l* N/A

      \[\leadsto x \cdot \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + {x}^{2} \cdot \left(\frac{1}{362880} \cdot {x}^{2} - \frac{1}{5040}\right)\right) - \frac{1}{6}\right)\right)} \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + {x}^{2} \cdot \left(\frac{1}{362880} \cdot {x}^{2} - \frac{1}{5040}\right)\right) - \frac{1}{6}\right)\right)}\right) \]

   5. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \left(\left(x \cdot x\right) \cdot \left(\color{blue}{{x}^{2} \cdot \left(\frac{1}{120} + {x}^{2} \cdot \left(\frac{1}{362880} \cdot {x}^{2} - \frac{1}{5040}\right)\right)} - \frac{1}{6}\right)\right)\right) \]

   6. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left(x \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + {x}^{2} \cdot \left(\frac{1}{362880} \cdot {x}^{2} - \frac{1}{5040}\right)\right) - \frac{1}{6}\right)\right)}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + {x}^{2} \cdot \left(\frac{1}{362880} \cdot {x}^{2} - \frac{1}{5040}\right)\right) - \frac{1}{6}\right)\right)}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{120} + {x}^{2} \cdot \left(\frac{1}{362880} \cdot {x}^{2} - \frac{1}{5040}\right)\right) - \frac{1}{6}\right)}\right)\right)\right) \]

   9. sub-neg N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \left(\frac{1}{120} + {x}^{2} \cdot \left(\frac{1}{362880} \cdot {x}^{2} - \frac{1}{5040}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}\right)\right)\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \left(\frac{1}{120} + {x}^{2} \cdot \left(\frac{1}{362880} \cdot {x}^{2} - \frac{1}{5040}\right)\right) + \frac{-1}{6}\right)\right)\right)\right) \]

   11. +-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{-1}{6} + \color{blue}{{x}^{2} \cdot \left(\frac{1}{120} + {x}^{2} \cdot \left(\frac{1}{362880} \cdot {x}^{2} - \frac{1}{5040}\right)\right)}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{6}, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{120} + {x}^{2} \cdot \left(\frac{1}{362880} \cdot {x}^{2} - \frac{1}{5040}\right)\right)\right)}\right)\right)\right)\right) \]

5. Simplified 99.3%

   \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(x \cdot \left(-0.16666666666666666 + \left(x \cdot x\right) \cdot \left(0.008333333333333333 + x \cdot \left(x \cdot \left(-0.0001984126984126984 + x \cdot \left(x \cdot 2.7557319223985893 \cdot 10^{-6}\right)\right)\right)\right)\right)\right)\right)} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\left(\frac{-1}{6} + \left(x \cdot x\right) \cdot \left(\frac{1}{120} + x \cdot \left(x \cdot \left(\frac{-1}{5040} + x \cdot \left(x \cdot \frac{1}{362880}\right)\right)\right)\right)\right) \cdot \color{blue}{x}\right)\right)\right) \]

   2. flip-+ N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{\frac{-1}{6} \cdot \frac{-1}{6} - \left(\left(x \cdot x\right) \cdot \left(\frac{1}{120} + x \cdot \left(x \cdot \left(\frac{-1}{5040} + x \cdot \left(x \cdot \frac{1}{362880}\right)\right)\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{120} + x \cdot \left(x \cdot \left(\frac{-1}{5040} + x \cdot \left(x \cdot \frac{1}{362880}\right)\right)\right)\right)\right)}{\frac{-1}{6} - \left(x \cdot x\right) \cdot \left(\frac{1}{120} + x \cdot \left(x \cdot \left(\frac{-1}{5040} + x \cdot \left(x \cdot \frac{1}{362880}\right)\right)\right)\right)} \cdot x\right)\right)\right) \]

   3. associate-*l/ N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{\left(\frac{-1}{6} \cdot \frac{-1}{6} - \left(\left(x \cdot x\right) \cdot \left(\frac{1}{120} + x \cdot \left(x \cdot \left(\frac{-1}{5040} + x \cdot \left(x \cdot \frac{1}{362880}\right)\right)\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{120} + x \cdot \left(x \cdot \left(\frac{-1}{5040} + x \cdot \left(x \cdot \frac{1}{362880}\right)\right)\right)\right)\right)\right) \cdot x}{\color{blue}{\frac{-1}{6} - \left(x \cdot x\right) \cdot \left(\frac{1}{120} + x \cdot \left(x \cdot \left(\frac{-1}{5040} + x \cdot \left(x \cdot \frac{1}{362880}\right)\right)\right)\right)}}\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\left(\left(\frac{-1}{6} \cdot \frac{-1}{6} - \left(\left(x \cdot x\right) \cdot \left(\frac{1}{120} + x \cdot \left(x \cdot \left(\frac{-1}{5040} + x \cdot \left(x \cdot \frac{1}{362880}\right)\right)\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{120} + x \cdot \left(x \cdot \left(\frac{-1}{5040} + x \cdot \left(x \cdot \frac{1}{362880}\right)\right)\right)\right)\right)\right) \cdot x\right), \color{blue}{\left(\frac{-1}{6} - \left(x \cdot x\right) \cdot \left(\frac{1}{120} + x \cdot \left(x \cdot \left(\frac{-1}{5040} + x \cdot \left(x \cdot \frac{1}{362880}\right)\right)\right)\right)\right)}\right)\right)\right) \]

7. Applied egg-rr 99.3%

   \[\leadsto x \cdot \left(x \cdot \color{blue}{\frac{\left(0.027777777777777776 - \left(\left(x \cdot x\right) \cdot \left(0.008333333333333333 + \left(x \cdot x\right) \cdot \left(-0.0001984126984126984 + x \cdot \left(x \cdot 2.7557319223985893 \cdot 10^{-6}\right)\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(0.008333333333333333 + \left(x \cdot x\right) \cdot \left(-0.0001984126984126984 + x \cdot \left(x \cdot 2.7557319223985893 \cdot 10^{-6}\right)\right)\right)\right)\right) \cdot x}{-0.16666666666666666 - \left(x \cdot x\right) \cdot \left(0.008333333333333333 + \left(x \cdot x\right) \cdot \left(-0.0001984126984126984 + x \cdot \left(x \cdot 2.7557319223985893 \cdot 10^{-6}\right)\right)\right)}}\right) \]
8. Step-by-step derivation

   1. clear-num N/A

      \[\leadsto \mathsf{*.f64}\left(x, \left(x \cdot \frac{1}{\color{blue}{\frac{\frac{-1}{6} - \left(x \cdot x\right) \cdot \left(\frac{1}{120} + \left(x \cdot x\right) \cdot \left(\frac{-1}{5040} + x \cdot \left(x \cdot \frac{1}{362880}\right)\right)\right)}{\left(\frac{1}{36} - \left(\left(x \cdot x\right) \cdot \left(\frac{1}{120} + \left(x \cdot x\right) \cdot \left(\frac{-1}{5040} + x \cdot \left(x \cdot \frac{1}{362880}\right)\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{120} + \left(x \cdot x\right) \cdot \left(\frac{-1}{5040} + x \cdot \left(x \cdot \frac{1}{362880}\right)\right)\right)\right)\right) \cdot x}}}\right)\right) \]

   2. un-div-inv N/A

      \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{x}{\color{blue}{\frac{\frac{-1}{6} - \left(x \cdot x\right) \cdot \left(\frac{1}{120} + \left(x \cdot x\right) \cdot \left(\frac{-1}{5040} + x \cdot \left(x \cdot \frac{1}{362880}\right)\right)\right)}{\left(\frac{1}{36} - \left(\left(x \cdot x\right) \cdot \left(\frac{1}{120} + \left(x \cdot x\right) \cdot \left(\frac{-1}{5040} + x \cdot \left(x \cdot \frac{1}{362880}\right)\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{120} + \left(x \cdot x\right) \cdot \left(\frac{-1}{5040} + x \cdot \left(x \cdot \frac{1}{362880}\right)\right)\right)\right)\right) \cdot x}}}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(x, \color{blue}{\left(\frac{\frac{-1}{6} - \left(x \cdot x\right) \cdot \left(\frac{1}{120} + \left(x \cdot x\right) \cdot \left(\frac{-1}{5040} + x \cdot \left(x \cdot \frac{1}{362880}\right)\right)\right)}{\left(\frac{1}{36} - \left(\left(x \cdot x\right) \cdot \left(\frac{1}{120} + \left(x \cdot x\right) \cdot \left(\frac{-1}{5040} + x \cdot \left(x \cdot \frac{1}{362880}\right)\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{120} + \left(x \cdot x\right) \cdot \left(\frac{-1}{5040} + x \cdot \left(x \cdot \frac{1}{362880}\right)\right)\right)\right)\right) \cdot x}\right)}\right)\right) \]

9. Applied egg-rr 99.3%

   \[\leadsto x \cdot \color{blue}{\frac{x}{\frac{1}{x \cdot \left(x \cdot \left(x \cdot \left(0.008333333333333333 + x \cdot \left(x \cdot \left(-0.0001984126984126984 + x \cdot \left(x \cdot 2.7557319223985893 \cdot 10^{-6}\right)\right)\right)\right)\right) + -0.16666666666666666\right)}}} \]

10. Taylor expanded in x around 0

    \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(x, \color{blue}{\left(\frac{{x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-17}{126000} \cdot {x}^{2} - \frac{11}{1400}\right) - \frac{3}{10}\right) - 6}{x}\right)}\right)\right) \]
11. Step-by-step derivation

    1. div-sub N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(x, \left(\frac{{x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-17}{126000} \cdot {x}^{2} - \frac{11}{1400}\right) - \frac{3}{10}\right)}{x} - \color{blue}{\frac{6}{x}}\right)\right)\right) \]

    2. sub-neg N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(x, \left(\frac{{x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-17}{126000} \cdot {x}^{2} - \frac{11}{1400}\right) - \frac{3}{10}\right)}{x} + \color{blue}{\left(\mathsf{neg}\left(\frac{6}{x}\right)\right)}\right)\right)\right) \]

    3. +-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(x, \left(\left(\mathsf{neg}\left(\frac{6}{x}\right)\right) + \color{blue}{\frac{{x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-17}{126000} \cdot {x}^{2} - \frac{11}{1400}\right) - \frac{3}{10}\right)}{x}}\right)\right)\right) \]

    4. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(x, \left(\left(\mathsf{neg}\left(\frac{6}{x}\right)\right) + \frac{\left({x}^{2} \cdot \left(\frac{-17}{126000} \cdot {x}^{2} - \frac{11}{1400}\right) - \frac{3}{10}\right) \cdot {x}^{2}}{x}\right)\right)\right) \]

    5. associate-/l* N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(x, \left(\left(\mathsf{neg}\left(\frac{6}{x}\right)\right) + \left({x}^{2} \cdot \left(\frac{-17}{126000} \cdot {x}^{2} - \frac{11}{1400}\right) - \frac{3}{10}\right) \cdot \color{blue}{\frac{{x}^{2}}{x}}\right)\right)\right) \]

    6. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(x, \left(\left(\mathsf{neg}\left(\frac{6}{x}\right)\right) + \left({x}^{2} \cdot \left(\frac{-17}{126000} \cdot {x}^{2} - \frac{11}{1400}\right) - \frac{3}{10}\right) \cdot \frac{x \cdot x}{x}\right)\right)\right) \]

    7. associate-/l* N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(x, \left(\left(\mathsf{neg}\left(\frac{6}{x}\right)\right) + \left({x}^{2} \cdot \left(\frac{-17}{126000} \cdot {x}^{2} - \frac{11}{1400}\right) - \frac{3}{10}\right) \cdot \left(x \cdot \color{blue}{\frac{x}{x}}\right)\right)\right)\right) \]

    8. *-inverses N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(x, \left(\left(\mathsf{neg}\left(\frac{6}{x}\right)\right) + \left({x}^{2} \cdot \left(\frac{-17}{126000} \cdot {x}^{2} - \frac{11}{1400}\right) - \frac{3}{10}\right) \cdot \left(x \cdot 1\right)\right)\right)\right) \]

    9. *-rgt-identity N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(x, \left(\left(\mathsf{neg}\left(\frac{6}{x}\right)\right) + \left({x}^{2} \cdot \left(\frac{-17}{126000} \cdot {x}^{2} - \frac{11}{1400}\right) - \frac{3}{10}\right) \cdot x\right)\right)\right) \]

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

        \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\left(\mathsf{neg}\left(\frac{6}{x}\right)\right), \color{blue}{\left(\left({x}^{2} \cdot \left(\frac{-17}{126000} \cdot {x}^{2} - \frac{11}{1400}\right) - \frac{3}{10}\right) \cdot x\right)}\right)\right)\right) \]

    11. distribute-neg-frac N/A

        \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\left(\frac{\mathsf{neg}\left(6\right)}{x}\right), \left(\color{blue}{\left({x}^{2} \cdot \left(\frac{-17}{126000} \cdot {x}^{2} - \frac{11}{1400}\right) - \frac{3}{10}\right)} \cdot x\right)\right)\right)\right) \]

    12. metadata-eval N/A

        \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\left(\frac{-6}{x}\right), \left(\left(\color{blue}{{x}^{2} \cdot \left(\frac{-17}{126000} \cdot {x}^{2} - \frac{11}{1400}\right)} - \frac{3}{10}\right) \cdot x\right)\right)\right)\right) \]

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

        \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(-6, x\right), \left(\color{blue}{\left({x}^{2} \cdot \left(\frac{-17}{126000} \cdot {x}^{2} - \frac{11}{1400}\right) - \frac{3}{10}\right)} \cdot x\right)\right)\right)\right) \]

    14. *-commutative N/A

        \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(-6, x\right), \left(x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{-17}{126000} \cdot {x}^{2} - \frac{11}{1400}\right) - \frac{3}{10}\right)}\right)\right)\right)\right) \]

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

        \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(-6, x\right), \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{2} \cdot \left(\frac{-17}{126000} \cdot {x}^{2} - \frac{11}{1400}\right) - \frac{3}{10}\right)}\right)\right)\right)\right) \]

    16. sub-neg N/A

        \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(-6, x\right), \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \left(\frac{-17}{126000} \cdot {x}^{2} - \frac{11}{1400}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{3}{10}\right)\right)}\right)\right)\right)\right)\right) \]

12. Simplified 99.4%

    \[\leadsto x \cdot \frac{x}{\color{blue}{\frac{-6}{x} + x \cdot \left(-0.3 + x \cdot \left(x \cdot \left(-0.007857142857142858 + \left(x \cdot x\right) \cdot -0.0001349206349206349\right)\right)\right)}} \]
13. Add Preprocessing

Alternative 2: 98.9% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ x \cdot \frac{x}{\frac{-6}{x} + x \cdot \left(-0.3 + x \cdot \left(x \cdot -0.007857142857142858\right)\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* x (/ x (+ (/ -6.0 x) (* x (+ -0.3 (* x (* x -0.007857142857142858))))))))

C

double code(double x) {
	return x * (x / ((-6.0 / x) + (x * (-0.3 + (x * (x * -0.007857142857142858))))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * (x / (((-6.0d0) / x) + (x * ((-0.3d0) + (x * (x * (-0.007857142857142858d0)))))))
end function

Java

public static double code(double x) {
	return x * (x / ((-6.0 / x) + (x * (-0.3 + (x * (x * -0.007857142857142858))))));
}

Python

def code(x):
	return x * (x / ((-6.0 / x) + (x * (-0.3 + (x * (x * -0.007857142857142858))))))

Julia

function code(x)
	return Float64(x * Float64(x / Float64(Float64(-6.0 / x) + Float64(x * Float64(-0.3 + Float64(x * Float64(x * -0.007857142857142858)))))))
end

MATLAB

function tmp = code(x)
	tmp = x * (x / ((-6.0 / x) + (x * (-0.3 + (x * (x * -0.007857142857142858))))));
end

Wolfram

code[x_] := N[(x * N[(x / N[(N[(-6.0 / x), $MachinePrecision] + N[(x * N[(-0.3 + N[(x * N[(x * -0.007857142857142858), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 67.7%

   \[\sin x - x \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{{x}^{3} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + {x}^{2} \cdot \left(\frac{1}{362880} \cdot {x}^{2} - \frac{1}{5040}\right)\right) - \frac{1}{6}\right)} \]
4. Step-by-step derivation

   1. cube-mult N/A

      \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\color{blue}{{x}^{2} \cdot \left(\frac{1}{120} + {x}^{2} \cdot \left(\frac{1}{362880} \cdot {x}^{2} - \frac{1}{5040}\right)\right)} - \frac{1}{6}\right) \]

   2. unpow2 N/A

      \[\leadsto \left(x \cdot {x}^{2}\right) \cdot \left({x}^{2} \cdot \color{blue}{\left(\frac{1}{120} + {x}^{2} \cdot \left(\frac{1}{362880} \cdot {x}^{2} - \frac{1}{5040}\right)\right)} - \frac{1}{6}\right) \]

   3. associate-*l* N/A

      \[\leadsto x \cdot \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + {x}^{2} \cdot \left(\frac{1}{362880} \cdot {x}^{2} - \frac{1}{5040}\right)\right) - \frac{1}{6}\right)\right)} \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + {x}^{2} \cdot \left(\frac{1}{362880} \cdot {x}^{2} - \frac{1}{5040}\right)\right) - \frac{1}{6}\right)\right)}\right) \]

   5. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \left(\left(x \cdot x\right) \cdot \left(\color{blue}{{x}^{2} \cdot \left(\frac{1}{120} + {x}^{2} \cdot \left(\frac{1}{362880} \cdot {x}^{2} - \frac{1}{5040}\right)\right)} - \frac{1}{6}\right)\right)\right) \]

   6. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left(x \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + {x}^{2} \cdot \left(\frac{1}{362880} \cdot {x}^{2} - \frac{1}{5040}\right)\right) - \frac{1}{6}\right)\right)}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + {x}^{2} \cdot \left(\frac{1}{362880} \cdot {x}^{2} - \frac{1}{5040}\right)\right) - \frac{1}{6}\right)\right)}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{120} + {x}^{2} \cdot \left(\frac{1}{362880} \cdot {x}^{2} - \frac{1}{5040}\right)\right) - \frac{1}{6}\right)}\right)\right)\right) \]

   9. sub-neg N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \left(\frac{1}{120} + {x}^{2} \cdot \left(\frac{1}{362880} \cdot {x}^{2} - \frac{1}{5040}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}\right)\right)\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \left(\frac{1}{120} + {x}^{2} \cdot \left(\frac{1}{362880} \cdot {x}^{2} - \frac{1}{5040}\right)\right) + \frac{-1}{6}\right)\right)\right)\right) \]

   11. +-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{-1}{6} + \color{blue}{{x}^{2} \cdot \left(\frac{1}{120} + {x}^{2} \cdot \left(\frac{1}{362880} \cdot {x}^{2} - \frac{1}{5040}\right)\right)}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{6}, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{120} + {x}^{2} \cdot \left(\frac{1}{362880} \cdot {x}^{2} - \frac{1}{5040}\right)\right)\right)}\right)\right)\right)\right) \]

5. Simplified 99.3%

   \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(x \cdot \left(-0.16666666666666666 + \left(x \cdot x\right) \cdot \left(0.008333333333333333 + x \cdot \left(x \cdot \left(-0.0001984126984126984 + x \cdot \left(x \cdot 2.7557319223985893 \cdot 10^{-6}\right)\right)\right)\right)\right)\right)\right)} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\left(\frac{-1}{6} + \left(x \cdot x\right) \cdot \left(\frac{1}{120} + x \cdot \left(x \cdot \left(\frac{-1}{5040} + x \cdot \left(x \cdot \frac{1}{362880}\right)\right)\right)\right)\right) \cdot \color{blue}{x}\right)\right)\right) \]

   2. flip-+ N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{\frac{-1}{6} \cdot \frac{-1}{6} - \left(\left(x \cdot x\right) \cdot \left(\frac{1}{120} + x \cdot \left(x \cdot \left(\frac{-1}{5040} + x \cdot \left(x \cdot \frac{1}{362880}\right)\right)\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{120} + x \cdot \left(x \cdot \left(\frac{-1}{5040} + x \cdot \left(x \cdot \frac{1}{362880}\right)\right)\right)\right)\right)}{\frac{-1}{6} - \left(x \cdot x\right) \cdot \left(\frac{1}{120} + x \cdot \left(x \cdot \left(\frac{-1}{5040} + x \cdot \left(x \cdot \frac{1}{362880}\right)\right)\right)\right)} \cdot x\right)\right)\right) \]

   3. associate-*l/ N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{\left(\frac{-1}{6} \cdot \frac{-1}{6} - \left(\left(x \cdot x\right) \cdot \left(\frac{1}{120} + x \cdot \left(x \cdot \left(\frac{-1}{5040} + x \cdot \left(x \cdot \frac{1}{362880}\right)\right)\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{120} + x \cdot \left(x \cdot \left(\frac{-1}{5040} + x \cdot \left(x \cdot \frac{1}{362880}\right)\right)\right)\right)\right)\right) \cdot x}{\color{blue}{\frac{-1}{6} - \left(x \cdot x\right) \cdot \left(\frac{1}{120} + x \cdot \left(x \cdot \left(\frac{-1}{5040} + x \cdot \left(x \cdot \frac{1}{362880}\right)\right)\right)\right)}}\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\left(\left(\frac{-1}{6} \cdot \frac{-1}{6} - \left(\left(x \cdot x\right) \cdot \left(\frac{1}{120} + x \cdot \left(x \cdot \left(\frac{-1}{5040} + x \cdot \left(x \cdot \frac{1}{362880}\right)\right)\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{120} + x \cdot \left(x \cdot \left(\frac{-1}{5040} + x \cdot \left(x \cdot \frac{1}{362880}\right)\right)\right)\right)\right)\right) \cdot x\right), \color{blue}{\left(\frac{-1}{6} - \left(x \cdot x\right) \cdot \left(\frac{1}{120} + x \cdot \left(x \cdot \left(\frac{-1}{5040} + x \cdot \left(x \cdot \frac{1}{362880}\right)\right)\right)\right)\right)}\right)\right)\right) \]

7. Applied egg-rr 99.3%

   \[\leadsto x \cdot \left(x \cdot \color{blue}{\frac{\left(0.027777777777777776 - \left(\left(x \cdot x\right) \cdot \left(0.008333333333333333 + \left(x \cdot x\right) \cdot \left(-0.0001984126984126984 + x \cdot \left(x \cdot 2.7557319223985893 \cdot 10^{-6}\right)\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(0.008333333333333333 + \left(x \cdot x\right) \cdot \left(-0.0001984126984126984 + x \cdot \left(x \cdot 2.7557319223985893 \cdot 10^{-6}\right)\right)\right)\right)\right) \cdot x}{-0.16666666666666666 - \left(x \cdot x\right) \cdot \left(0.008333333333333333 + \left(x \cdot x\right) \cdot \left(-0.0001984126984126984 + x \cdot \left(x \cdot 2.7557319223985893 \cdot 10^{-6}\right)\right)\right)}}\right) \]
8. Step-by-step derivation

   1. clear-num N/A

      \[\leadsto \mathsf{*.f64}\left(x, \left(x \cdot \frac{1}{\color{blue}{\frac{\frac{-1}{6} - \left(x \cdot x\right) \cdot \left(\frac{1}{120} + \left(x \cdot x\right) \cdot \left(\frac{-1}{5040} + x \cdot \left(x \cdot \frac{1}{362880}\right)\right)\right)}{\left(\frac{1}{36} - \left(\left(x \cdot x\right) \cdot \left(\frac{1}{120} + \left(x \cdot x\right) \cdot \left(\frac{-1}{5040} + x \cdot \left(x \cdot \frac{1}{362880}\right)\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{120} + \left(x \cdot x\right) \cdot \left(\frac{-1}{5040} + x \cdot \left(x \cdot \frac{1}{362880}\right)\right)\right)\right)\right) \cdot x}}}\right)\right) \]

   2. un-div-inv N/A

      \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{x}{\color{blue}{\frac{\frac{-1}{6} - \left(x \cdot x\right) \cdot \left(\frac{1}{120} + \left(x \cdot x\right) \cdot \left(\frac{-1}{5040} + x \cdot \left(x \cdot \frac{1}{362880}\right)\right)\right)}{\left(\frac{1}{36} - \left(\left(x \cdot x\right) \cdot \left(\frac{1}{120} + \left(x \cdot x\right) \cdot \left(\frac{-1}{5040} + x \cdot \left(x \cdot \frac{1}{362880}\right)\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{120} + \left(x \cdot x\right) \cdot \left(\frac{-1}{5040} + x \cdot \left(x \cdot \frac{1}{362880}\right)\right)\right)\right)\right) \cdot x}}}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(x, \color{blue}{\left(\frac{\frac{-1}{6} - \left(x \cdot x\right) \cdot \left(\frac{1}{120} + \left(x \cdot x\right) \cdot \left(\frac{-1}{5040} + x \cdot \left(x \cdot \frac{1}{362880}\right)\right)\right)}{\left(\frac{1}{36} - \left(\left(x \cdot x\right) \cdot \left(\frac{1}{120} + \left(x \cdot x\right) \cdot \left(\frac{-1}{5040} + x \cdot \left(x \cdot \frac{1}{362880}\right)\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{120} + \left(x \cdot x\right) \cdot \left(\frac{-1}{5040} + x \cdot \left(x \cdot \frac{1}{362880}\right)\right)\right)\right)\right) \cdot x}\right)}\right)\right) \]

9. Applied egg-rr 99.3%

   \[\leadsto x \cdot \color{blue}{\frac{x}{\frac{1}{x \cdot \left(x \cdot \left(x \cdot \left(0.008333333333333333 + x \cdot \left(x \cdot \left(-0.0001984126984126984 + x \cdot \left(x \cdot 2.7557319223985893 \cdot 10^{-6}\right)\right)\right)\right)\right) + -0.16666666666666666\right)}}} \]

10. Taylor expanded in x around 0

    \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(x, \color{blue}{\left(\frac{{x}^{2} \cdot \left(\frac{-11}{1400} \cdot {x}^{2} - \frac{3}{10}\right) - 6}{x}\right)}\right)\right) \]
11. Step-by-step derivation

    1. div-sub N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(x, \left(\frac{{x}^{2} \cdot \left(\frac{-11}{1400} \cdot {x}^{2} - \frac{3}{10}\right)}{x} - \color{blue}{\frac{6}{x}}\right)\right)\right) \]

    2. sub-neg N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(x, \left(\frac{{x}^{2} \cdot \left(\frac{-11}{1400} \cdot {x}^{2} - \frac{3}{10}\right)}{x} + \color{blue}{\left(\mathsf{neg}\left(\frac{6}{x}\right)\right)}\right)\right)\right) \]

    3. +-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(x, \left(\left(\mathsf{neg}\left(\frac{6}{x}\right)\right) + \color{blue}{\frac{{x}^{2} \cdot \left(\frac{-11}{1400} \cdot {x}^{2} - \frac{3}{10}\right)}{x}}\right)\right)\right) \]

    4. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(x, \left(\left(\mathsf{neg}\left(\frac{6}{x}\right)\right) + \frac{\left(\frac{-11}{1400} \cdot {x}^{2} - \frac{3}{10}\right) \cdot {x}^{2}}{x}\right)\right)\right) \]

    5. associate-/l* N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(x, \left(\left(\mathsf{neg}\left(\frac{6}{x}\right)\right) + \left(\frac{-11}{1400} \cdot {x}^{2} - \frac{3}{10}\right) \cdot \color{blue}{\frac{{x}^{2}}{x}}\right)\right)\right) \]

    6. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(x, \left(\left(\mathsf{neg}\left(\frac{6}{x}\right)\right) + \left(\frac{-11}{1400} \cdot {x}^{2} - \frac{3}{10}\right) \cdot \frac{x \cdot x}{x}\right)\right)\right) \]

    7. associate-/l* N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(x, \left(\left(\mathsf{neg}\left(\frac{6}{x}\right)\right) + \left(\frac{-11}{1400} \cdot {x}^{2} - \frac{3}{10}\right) \cdot \left(x \cdot \color{blue}{\frac{x}{x}}\right)\right)\right)\right) \]

    8. *-inverses N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(x, \left(\left(\mathsf{neg}\left(\frac{6}{x}\right)\right) + \left(\frac{-11}{1400} \cdot {x}^{2} - \frac{3}{10}\right) \cdot \left(x \cdot 1\right)\right)\right)\right) \]

    9. *-rgt-identity N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(x, \left(\left(\mathsf{neg}\left(\frac{6}{x}\right)\right) + \left(\frac{-11}{1400} \cdot {x}^{2} - \frac{3}{10}\right) \cdot x\right)\right)\right) \]

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

        \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\left(\mathsf{neg}\left(\frac{6}{x}\right)\right), \color{blue}{\left(\left(\frac{-11}{1400} \cdot {x}^{2} - \frac{3}{10}\right) \cdot x\right)}\right)\right)\right) \]

    11. distribute-neg-frac N/A

        \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\left(\frac{\mathsf{neg}\left(6\right)}{x}\right), \left(\color{blue}{\left(\frac{-11}{1400} \cdot {x}^{2} - \frac{3}{10}\right)} \cdot x\right)\right)\right)\right) \]

    12. metadata-eval N/A

        \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\left(\frac{-6}{x}\right), \left(\left(\color{blue}{\frac{-11}{1400} \cdot {x}^{2}} - \frac{3}{10}\right) \cdot x\right)\right)\right)\right) \]

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

        \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(-6, x\right), \left(\color{blue}{\left(\frac{-11}{1400} \cdot {x}^{2} - \frac{3}{10}\right)} \cdot x\right)\right)\right)\right) \]

    14. *-commutative N/A

        \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(-6, x\right), \left(x \cdot \color{blue}{\left(\frac{-11}{1400} \cdot {x}^{2} - \frac{3}{10}\right)}\right)\right)\right)\right) \]

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

        \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(-6, x\right), \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{-11}{1400} \cdot {x}^{2} - \frac{3}{10}\right)}\right)\right)\right)\right) \]

    16. sub-neg N/A

        \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(-6, x\right), \mathsf{*.f64}\left(x, \left(\frac{-11}{1400} \cdot {x}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{3}{10}\right)\right)}\right)\right)\right)\right)\right) \]

    17. metadata-eval N/A

        \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(-6, x\right), \mathsf{*.f64}\left(x, \left(\frac{-11}{1400} \cdot {x}^{2} + \frac{-3}{10}\right)\right)\right)\right)\right) \]

    18. +-commutative N/A

        \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(-6, x\right), \mathsf{*.f64}\left(x, \left(\frac{-3}{10} + \color{blue}{\frac{-11}{1400} \cdot {x}^{2}}\right)\right)\right)\right)\right) \]

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

        \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(-6, x\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-3}{10}, \color{blue}{\left(\frac{-11}{1400} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right) \]

    20. unpow2 N/A

        \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(-6, x\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-3}{10}, \left(\frac{-11}{1400} \cdot \left(x \cdot \color{blue}{x}\right)\right)\right)\right)\right)\right)\right) \]

    21. associate-*r* N/A

        \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(-6, x\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-3}{10}, \left(\left(\frac{-11}{1400} \cdot x\right) \cdot \color{blue}{x}\right)\right)\right)\right)\right)\right) \]

    22. *-commutative N/A

        \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(-6, x\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-3}{10}, \left(x \cdot \color{blue}{\left(\frac{-11}{1400} \cdot x\right)}\right)\right)\right)\right)\right)\right) \]

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

        \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(-6, x\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-3}{10}, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{-11}{1400} \cdot x\right)}\right)\right)\right)\right)\right)\right) \]

    24. *-commutative N/A

        \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(-6, x\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-3}{10}, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\frac{-11}{1400}}\right)\right)\right)\right)\right)\right)\right) \]

    25. *-lowering-*.f64 99.4%

        \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(-6, x\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-3}{10}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\frac{-11}{1400}}\right)\right)\right)\right)\right)\right)\right) \]

12. Simplified 99.4%

    \[\leadsto x \cdot \frac{x}{\color{blue}{\frac{-6}{x} + x \cdot \left(-0.3 + x \cdot \left(x \cdot -0.007857142857142858\right)\right)}} \]
13. Add Preprocessing

Alternative 3: 98.7% accurate, 9.4× speedup

Math

\[\begin{array}{l} \\ x \cdot \frac{x}{\frac{-6}{x} + x \cdot -0.3} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* x (/ x (+ (/ -6.0 x) (* x -0.3)))))

C

double code(double x) {
	return x * (x / ((-6.0 / x) + (x * -0.3)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * (x / (((-6.0d0) / x) + (x * (-0.3d0))))
end function

Java

public static double code(double x) {
	return x * (x / ((-6.0 / x) + (x * -0.3)));
}

Python

def code(x):
	return x * (x / ((-6.0 / x) + (x * -0.3)))

Julia

function code(x)
	return Float64(x * Float64(x / Float64(Float64(-6.0 / x) + Float64(x * -0.3))))
end

MATLAB

function tmp = code(x)
	tmp = x * (x / ((-6.0 / x) + (x * -0.3)));
end

Wolfram

code[x_] := N[(x * N[(x / N[(N[(-6.0 / x), $MachinePrecision] + N[(x * -0.3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 67.7%

   \[\sin x - x \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{{x}^{3} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + {x}^{2} \cdot \left(\frac{1}{362880} \cdot {x}^{2} - \frac{1}{5040}\right)\right) - \frac{1}{6}\right)} \]
4. Step-by-step derivation

   1. cube-mult N/A

      \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\color{blue}{{x}^{2} \cdot \left(\frac{1}{120} + {x}^{2} \cdot \left(\frac{1}{362880} \cdot {x}^{2} - \frac{1}{5040}\right)\right)} - \frac{1}{6}\right) \]

   2. unpow2 N/A

      \[\leadsto \left(x \cdot {x}^{2}\right) \cdot \left({x}^{2} \cdot \color{blue}{\left(\frac{1}{120} + {x}^{2} \cdot \left(\frac{1}{362880} \cdot {x}^{2} - \frac{1}{5040}\right)\right)} - \frac{1}{6}\right) \]

   3. associate-*l* N/A

      \[\leadsto x \cdot \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + {x}^{2} \cdot \left(\frac{1}{362880} \cdot {x}^{2} - \frac{1}{5040}\right)\right) - \frac{1}{6}\right)\right)} \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + {x}^{2} \cdot \left(\frac{1}{362880} \cdot {x}^{2} - \frac{1}{5040}\right)\right) - \frac{1}{6}\right)\right)}\right) \]

   5. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \left(\left(x \cdot x\right) \cdot \left(\color{blue}{{x}^{2} \cdot \left(\frac{1}{120} + {x}^{2} \cdot \left(\frac{1}{362880} \cdot {x}^{2} - \frac{1}{5040}\right)\right)} - \frac{1}{6}\right)\right)\right) \]

   6. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left(x \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + {x}^{2} \cdot \left(\frac{1}{362880} \cdot {x}^{2} - \frac{1}{5040}\right)\right) - \frac{1}{6}\right)\right)}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + {x}^{2} \cdot \left(\frac{1}{362880} \cdot {x}^{2} - \frac{1}{5040}\right)\right) - \frac{1}{6}\right)\right)}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{120} + {x}^{2} \cdot \left(\frac{1}{362880} \cdot {x}^{2} - \frac{1}{5040}\right)\right) - \frac{1}{6}\right)}\right)\right)\right) \]

   9. sub-neg N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \left(\frac{1}{120} + {x}^{2} \cdot \left(\frac{1}{362880} \cdot {x}^{2} - \frac{1}{5040}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}\right)\right)\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \left(\frac{1}{120} + {x}^{2} \cdot \left(\frac{1}{362880} \cdot {x}^{2} - \frac{1}{5040}\right)\right) + \frac{-1}{6}\right)\right)\right)\right) \]

   11. +-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{-1}{6} + \color{blue}{{x}^{2} \cdot \left(\frac{1}{120} + {x}^{2} \cdot \left(\frac{1}{362880} \cdot {x}^{2} - \frac{1}{5040}\right)\right)}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{6}, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{120} + {x}^{2} \cdot \left(\frac{1}{362880} \cdot {x}^{2} - \frac{1}{5040}\right)\right)\right)}\right)\right)\right)\right) \]

5. Simplified 99.3%

   \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(x \cdot \left(-0.16666666666666666 + \left(x \cdot x\right) \cdot \left(0.008333333333333333 + x \cdot \left(x \cdot \left(-0.0001984126984126984 + x \cdot \left(x \cdot 2.7557319223985893 \cdot 10^{-6}\right)\right)\right)\right)\right)\right)\right)} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\left(\frac{-1}{6} + \left(x \cdot x\right) \cdot \left(\frac{1}{120} + x \cdot \left(x \cdot \left(\frac{-1}{5040} + x \cdot \left(x \cdot \frac{1}{362880}\right)\right)\right)\right)\right) \cdot \color{blue}{x}\right)\right)\right) \]

   2. flip-+ N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{\frac{-1}{6} \cdot \frac{-1}{6} - \left(\left(x \cdot x\right) \cdot \left(\frac{1}{120} + x \cdot \left(x \cdot \left(\frac{-1}{5040} + x \cdot \left(x \cdot \frac{1}{362880}\right)\right)\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{120} + x \cdot \left(x \cdot \left(\frac{-1}{5040} + x \cdot \left(x \cdot \frac{1}{362880}\right)\right)\right)\right)\right)}{\frac{-1}{6} - \left(x \cdot x\right) \cdot \left(\frac{1}{120} + x \cdot \left(x \cdot \left(\frac{-1}{5040} + x \cdot \left(x \cdot \frac{1}{362880}\right)\right)\right)\right)} \cdot x\right)\right)\right) \]

   3. associate-*l/ N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{\left(\frac{-1}{6} \cdot \frac{-1}{6} - \left(\left(x \cdot x\right) \cdot \left(\frac{1}{120} + x \cdot \left(x \cdot \left(\frac{-1}{5040} + x \cdot \left(x \cdot \frac{1}{362880}\right)\right)\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{120} + x \cdot \left(x \cdot \left(\frac{-1}{5040} + x \cdot \left(x \cdot \frac{1}{362880}\right)\right)\right)\right)\right)\right) \cdot x}{\color{blue}{\frac{-1}{6} - \left(x \cdot x\right) \cdot \left(\frac{1}{120} + x \cdot \left(x \cdot \left(\frac{-1}{5040} + x \cdot \left(x \cdot \frac{1}{362880}\right)\right)\right)\right)}}\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\left(\left(\frac{-1}{6} \cdot \frac{-1}{6} - \left(\left(x \cdot x\right) \cdot \left(\frac{1}{120} + x \cdot \left(x \cdot \left(\frac{-1}{5040} + x \cdot \left(x \cdot \frac{1}{362880}\right)\right)\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{120} + x \cdot \left(x \cdot \left(\frac{-1}{5040} + x \cdot \left(x \cdot \frac{1}{362880}\right)\right)\right)\right)\right)\right) \cdot x\right), \color{blue}{\left(\frac{-1}{6} - \left(x \cdot x\right) \cdot \left(\frac{1}{120} + x \cdot \left(x \cdot \left(\frac{-1}{5040} + x \cdot \left(x \cdot \frac{1}{362880}\right)\right)\right)\right)\right)}\right)\right)\right) \]

7. Applied egg-rr 99.3%

   \[\leadsto x \cdot \left(x \cdot \color{blue}{\frac{\left(0.027777777777777776 - \left(\left(x \cdot x\right) \cdot \left(0.008333333333333333 + \left(x \cdot x\right) \cdot \left(-0.0001984126984126984 + x \cdot \left(x \cdot 2.7557319223985893 \cdot 10^{-6}\right)\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(0.008333333333333333 + \left(x \cdot x\right) \cdot \left(-0.0001984126984126984 + x \cdot \left(x \cdot 2.7557319223985893 \cdot 10^{-6}\right)\right)\right)\right)\right) \cdot x}{-0.16666666666666666 - \left(x \cdot x\right) \cdot \left(0.008333333333333333 + \left(x \cdot x\right) \cdot \left(-0.0001984126984126984 + x \cdot \left(x \cdot 2.7557319223985893 \cdot 10^{-6}\right)\right)\right)}}\right) \]
8. Step-by-step derivation

   1. clear-num N/A

      \[\leadsto \mathsf{*.f64}\left(x, \left(x \cdot \frac{1}{\color{blue}{\frac{\frac{-1}{6} - \left(x \cdot x\right) \cdot \left(\frac{1}{120} + \left(x \cdot x\right) \cdot \left(\frac{-1}{5040} + x \cdot \left(x \cdot \frac{1}{362880}\right)\right)\right)}{\left(\frac{1}{36} - \left(\left(x \cdot x\right) \cdot \left(\frac{1}{120} + \left(x \cdot x\right) \cdot \left(\frac{-1}{5040} + x \cdot \left(x \cdot \frac{1}{362880}\right)\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{120} + \left(x \cdot x\right) \cdot \left(\frac{-1}{5040} + x \cdot \left(x \cdot \frac{1}{362880}\right)\right)\right)\right)\right) \cdot x}}}\right)\right) \]

   2. un-div-inv N/A

      \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{x}{\color{blue}{\frac{\frac{-1}{6} - \left(x \cdot x\right) \cdot \left(\frac{1}{120} + \left(x \cdot x\right) \cdot \left(\frac{-1}{5040} + x \cdot \left(x \cdot \frac{1}{362880}\right)\right)\right)}{\left(\frac{1}{36} - \left(\left(x \cdot x\right) \cdot \left(\frac{1}{120} + \left(x \cdot x\right) \cdot \left(\frac{-1}{5040} + x \cdot \left(x \cdot \frac{1}{362880}\right)\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{120} + \left(x \cdot x\right) \cdot \left(\frac{-1}{5040} + x \cdot \left(x \cdot \frac{1}{362880}\right)\right)\right)\right)\right) \cdot x}}}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(x, \color{blue}{\left(\frac{\frac{-1}{6} - \left(x \cdot x\right) \cdot \left(\frac{1}{120} + \left(x \cdot x\right) \cdot \left(\frac{-1}{5040} + x \cdot \left(x \cdot \frac{1}{362880}\right)\right)\right)}{\left(\frac{1}{36} - \left(\left(x \cdot x\right) \cdot \left(\frac{1}{120} + \left(x \cdot x\right) \cdot \left(\frac{-1}{5040} + x \cdot \left(x \cdot \frac{1}{362880}\right)\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{120} + \left(x \cdot x\right) \cdot \left(\frac{-1}{5040} + x \cdot \left(x \cdot \frac{1}{362880}\right)\right)\right)\right)\right) \cdot x}\right)}\right)\right) \]

9. Applied egg-rr 99.3%

   \[\leadsto x \cdot \color{blue}{\frac{x}{\frac{1}{x \cdot \left(x \cdot \left(x \cdot \left(0.008333333333333333 + x \cdot \left(x \cdot \left(-0.0001984126984126984 + x \cdot \left(x \cdot 2.7557319223985893 \cdot 10^{-6}\right)\right)\right)\right)\right) + -0.16666666666666666\right)}}} \]

10. Taylor expanded in x around 0

    \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(x, \color{blue}{\left(\frac{\frac{-3}{10} \cdot {x}^{2} - 6}{x}\right)}\right)\right) \]
11. Step-by-step derivation

    1. div-sub N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(x, \left(\frac{\frac{-3}{10} \cdot {x}^{2}}{x} - \color{blue}{\frac{6}{x}}\right)\right)\right) \]

    2. sub-neg N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(x, \left(\frac{\frac{-3}{10} \cdot {x}^{2}}{x} + \color{blue}{\left(\mathsf{neg}\left(\frac{6}{x}\right)\right)}\right)\right)\right) \]

    3. +-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(x, \left(\left(\mathsf{neg}\left(\frac{6}{x}\right)\right) + \color{blue}{\frac{\frac{-3}{10} \cdot {x}^{2}}{x}}\right)\right)\right) \]

    4. associate-/l* N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(x, \left(\left(\mathsf{neg}\left(\frac{6}{x}\right)\right) + \frac{-3}{10} \cdot \color{blue}{\frac{{x}^{2}}{x}}\right)\right)\right) \]

    5. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(x, \left(\left(\mathsf{neg}\left(\frac{6}{x}\right)\right) + \frac{-3}{10} \cdot \frac{x \cdot x}{x}\right)\right)\right) \]

    6. associate-/l* N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(x, \left(\left(\mathsf{neg}\left(\frac{6}{x}\right)\right) + \frac{-3}{10} \cdot \left(x \cdot \color{blue}{\frac{x}{x}}\right)\right)\right)\right) \]

    7. *-inverses N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(x, \left(\left(\mathsf{neg}\left(\frac{6}{x}\right)\right) + \frac{-3}{10} \cdot \left(x \cdot 1\right)\right)\right)\right) \]

    8. *-rgt-identity N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(x, \left(\left(\mathsf{neg}\left(\frac{6}{x}\right)\right) + \frac{-3}{10} \cdot x\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\left(\mathsf{neg}\left(\frac{6}{x}\right)\right), \color{blue}{\left(\frac{-3}{10} \cdot x\right)}\right)\right)\right) \]

    10. distribute-neg-frac N/A

        \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\left(\frac{\mathsf{neg}\left(6\right)}{x}\right), \left(\color{blue}{\frac{-3}{10}} \cdot x\right)\right)\right)\right) \]

    11. metadata-eval N/A

        \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\left(\frac{-6}{x}\right), \left(\frac{-3}{10} \cdot x\right)\right)\right)\right) \]

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

        \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(-6, x\right), \left(\color{blue}{\frac{-3}{10}} \cdot x\right)\right)\right)\right) \]

    13. *-commutative N/A

        \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(-6, x\right), \left(x \cdot \color{blue}{\frac{-3}{10}}\right)\right)\right)\right) \]

    14. *-lowering-*.f64 99.3%

        \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\mathsf{/.f64}\left(-6, x\right), \mathsf{*.f64}\left(x, \color{blue}{\frac{-3}{10}}\right)\right)\right)\right) \]

12. Simplified 99.3%

    \[\leadsto x \cdot \frac{x}{\color{blue}{\frac{-6}{x} + x \cdot -0.3}} \]
13. Add Preprocessing

Alternative 4: 98.2% accurate, 14.7× speedup

Math

\[\begin{array}{l} \\ x \cdot \frac{x}{\frac{-6}{x}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* x (/ x (/ -6.0 x))))

C

double code(double x) {
	return x * (x / (-6.0 / x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * (x / ((-6.0d0) / x))
end function

Java

public static double code(double x) {
	return x * (x / (-6.0 / x));
}

Python

def code(x):
	return x * (x / (-6.0 / x))

Julia

function code(x)
	return Float64(x * Float64(x / Float64(-6.0 / x)))
end

MATLAB

function tmp = code(x)
	tmp = x * (x / (-6.0 / x));
end

Wolfram

code[x_] := N[(x * N[(x / N[(-6.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 67.7%

   \[\sin x - x \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{{x}^{3} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + {x}^{2} \cdot \left(\frac{1}{362880} \cdot {x}^{2} - \frac{1}{5040}\right)\right) - \frac{1}{6}\right)} \]
4. Step-by-step derivation

   1. cube-mult N/A

      \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\color{blue}{{x}^{2} \cdot \left(\frac{1}{120} + {x}^{2} \cdot \left(\frac{1}{362880} \cdot {x}^{2} - \frac{1}{5040}\right)\right)} - \frac{1}{6}\right) \]

   2. unpow2 N/A

      \[\leadsto \left(x \cdot {x}^{2}\right) \cdot \left({x}^{2} \cdot \color{blue}{\left(\frac{1}{120} + {x}^{2} \cdot \left(\frac{1}{362880} \cdot {x}^{2} - \frac{1}{5040}\right)\right)} - \frac{1}{6}\right) \]

   3. associate-*l* N/A

      \[\leadsto x \cdot \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + {x}^{2} \cdot \left(\frac{1}{362880} \cdot {x}^{2} - \frac{1}{5040}\right)\right) - \frac{1}{6}\right)\right)} \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + {x}^{2} \cdot \left(\frac{1}{362880} \cdot {x}^{2} - \frac{1}{5040}\right)\right) - \frac{1}{6}\right)\right)}\right) \]

   5. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \left(\left(x \cdot x\right) \cdot \left(\color{blue}{{x}^{2} \cdot \left(\frac{1}{120} + {x}^{2} \cdot \left(\frac{1}{362880} \cdot {x}^{2} - \frac{1}{5040}\right)\right)} - \frac{1}{6}\right)\right)\right) \]

   6. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left(x \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + {x}^{2} \cdot \left(\frac{1}{362880} \cdot {x}^{2} - \frac{1}{5040}\right)\right) - \frac{1}{6}\right)\right)}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + {x}^{2} \cdot \left(\frac{1}{362880} \cdot {x}^{2} - \frac{1}{5040}\right)\right) - \frac{1}{6}\right)\right)}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{120} + {x}^{2} \cdot \left(\frac{1}{362880} \cdot {x}^{2} - \frac{1}{5040}\right)\right) - \frac{1}{6}\right)}\right)\right)\right) \]

   9. sub-neg N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \left(\frac{1}{120} + {x}^{2} \cdot \left(\frac{1}{362880} \cdot {x}^{2} - \frac{1}{5040}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}\right)\right)\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \left(\frac{1}{120} + {x}^{2} \cdot \left(\frac{1}{362880} \cdot {x}^{2} - \frac{1}{5040}\right)\right) + \frac{-1}{6}\right)\right)\right)\right) \]

   11. +-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{-1}{6} + \color{blue}{{x}^{2} \cdot \left(\frac{1}{120} + {x}^{2} \cdot \left(\frac{1}{362880} \cdot {x}^{2} - \frac{1}{5040}\right)\right)}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{6}, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{120} + {x}^{2} \cdot \left(\frac{1}{362880} \cdot {x}^{2} - \frac{1}{5040}\right)\right)\right)}\right)\right)\right)\right) \]

5. Simplified 99.3%

   \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(x \cdot \left(-0.16666666666666666 + \left(x \cdot x\right) \cdot \left(0.008333333333333333 + x \cdot \left(x \cdot \left(-0.0001984126984126984 + x \cdot \left(x \cdot 2.7557319223985893 \cdot 10^{-6}\right)\right)\right)\right)\right)\right)\right)} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\left(\frac{-1}{6} + \left(x \cdot x\right) \cdot \left(\frac{1}{120} + x \cdot \left(x \cdot \left(\frac{-1}{5040} + x \cdot \left(x \cdot \frac{1}{362880}\right)\right)\right)\right)\right) \cdot \color{blue}{x}\right)\right)\right) \]

   2. flip-+ N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{\frac{-1}{6} \cdot \frac{-1}{6} - \left(\left(x \cdot x\right) \cdot \left(\frac{1}{120} + x \cdot \left(x \cdot \left(\frac{-1}{5040} + x \cdot \left(x \cdot \frac{1}{362880}\right)\right)\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{120} + x \cdot \left(x \cdot \left(\frac{-1}{5040} + x \cdot \left(x \cdot \frac{1}{362880}\right)\right)\right)\right)\right)}{\frac{-1}{6} - \left(x \cdot x\right) \cdot \left(\frac{1}{120} + x \cdot \left(x \cdot \left(\frac{-1}{5040} + x \cdot \left(x \cdot \frac{1}{362880}\right)\right)\right)\right)} \cdot x\right)\right)\right) \]

   3. associate-*l/ N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{\left(\frac{-1}{6} \cdot \frac{-1}{6} - \left(\left(x \cdot x\right) \cdot \left(\frac{1}{120} + x \cdot \left(x \cdot \left(\frac{-1}{5040} + x \cdot \left(x \cdot \frac{1}{362880}\right)\right)\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{120} + x \cdot \left(x \cdot \left(\frac{-1}{5040} + x \cdot \left(x \cdot \frac{1}{362880}\right)\right)\right)\right)\right)\right) \cdot x}{\color{blue}{\frac{-1}{6} - \left(x \cdot x\right) \cdot \left(\frac{1}{120} + x \cdot \left(x \cdot \left(\frac{-1}{5040} + x \cdot \left(x \cdot \frac{1}{362880}\right)\right)\right)\right)}}\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\left(\left(\frac{-1}{6} \cdot \frac{-1}{6} - \left(\left(x \cdot x\right) \cdot \left(\frac{1}{120} + x \cdot \left(x \cdot \left(\frac{-1}{5040} + x \cdot \left(x \cdot \frac{1}{362880}\right)\right)\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{120} + x \cdot \left(x \cdot \left(\frac{-1}{5040} + x \cdot \left(x \cdot \frac{1}{362880}\right)\right)\right)\right)\right)\right) \cdot x\right), \color{blue}{\left(\frac{-1}{6} - \left(x \cdot x\right) \cdot \left(\frac{1}{120} + x \cdot \left(x \cdot \left(\frac{-1}{5040} + x \cdot \left(x \cdot \frac{1}{362880}\right)\right)\right)\right)\right)}\right)\right)\right) \]

7. Applied egg-rr 99.3%

   \[\leadsto x \cdot \left(x \cdot \color{blue}{\frac{\left(0.027777777777777776 - \left(\left(x \cdot x\right) \cdot \left(0.008333333333333333 + \left(x \cdot x\right) \cdot \left(-0.0001984126984126984 + x \cdot \left(x \cdot 2.7557319223985893 \cdot 10^{-6}\right)\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(0.008333333333333333 + \left(x \cdot x\right) \cdot \left(-0.0001984126984126984 + x \cdot \left(x \cdot 2.7557319223985893 \cdot 10^{-6}\right)\right)\right)\right)\right) \cdot x}{-0.16666666666666666 - \left(x \cdot x\right) \cdot \left(0.008333333333333333 + \left(x \cdot x\right) \cdot \left(-0.0001984126984126984 + x \cdot \left(x \cdot 2.7557319223985893 \cdot 10^{-6}\right)\right)\right)}}\right) \]
8. Step-by-step derivation

   1. clear-num N/A

      \[\leadsto \mathsf{*.f64}\left(x, \left(x \cdot \frac{1}{\color{blue}{\frac{\frac{-1}{6} - \left(x \cdot x\right) \cdot \left(\frac{1}{120} + \left(x \cdot x\right) \cdot \left(\frac{-1}{5040} + x \cdot \left(x \cdot \frac{1}{362880}\right)\right)\right)}{\left(\frac{1}{36} - \left(\left(x \cdot x\right) \cdot \left(\frac{1}{120} + \left(x \cdot x\right) \cdot \left(\frac{-1}{5040} + x \cdot \left(x \cdot \frac{1}{362880}\right)\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{120} + \left(x \cdot x\right) \cdot \left(\frac{-1}{5040} + x \cdot \left(x \cdot \frac{1}{362880}\right)\right)\right)\right)\right) \cdot x}}}\right)\right) \]

   2. un-div-inv N/A

      \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{x}{\color{blue}{\frac{\frac{-1}{6} - \left(x \cdot x\right) \cdot \left(\frac{1}{120} + \left(x \cdot x\right) \cdot \left(\frac{-1}{5040} + x \cdot \left(x \cdot \frac{1}{362880}\right)\right)\right)}{\left(\frac{1}{36} - \left(\left(x \cdot x\right) \cdot \left(\frac{1}{120} + \left(x \cdot x\right) \cdot \left(\frac{-1}{5040} + x \cdot \left(x \cdot \frac{1}{362880}\right)\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{120} + \left(x \cdot x\right) \cdot \left(\frac{-1}{5040} + x \cdot \left(x \cdot \frac{1}{362880}\right)\right)\right)\right)\right) \cdot x}}}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(x, \color{blue}{\left(\frac{\frac{-1}{6} - \left(x \cdot x\right) \cdot \left(\frac{1}{120} + \left(x \cdot x\right) \cdot \left(\frac{-1}{5040} + x \cdot \left(x \cdot \frac{1}{362880}\right)\right)\right)}{\left(\frac{1}{36} - \left(\left(x \cdot x\right) \cdot \left(\frac{1}{120} + \left(x \cdot x\right) \cdot \left(\frac{-1}{5040} + x \cdot \left(x \cdot \frac{1}{362880}\right)\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{120} + \left(x \cdot x\right) \cdot \left(\frac{-1}{5040} + x \cdot \left(x \cdot \frac{1}{362880}\right)\right)\right)\right)\right) \cdot x}\right)}\right)\right) \]

9. Applied egg-rr 99.3%

   \[\leadsto x \cdot \color{blue}{\frac{x}{\frac{1}{x \cdot \left(x \cdot \left(x \cdot \left(0.008333333333333333 + x \cdot \left(x \cdot \left(-0.0001984126984126984 + x \cdot \left(x \cdot 2.7557319223985893 \cdot 10^{-6}\right)\right)\right)\right)\right) + -0.16666666666666666\right)}}} \]

10. Taylor expanded in x around 0

    \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(x, \color{blue}{\left(\frac{-6}{x}\right)}\right)\right) \]
11. Step-by-step derivation

    1. /-lowering-/.f64 98.8%

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(-6, \color{blue}{x}\right)\right)\right) \]

12. Simplified 98.8%

    \[\leadsto x \cdot \frac{x}{\color{blue}{\frac{-6}{x}}} \]
13. Add Preprocessing

Alternative 5: 98.2% accurate, 14.7× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(\left(x \cdot x\right) \cdot -0.16666666666666666\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* x (* (* x x) -0.16666666666666666)))

C

double code(double x) {
	return x * ((x * x) * -0.16666666666666666);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * ((x * x) * (-0.16666666666666666d0))
end function

Java

public static double code(double x) {
	return x * ((x * x) * -0.16666666666666666);
}

Python

def code(x):
	return x * ((x * x) * -0.16666666666666666)

Julia

function code(x)
	return Float64(x * Float64(Float64(x * x) * -0.16666666666666666))
end

MATLAB

function tmp = code(x)
	tmp = x * ((x * x) * -0.16666666666666666);
end

Wolfram

code[x_] := N[(x * N[(N[(x * x), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 67.7%

   \[\sin x - x \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{-1}{6} \cdot {x}^{3}} \]
4. Step-by-step derivation

   1. unpow3 N/A

      \[\leadsto \frac{-1}{6} \cdot \left(\left(x \cdot x\right) \cdot \color{blue}{x}\right) \]

   2. unpow2 N/A

      \[\leadsto \frac{-1}{6} \cdot \left({x}^{2} \cdot x\right) \]

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

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

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{-1}{6} \cdot {x}^{2}\right)}\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{-1}{6}, \color{blue}{\left({x}^{2}\right)}\right)\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{-1}{6}, \left(x \cdot \color{blue}{x}\right)\right)\right) \]

   8. *-lowering-*.f64 98.7%

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right) \]

5. Simplified 98.7%

   \[\leadsto \color{blue}{x \cdot \left(-0.16666666666666666 \cdot \left(x \cdot x\right)\right)} \]

6. Final simplification 98.7%

   \[\leadsto x \cdot \left(\left(x \cdot x\right) \cdot -0.16666666666666666\right) \]
7. Add Preprocessing

Alternative 6: 67.0% accurate, 103.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 0.0)

C

double code(double x) {
	return 0.0;
}

Fortran

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

Java

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

Python

def code(x):
	return 0.0

Julia

function code(x)
	return 0.0
end

MATLAB

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

Wolfram

code[x_] := 0.0

Derivation

1. Initial program 67.7%

   \[\sin x - x \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \mathsf{\_.f64}\left(\color{blue}{x}, x\right) \]
4. Step-by-step derivation

5. Simplified 66.2%

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

   1. +-inverses 66.2%

      \[\leadsto 0 \]

7. Applied egg-rr 66.2%

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

Developer Target 1: 99.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| < 0.07:\\ \;\;\;\;-\left(\left(\frac{{x}^{3}}{6} - \frac{{x}^{5}}{120}\right) + \frac{{x}^{7}}{5040}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin x - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (< (fabs x) 0.07)
   (- (+ (- (/ (pow x 3.0) 6.0) (/ (pow x 5.0) 120.0)) (/ (pow x 7.0) 5040.0)))
   (- (sin x) x)))

C

double code(double x) {
	double tmp;
	if (fabs(x) < 0.07) {
		tmp = -(((pow(x, 3.0) / 6.0) - (pow(x, 5.0) / 120.0)) + (pow(x, 7.0) / 5040.0));
	} else {
		tmp = sin(x) - x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (abs(x) < 0.07d0) then
        tmp = -((((x ** 3.0d0) / 6.0d0) - ((x ** 5.0d0) / 120.0d0)) + ((x ** 7.0d0) / 5040.0d0))
    else
        tmp = sin(x) - x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (Math.abs(x) < 0.07) {
		tmp = -(((Math.pow(x, 3.0) / 6.0) - (Math.pow(x, 5.0) / 120.0)) + (Math.pow(x, 7.0) / 5040.0));
	} else {
		tmp = Math.sin(x) - x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.fabs(x) < 0.07:
		tmp = -(((math.pow(x, 3.0) / 6.0) - (math.pow(x, 5.0) / 120.0)) + (math.pow(x, 7.0) / 5040.0))
	else:
		tmp = math.sin(x) - x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (abs(x) < 0.07)
		tmp = Float64(-Float64(Float64(Float64((x ^ 3.0) / 6.0) - Float64((x ^ 5.0) / 120.0)) + Float64((x ^ 7.0) / 5040.0)));
	else
		tmp = Float64(sin(x) - x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (abs(x) < 0.07)
		tmp = -((((x ^ 3.0) / 6.0) - ((x ^ 5.0) / 120.0)) + ((x ^ 7.0) / 5040.0));
	else
		tmp = sin(x) - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Less[N[Abs[x], $MachinePrecision], 0.07], (-N[(N[(N[(N[Power[x, 3.0], $MachinePrecision] / 6.0), $MachinePrecision] - N[(N[Power[x, 5.0], $MachinePrecision] / 120.0), $MachinePrecision]), $MachinePrecision] + N[(N[Power[x, 7.0], $MachinePrecision] / 5040.0), $MachinePrecision]), $MachinePrecision]), N[(N[Sin[x], $MachinePrecision] - x), $MachinePrecision]]

Reproduce

herbie shell --seed 2024161 
(FPCore (x)
  :name "bug500 (missed optimization)"
  :precision binary64
  :pre (and (< -1000.0 x) (< x 1000.0))

  :alt
  (! :herbie-platform default (if (< (fabs x) 7/100) (- (+ (- (/ (pow x 3) 6) (/ (pow x 5) 120)) (/ (pow x 7) 5040))) (- (sin x) x)))

  (- (sin x) x))