bug500 (missed optimization)

Percentage Accurate: 69.6% → 99.1%

Time: 9.1s

Alternatives: 13

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: 69.6% 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: 99.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot \left(0.008333333333333333 + \left(-0.0001984126984126984 + x \cdot \left(x \cdot 2.7557319223985893 \cdot 10^{-6}\right)\right) \cdot \left(x \cdot x\right)\right)\right)\\ x \cdot \frac{x}{\frac{0.027777777777777776 + t\_0 \cdot \left(t\_0 + 0.16666666666666666\right)}{x \cdot \left(-0.004629629629629629 + \left(5.787037037037037 \cdot 10^{-7} + x \cdot \left(x \cdot -4.1335978835978836 \cdot 10^{-8}\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\right)}} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0
         (*
          x
          (*
           x
           (+
            0.008333333333333333
            (*
             (+ -0.0001984126984126984 (* x (* x 2.7557319223985893e-6)))
             (* x x)))))))
   (*
    x
    (/
     x
     (/
      (+ 0.027777777777777776 (* t_0 (+ t_0 0.16666666666666666)))
      (*
       x
       (+
        -0.004629629629629629
        (*
         (+ 5.787037037037037e-7 (* x (* x -4.1335978835978836e-8)))
         (* (* x x) (* (* x x) (* x x)))))))))))

C

double code(double x) {
	double t_0 = x * (x * (0.008333333333333333 + ((-0.0001984126984126984 + (x * (x * 2.7557319223985893e-6))) * (x * x))));
	return x * (x / ((0.027777777777777776 + (t_0 * (t_0 + 0.16666666666666666))) / (x * (-0.004629629629629629 + ((5.787037037037037e-7 + (x * (x * -4.1335978835978836e-8))) * ((x * x) * ((x * x) * (x * x))))))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = x * (x * (0.008333333333333333d0 + (((-0.0001984126984126984d0) + (x * (x * 2.7557319223985893d-6))) * (x * x))))
    code = x * (x / ((0.027777777777777776d0 + (t_0 * (t_0 + 0.16666666666666666d0))) / (x * ((-0.004629629629629629d0) + ((5.787037037037037d-7 + (x * (x * (-4.1335978835978836d-8)))) * ((x * x) * ((x * x) * (x * x))))))))
end function

Java

public static double code(double x) {
	double t_0 = x * (x * (0.008333333333333333 + ((-0.0001984126984126984 + (x * (x * 2.7557319223985893e-6))) * (x * x))));
	return x * (x / ((0.027777777777777776 + (t_0 * (t_0 + 0.16666666666666666))) / (x * (-0.004629629629629629 + ((5.787037037037037e-7 + (x * (x * -4.1335978835978836e-8))) * ((x * x) * ((x * x) * (x * x))))))));
}

Python

def code(x):
	t_0 = x * (x * (0.008333333333333333 + ((-0.0001984126984126984 + (x * (x * 2.7557319223985893e-6))) * (x * x))))
	return x * (x / ((0.027777777777777776 + (t_0 * (t_0 + 0.16666666666666666))) / (x * (-0.004629629629629629 + ((5.787037037037037e-7 + (x * (x * -4.1335978835978836e-8))) * ((x * x) * ((x * x) * (x * x))))))))

Julia

function code(x)
	t_0 = Float64(x * Float64(x * Float64(0.008333333333333333 + Float64(Float64(-0.0001984126984126984 + Float64(x * Float64(x * 2.7557319223985893e-6))) * Float64(x * x)))))
	return Float64(x * Float64(x / Float64(Float64(0.027777777777777776 + Float64(t_0 * Float64(t_0 + 0.16666666666666666))) / Float64(x * Float64(-0.004629629629629629 + Float64(Float64(5.787037037037037e-7 + Float64(x * Float64(x * -4.1335978835978836e-8))) * Float64(Float64(x * x) * Float64(Float64(x * x) * Float64(x * x)))))))))
end

MATLAB

function tmp = code(x)
	t_0 = x * (x * (0.008333333333333333 + ((-0.0001984126984126984 + (x * (x * 2.7557319223985893e-6))) * (x * x))));
	tmp = x * (x / ((0.027777777777777776 + (t_0 * (t_0 + 0.16666666666666666))) / (x * (-0.004629629629629629 + ((5.787037037037037e-7 + (x * (x * -4.1335978835978836e-8))) * ((x * x) * ((x * x) * (x * x))))))));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * N[(x * N[(0.008333333333333333 + N[(N[(-0.0001984126984126984 + N[(x * N[(x * 2.7557319223985893e-6), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x * N[(x / N[(N[(0.027777777777777776 + N[(t$95$0 * N[(t$95$0 + 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * N[(-0.004629629629629629 + N[(N[(5.787037037037037e-7 + N[(x * N[(x * -4.1335978835978836e-8), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 66.1%

   \[\sin x - x \]
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%

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

   1. *-commutative N/A

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

   2. flip3-+ N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{{\frac{-1}{6}}^{3} + {\left(\left(x \cdot x\right) \cdot \left(\frac{1}{120} + \left(x \cdot x\right) \cdot \left(\frac{-1}{5040} + x \cdot \left(x \cdot \frac{1}{362880}\right)\right)\right)\right)}^{3}}{\frac{-1}{6} \cdot \frac{-1}{6} + \left(\left(\left(x \cdot x\right) \cdot \left(\frac{1}{120} + \left(x \cdot x\right) \cdot \left(\frac{-1}{5040} + x \cdot \left(x \cdot \frac{1}{362880}\right)\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{120} + \left(x \cdot x\right) \cdot \left(\frac{-1}{5040} + x \cdot \left(x \cdot \frac{1}{362880}\right)\right)\right)\right) - \frac{-1}{6} \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{120} + \left(x \cdot x\right) \cdot \left(\frac{-1}{5040} + x \cdot \left(x \cdot \frac{1}{362880}\right)\right)\right)\right)\right)} \cdot x\right)\right)\right) \]

   3. associate-*l/ N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{\left({\frac{-1}{6}}^{3} + {\left(\left(x \cdot x\right) \cdot \left(\frac{1}{120} + \left(x \cdot x\right) \cdot \left(\frac{-1}{5040} + x \cdot \left(x \cdot \frac{1}{362880}\right)\right)\right)\right)}^{3}\right) \cdot x}{\color{blue}{\frac{-1}{6} \cdot \frac{-1}{6} + \left(\left(\left(x \cdot x\right) \cdot \left(\frac{1}{120} + \left(x \cdot x\right) \cdot \left(\frac{-1}{5040} + x \cdot \left(x \cdot \frac{1}{362880}\right)\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{120} + \left(x \cdot x\right) \cdot \left(\frac{-1}{5040} + x \cdot \left(x \cdot \frac{1}{362880}\right)\right)\right)\right) - \frac{-1}{6} \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{120} + \left(x \cdot x\right) \cdot \left(\frac{-1}{5040} + x \cdot \left(x \cdot \frac{1}{362880}\right)\right)\right)\right)\right)}}\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\left(\left({\frac{-1}{6}}^{3} + {\left(\left(x \cdot x\right) \cdot \left(\frac{1}{120} + \left(x \cdot x\right) \cdot \left(\frac{-1}{5040} + x \cdot \left(x \cdot \frac{1}{362880}\right)\right)\right)\right)}^{3}\right) \cdot x\right), \color{blue}{\left(\frac{-1}{6} \cdot \frac{-1}{6} + \left(\left(\left(x \cdot x\right) \cdot \left(\frac{1}{120} + \left(x \cdot x\right) \cdot \left(\frac{-1}{5040} + x \cdot \left(x \cdot \frac{1}{362880}\right)\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{120} + \left(x \cdot x\right) \cdot \left(\frac{-1}{5040} + x \cdot \left(x \cdot \frac{1}{362880}\right)\right)\right)\right) - \frac{-1}{6} \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{120} + \left(x \cdot x\right) \cdot \left(\frac{-1}{5040} + x \cdot \left(x \cdot \frac{1}{362880}\right)\right)\right)\right)\right)\right)}\right)\right)\right) \]

7. Applied egg-rr 99.4%

   \[\leadsto x \cdot \left(x \cdot \color{blue}{\frac{\left(-0.004629629629629629 + \left(x \cdot \left(x \cdot \left(0.008333333333333333 + x \cdot \left(x \cdot \left(-0.0001984126984126984 + x \cdot \left(x \cdot 2.7557319223985893 \cdot 10^{-6}\right)\right)\right)\right)\right)\right) \cdot \left(\left(x \cdot \left(x \cdot \left(0.008333333333333333 + x \cdot \left(x \cdot \left(-0.0001984126984126984 + x \cdot \left(x \cdot 2.7557319223985893 \cdot 10^{-6}\right)\right)\right)\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(0.008333333333333333 + x \cdot \left(x \cdot \left(-0.0001984126984126984 + x \cdot \left(x \cdot 2.7557319223985893 \cdot 10^{-6}\right)\right)\right)\right)\right)\right)\right)\right) \cdot x}{0.027777777777777776 + \left(x \cdot \left(x \cdot \left(0.008333333333333333 + x \cdot \left(x \cdot \left(-0.0001984126984126984 + x \cdot \left(x \cdot 2.7557319223985893 \cdot 10^{-6}\right)\right)\right)\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(0.008333333333333333 + x \cdot \left(x \cdot \left(-0.0001984126984126984 + x \cdot \left(x \cdot 2.7557319223985893 \cdot 10^{-6}\right)\right)\right)\right)\right) - -0.16666666666666666\right)}}\right) \]

8. Taylor expanded in x around 0

   \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\color{blue}{\left(x \cdot \left({x}^{6} \cdot \left(\frac{1}{1728000} + \frac{-1}{24192000} \cdot {x}^{2}\right) - \frac{1}{216}\right)\right)}, \mathsf{+.f64}\left(\frac{1}{36}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{5040}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{362880}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{5040}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{362880}\right)\right)\right)\right)\right)\right)\right)\right), \frac{-1}{6}\right)\right)\right)\right)\right)\right) \]
9. Step-by-step derivation

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \left({x}^{6} \cdot \left(\frac{1}{1728000} + \frac{-1}{24192000} \cdot {x}^{2}\right) - \frac{1}{216}\right)\right), \mathsf{+.f64}\left(\color{blue}{\frac{1}{36}}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{5040}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{362880}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{5040}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{362880}\right)\right)\right)\right)\right)\right)\right)\right), \frac{-1}{6}\right)\right)\right)\right)\right)\right) \]

   2. sub-neg N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \left({x}^{6} \cdot \left(\frac{1}{1728000} + \frac{-1}{24192000} \cdot {x}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{216}\right)\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{36}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{5040}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{362880}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{5040}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{362880}\right)\right)\right)\right)\right)\right)\right)\right), \frac{-1}{6}\right)\right)\right)\right)\right)\right) \]

   3. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \left({x}^{6} \cdot \left(\frac{1}{1728000} + \frac{-1}{24192000} \cdot {x}^{2}\right) + \frac{-1}{216}\right)\right), \mathsf{+.f64}\left(\frac{1}{36}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{5040}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{362880}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{5040}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{362880}\right)\right)\right)\right)\right)\right)\right)\right), \frac{-1}{6}\right)\right)\right)\right)\right)\right) \]

   4. +-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{-1}{216} + {x}^{6} \cdot \left(\frac{1}{1728000} + \frac{-1}{24192000} \cdot {x}^{2}\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{36}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{5040}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{362880}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{5040}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{362880}\right)\right)\right)\right)\right)\right)\right)\right), \frac{-1}{6}\right)\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{216}, \left({x}^{6} \cdot \left(\frac{1}{1728000} + \frac{-1}{24192000} \cdot {x}^{2}\right)\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{36}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{5040}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{362880}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{5040}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{362880}\right)\right)\right)\right)\right)\right)\right)\right), \frac{-1}{6}\right)\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{216}, \mathsf{*.f64}\left(\left({x}^{6}\right), \left(\frac{1}{1728000} + \frac{-1}{24192000} \cdot {x}^{2}\right)\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{36}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{5040}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{362880}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{5040}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{362880}\right)\right)\right)\right)\right)\right)\right)\right), \frac{-1}{6}\right)\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{216}, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 6\right), \left(\frac{1}{1728000} + \frac{-1}{24192000} \cdot {x}^{2}\right)\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{36}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{5040}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{362880}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{5040}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{362880}\right)\right)\right)\right)\right)\right)\right)\right), \frac{-1}{6}\right)\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{216}, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 6\right), \mathsf{+.f64}\left(\frac{1}{1728000}, \left(\frac{-1}{24192000} \cdot {x}^{2}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{36}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{5040}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{362880}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{5040}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{362880}\right)\right)\right)\right)\right)\right)\right)\right), \frac{-1}{6}\right)\right)\right)\right)\right)\right) \]

   9. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{216}, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 6\right), \mathsf{+.f64}\left(\frac{1}{1728000}, \left(\frac{-1}{24192000} \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{36}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{5040}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{362880}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{5040}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{362880}\right)\right)\right)\right)\right)\right)\right)\right), \frac{-1}{6}\right)\right)\right)\right)\right)\right) \]

   10. associate-*r* N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{216}, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 6\right), \mathsf{+.f64}\left(\frac{1}{1728000}, \left(\left(\frac{-1}{24192000} \cdot x\right) \cdot x\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{36}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{5040}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{362880}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{5040}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{362880}\right)\right)\right)\right)\right)\right)\right)\right), \frac{-1}{6}\right)\right)\right)\right)\right)\right) \]

   11. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{216}, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 6\right), \mathsf{+.f64}\left(\frac{1}{1728000}, \left(x \cdot \left(\frac{-1}{24192000} \cdot x\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{36}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{5040}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{362880}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{5040}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{362880}\right)\right)\right)\right)\right)\right)\right)\right), \frac{-1}{6}\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{216}, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 6\right), \mathsf{+.f64}\left(\frac{1}{1728000}, \mathsf{*.f64}\left(x, \left(\frac{-1}{24192000} \cdot x\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{36}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{5040}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{362880}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{5040}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{362880}\right)\right)\right)\right)\right)\right)\right)\right), \frac{-1}{6}\right)\right)\right)\right)\right)\right) \]

   13. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{216}, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 6\right), \mathsf{+.f64}\left(\frac{1}{1728000}, \mathsf{*.f64}\left(x, \left(x \cdot \frac{-1}{24192000}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{36}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{5040}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{362880}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{5040}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{362880}\right)\right)\right)\right)\right)\right)\right)\right), \frac{-1}{6}\right)\right)\right)\right)\right)\right) \]

   14. *-lowering-*.f64 99.4%

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{216}, \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, 6\right), \mathsf{+.f64}\left(\frac{1}{1728000}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{-1}{24192000}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(\frac{1}{36}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{5040}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{362880}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{5040}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{362880}\right)\right)\right)\right)\right)\right)\right)\right), \frac{-1}{6}\right)\right)\right)\right)\right)\right) \]

10. Simplified 99.4%

    \[\leadsto x \cdot \left(x \cdot \frac{\color{blue}{x \cdot \left(-0.004629629629629629 + {x}^{6} \cdot \left(5.787037037037037 \cdot 10^{-7} + x \cdot \left(x \cdot -4.1335978835978836 \cdot 10^{-8}\right)\right)\right)}}{0.027777777777777776 + \left(x \cdot \left(x \cdot \left(0.008333333333333333 + x \cdot \left(x \cdot \left(-0.0001984126984126984 + x \cdot \left(x \cdot 2.7557319223985893 \cdot 10^{-6}\right)\right)\right)\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(0.008333333333333333 + x \cdot \left(x \cdot \left(-0.0001984126984126984 + x \cdot \left(x \cdot 2.7557319223985893 \cdot 10^{-6}\right)\right)\right)\right)\right) - -0.16666666666666666\right)}\right) \]

12. Applied egg-rr 99.4%

    \[\leadsto \color{blue}{\frac{x}{\frac{0.027777777777777776 + \left(x \cdot \left(x \cdot \left(0.008333333333333333 + \left(-0.0001984126984126984 + x \cdot \left(x \cdot 2.7557319223985893 \cdot 10^{-6}\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(0.008333333333333333 + \left(-0.0001984126984126984 + x \cdot \left(x \cdot 2.7557319223985893 \cdot 10^{-6}\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.16666666666666666\right)}{x \cdot \left(-0.004629629629629629 + \left(5.787037037037037 \cdot 10^{-7} + x \cdot \left(x \cdot -4.1335978835978836 \cdot 10^{-8}\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\right)}} \cdot x} \]

13. Final simplification 99.4%

    \[\leadsto x \cdot \frac{x}{\frac{0.027777777777777776 + \left(x \cdot \left(x \cdot \left(0.008333333333333333 + \left(-0.0001984126984126984 + x \cdot \left(x \cdot 2.7557319223985893 \cdot 10^{-6}\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(0.008333333333333333 + \left(-0.0001984126984126984 + x \cdot \left(x \cdot 2.7557319223985893 \cdot 10^{-6}\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.16666666666666666\right)}{x \cdot \left(-0.004629629629629629 + \left(5.787037037037037 \cdot 10^{-7} + x \cdot \left(x \cdot -4.1335978835978836 \cdot 10^{-8}\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\right)}} \]
14. Add Preprocessing

Alternative 2: 99.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(x \cdot \frac{x \cdot \left(0.027777777777777776 + \left(-6.944444444444444 \cdot 10^{-5} + \left(x \cdot x\right) \cdot \left(3.306878306878307 \cdot 10^{-6} + \left(x \cdot x\right) \cdot -8.529646426471823 \cdot 10^{-8}\right)\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)}{-0.16666666666666666 - x \cdot \left(x \cdot \left(0.008333333333333333 + x \cdot \left(x \cdot \left(-0.0001984126984126984 + x \cdot \left(x \cdot 2.7557319223985893 \cdot 10^{-6}\right)\right)\right)\right)\right)}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  x
  (*
   x
   (/
    (*
     x
     (+
      0.027777777777777776
      (*
       (+
        -6.944444444444444e-5
        (* (* x x) (+ 3.306878306878307e-6 (* (* x x) -8.529646426471823e-8))))
       (* x (* x (* x x))))))
    (-
     -0.16666666666666666
     (*
      x
      (*
       x
       (+
        0.008333333333333333
        (*
         x
         (*
          x
          (+
           -0.0001984126984126984
           (* x (* x 2.7557319223985893e-6)))))))))))))

C

double code(double x) {
	return x * (x * ((x * (0.027777777777777776 + ((-6.944444444444444e-5 + ((x * x) * (3.306878306878307e-6 + ((x * x) * -8.529646426471823e-8)))) * (x * (x * (x * x)))))) / (-0.16666666666666666 - (x * (x * (0.008333333333333333 + (x * (x * (-0.0001984126984126984 + (x * (x * 2.7557319223985893e-6)))))))))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * (x * ((x * (0.027777777777777776d0 + (((-6.944444444444444d-5) + ((x * x) * (3.306878306878307d-6 + ((x * x) * (-8.529646426471823d-8))))) * (x * (x * (x * x)))))) / ((-0.16666666666666666d0) - (x * (x * (0.008333333333333333d0 + (x * (x * ((-0.0001984126984126984d0) + (x * (x * 2.7557319223985893d-6)))))))))))
end function

Java

public static double code(double x) {
	return x * (x * ((x * (0.027777777777777776 + ((-6.944444444444444e-5 + ((x * x) * (3.306878306878307e-6 + ((x * x) * -8.529646426471823e-8)))) * (x * (x * (x * x)))))) / (-0.16666666666666666 - (x * (x * (0.008333333333333333 + (x * (x * (-0.0001984126984126984 + (x * (x * 2.7557319223985893e-6)))))))))));
}

Python

def code(x):
	return x * (x * ((x * (0.027777777777777776 + ((-6.944444444444444e-5 + ((x * x) * (3.306878306878307e-6 + ((x * x) * -8.529646426471823e-8)))) * (x * (x * (x * x)))))) / (-0.16666666666666666 - (x * (x * (0.008333333333333333 + (x * (x * (-0.0001984126984126984 + (x * (x * 2.7557319223985893e-6)))))))))))

Julia

function code(x)
	return Float64(x * Float64(x * Float64(Float64(x * Float64(0.027777777777777776 + Float64(Float64(-6.944444444444444e-5 + Float64(Float64(x * x) * Float64(3.306878306878307e-6 + Float64(Float64(x * x) * -8.529646426471823e-8)))) * Float64(x * Float64(x * Float64(x * x)))))) / Float64(-0.16666666666666666 - Float64(x * Float64(x * Float64(0.008333333333333333 + Float64(x * Float64(x * Float64(-0.0001984126984126984 + Float64(x * Float64(x * 2.7557319223985893e-6))))))))))))
end

MATLAB

function tmp = code(x)
	tmp = x * (x * ((x * (0.027777777777777776 + ((-6.944444444444444e-5 + ((x * x) * (3.306878306878307e-6 + ((x * x) * -8.529646426471823e-8)))) * (x * (x * (x * x)))))) / (-0.16666666666666666 - (x * (x * (0.008333333333333333 + (x * (x * (-0.0001984126984126984 + (x * (x * 2.7557319223985893e-6)))))))))));
end

Wolfram

code[x_] := N[(x * N[(x * N[(N[(x * N[(0.027777777777777776 + N[(N[(-6.944444444444444e-5 + N[(N[(x * x), $MachinePrecision] * N[(3.306878306878307e-6 + N[(N[(x * x), $MachinePrecision] * -8.529646426471823e-8), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-0.16666666666666666 - N[(x * N[(x * N[(0.008333333333333333 + N[(x * N[(x * N[(-0.0001984126984126984 + N[(x * N[(x * 2.7557319223985893e-6), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 66.1%

   \[\sin x - x \]
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%

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

   1. *-commutative N/A

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

   2. flip-+ N/A

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

   3. associate-*l/ N/A

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

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

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

7. Applied egg-rr 99.4%

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

8. Taylor expanded in x around 0

   \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\color{blue}{\left(x \cdot \left(\frac{1}{36} + {x}^{4} \cdot \left({x}^{2} \cdot \left(\frac{1}{302400} + \frac{-13}{152409600} \cdot {x}^{2}\right) - \frac{1}{14400}\right)\right)\right)}, \mathsf{\_.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{5040}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{362880}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]
9. Step-by-step derivation

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{1}{36} + {x}^{4} \cdot \left({x}^{2} \cdot \left(\frac{1}{302400} + \frac{-13}{152409600} \cdot {x}^{2}\right) - \frac{1}{14400}\right)\right)\right), \mathsf{\_.f64}\left(\color{blue}{\frac{-1}{6}}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{5040}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{362880}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{36}, \left({x}^{4} \cdot \left({x}^{2} \cdot \left(\frac{1}{302400} + \frac{-13}{152409600} \cdot {x}^{2}\right) - \frac{1}{14400}\right)\right)\right)\right), \mathsf{\_.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{5040}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{362880}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   3. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{36}, \left({x}^{\left(2 \cdot 2\right)} \cdot \left({x}^{2} \cdot \left(\frac{1}{302400} + \frac{-13}{152409600} \cdot {x}^{2}\right) - \frac{1}{14400}\right)\right)\right)\right), \mathsf{\_.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{5040}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{362880}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   4. pow-sqr N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{36}, \left(\left({x}^{2} \cdot {x}^{2}\right) \cdot \left({x}^{2} \cdot \left(\frac{1}{302400} + \frac{-13}{152409600} \cdot {x}^{2}\right) - \frac{1}{14400}\right)\right)\right)\right), \mathsf{\_.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{5040}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{362880}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{36}, \left(\left({x}^{2} \cdot \left(\frac{1}{302400} + \frac{-13}{152409600} \cdot {x}^{2}\right) - \frac{1}{14400}\right) \cdot \left({x}^{2} \cdot {x}^{2}\right)\right)\right)\right), \mathsf{\_.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{5040}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{362880}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{36}, \mathsf{*.f64}\left(\left({x}^{2} \cdot \left(\frac{1}{302400} + \frac{-13}{152409600} \cdot {x}^{2}\right) - \frac{1}{14400}\right), \left({x}^{2} \cdot {x}^{2}\right)\right)\right)\right), \mathsf{\_.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{5040}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{362880}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

10. Simplified 99.4%

    \[\leadsto x \cdot \left(x \cdot \frac{\color{blue}{x \cdot \left(0.027777777777777776 + \left(-6.944444444444444 \cdot 10^{-5} + \left(x \cdot x\right) \cdot \left(3.306878306878307 \cdot 10^{-6} + \left(x \cdot x\right) \cdot -8.529646426471823 \cdot 10^{-8}\right)\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)}}{-0.16666666666666666 - x \cdot \left(x \cdot \left(0.008333333333333333 + x \cdot \left(x \cdot \left(-0.0001984126984126984 + x \cdot \left(x \cdot 2.7557319223985893 \cdot 10^{-6}\right)\right)\right)\right)\right)}\right) \]
11. Add Preprocessing

Alternative 3: 99.0% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(\left(x \cdot x\right) \cdot \left(-0.16666666666666666 + x \cdot \left(x \cdot \left(0.008333333333333333 + x \cdot \left(x \cdot \left(-0.0001984126984126984 + x \cdot \left(x \cdot 2.7557319223985893 \cdot 10^{-6}\right)\right)\right)\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  x
  (*
   (* x x)
   (+
    -0.16666666666666666
    (*
     x
     (*
      x
      (+
       0.008333333333333333
       (*
        x
        (*
         x
         (+ -0.0001984126984126984 (* x (* x 2.7557319223985893e-6))))))))))))

C

double code(double x) {
	return x * ((x * x) * (-0.16666666666666666 + (x * (x * (0.008333333333333333 + (x * (x * (-0.0001984126984126984 + (x * (x * 2.7557319223985893e-6))))))))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * ((x * x) * ((-0.16666666666666666d0) + (x * (x * (0.008333333333333333d0 + (x * (x * ((-0.0001984126984126984d0) + (x * (x * 2.7557319223985893d-6))))))))))
end function

Java

public static double code(double x) {
	return x * ((x * x) * (-0.16666666666666666 + (x * (x * (0.008333333333333333 + (x * (x * (-0.0001984126984126984 + (x * (x * 2.7557319223985893e-6))))))))));
}

Python

def code(x):
	return x * ((x * x) * (-0.16666666666666666 + (x * (x * (0.008333333333333333 + (x * (x * (-0.0001984126984126984 + (x * (x * 2.7557319223985893e-6))))))))))

Julia

function code(x)
	return Float64(x * Float64(Float64(x * x) * Float64(-0.16666666666666666 + Float64(x * Float64(x * Float64(0.008333333333333333 + Float64(x * Float64(x * Float64(-0.0001984126984126984 + Float64(x * Float64(x * 2.7557319223985893e-6)))))))))))
end

MATLAB

function tmp = code(x)
	tmp = x * ((x * x) * (-0.16666666666666666 + (x * (x * (0.008333333333333333 + (x * (x * (-0.0001984126984126984 + (x * (x * 2.7557319223985893e-6))))))))));
end

Wolfram

code[x_] := N[(x * N[(N[(x * x), $MachinePrecision] * N[(-0.16666666666666666 + N[(x * N[(x * N[(0.008333333333333333 + N[(x * N[(x * N[(-0.0001984126984126984 + N[(x * N[(x * 2.7557319223985893e-6), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 66.1%

   \[\sin x - x \]
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%

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

   1. *-commutative N/A

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

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

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

7. Applied egg-rr 99.4%

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

8. Final simplification 99.4%

   \[\leadsto x \cdot \left(\left(x \cdot x\right) \cdot \left(-0.16666666666666666 + x \cdot \left(x \cdot \left(0.008333333333333333 + x \cdot \left(x \cdot \left(-0.0001984126984126984 + x \cdot \left(x \cdot 2.7557319223985893 \cdot 10^{-6}\right)\right)\right)\right)\right)\right)\right) \]
9. Add Preprocessing

Alternative 4: 99.0% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(x \cdot \left(x \cdot \left(-0.16666666666666666 + \left(x \cdot x\right) \cdot \left(0.008333333333333333 + \left(-0.0001984126984126984 + x \cdot \left(x \cdot 2.7557319223985893 \cdot 10^{-6}\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  x
  (*
   x
   (*
    x
    (+
     -0.16666666666666666
     (*
      (* x x)
      (+
       0.008333333333333333
       (*
        (+ -0.0001984126984126984 (* x (* x 2.7557319223985893e-6)))
        (* x x)))))))))

C

double code(double x) {
	return x * (x * (x * (-0.16666666666666666 + ((x * x) * (0.008333333333333333 + ((-0.0001984126984126984 + (x * (x * 2.7557319223985893e-6))) * (x * x)))))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * (x * (x * ((-0.16666666666666666d0) + ((x * x) * (0.008333333333333333d0 + (((-0.0001984126984126984d0) + (x * (x * 2.7557319223985893d-6))) * (x * x)))))))
end function

Java

public static double code(double x) {
	return x * (x * (x * (-0.16666666666666666 + ((x * x) * (0.008333333333333333 + ((-0.0001984126984126984 + (x * (x * 2.7557319223985893e-6))) * (x * x)))))));
}

Python

def code(x):
	return x * (x * (x * (-0.16666666666666666 + ((x * x) * (0.008333333333333333 + ((-0.0001984126984126984 + (x * (x * 2.7557319223985893e-6))) * (x * x)))))))

Julia

function code(x)
	return Float64(x * Float64(x * Float64(x * Float64(-0.16666666666666666 + Float64(Float64(x * x) * Float64(0.008333333333333333 + Float64(Float64(-0.0001984126984126984 + Float64(x * Float64(x * 2.7557319223985893e-6))) * Float64(x * x))))))))
end

MATLAB

function tmp = code(x)
	tmp = x * (x * (x * (-0.16666666666666666 + ((x * x) * (0.008333333333333333 + ((-0.0001984126984126984 + (x * (x * 2.7557319223985893e-6))) * (x * x)))))));
end

Wolfram

code[x_] := N[(x * N[(x * N[(x * N[(-0.16666666666666666 + N[(N[(x * x), $MachinePrecision] * N[(0.008333333333333333 + N[(N[(-0.0001984126984126984 + N[(x * N[(x * 2.7557319223985893e-6), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 66.1%

   \[\sin x - x \]
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%

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

6. Final simplification 99.3%

   \[\leadsto x \cdot \left(x \cdot \left(x \cdot \left(-0.16666666666666666 + \left(x \cdot x\right) \cdot \left(0.008333333333333333 + \left(-0.0001984126984126984 + x \cdot \left(x \cdot 2.7557319223985893 \cdot 10^{-6}\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right)\right) \]
7. Add Preprocessing

Alternative 5: 99.0% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(0.008333333333333333 + -0.0001984126984126984 \cdot \left(x \cdot x\right)\right)\right)\right) + \left(x \cdot x\right) \cdot -0.16666666666666666\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  x
  (+
   (*
    (* x x)
    (* x (* x (+ 0.008333333333333333 (* -0.0001984126984126984 (* x x))))))
   (* (* x x) -0.16666666666666666))))

C

double code(double x) {
	return x * (((x * x) * (x * (x * (0.008333333333333333 + (-0.0001984126984126984 * (x * x)))))) + ((x * x) * -0.16666666666666666));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * (((x * x) * (x * (x * (0.008333333333333333d0 + ((-0.0001984126984126984d0) * (x * x)))))) + ((x * x) * (-0.16666666666666666d0)))
end function

Java

public static double code(double x) {
	return x * (((x * x) * (x * (x * (0.008333333333333333 + (-0.0001984126984126984 * (x * x)))))) + ((x * x) * -0.16666666666666666));
}

Python

def code(x):
	return x * (((x * x) * (x * (x * (0.008333333333333333 + (-0.0001984126984126984 * (x * x)))))) + ((x * x) * -0.16666666666666666))

Julia

function code(x)
	return Float64(x * Float64(Float64(Float64(x * x) * Float64(x * Float64(x * Float64(0.008333333333333333 + Float64(-0.0001984126984126984 * Float64(x * x)))))) + Float64(Float64(x * x) * -0.16666666666666666)))
end

MATLAB

function tmp = code(x)
	tmp = x * (((x * x) * (x * (x * (0.008333333333333333 + (-0.0001984126984126984 * (x * x)))))) + ((x * x) * -0.16666666666666666));
end

Wolfram

code[x_] := N[(x * N[(N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * N[(0.008333333333333333 + N[(-0.0001984126984126984 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(x * x), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 66.1%

   \[\sin x - x \]
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%

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

6. Taylor expanded in x around 0

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

   1. unpow2 N/A

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

   2. associate-*l* N/A

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

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

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

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

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

   5. sub-neg N/A

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

   6. metadata-eval N/A

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

   7. +-commutative N/A

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

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

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

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

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

   10. unpow2 N/A

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

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

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

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

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

   13. *-commutative N/A

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

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

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

   15. unpow2 N/A

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

   16. *-lowering-*.f64 99.3%

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

8. Simplified 99.3%

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

   1. associate-*r* N/A

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

   2. +-commutative N/A

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

   3. distribute-rgt-in N/A

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

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

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

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

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

   6. associate-*l* N/A

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

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

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

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

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

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

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

   10. *-commutative N/A

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

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

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

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

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

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

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

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

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

   15. *-lowering-*.f64 99.3%

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

10. Applied egg-rr 99.3%

    \[\leadsto x \cdot \color{blue}{\left(\left(x \cdot \left(x \cdot \left(0.008333333333333333 + -0.0001984126984126984 \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(x \cdot x\right) + -0.16666666666666666 \cdot \left(x \cdot x\right)\right)} \]

11. Final simplification 99.3%

    \[\leadsto x \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(0.008333333333333333 + -0.0001984126984126984 \cdot \left(x \cdot x\right)\right)\right)\right) + \left(x \cdot x\right) \cdot -0.16666666666666666\right) \]
12. Add Preprocessing

Alternative 6: 99.0% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(\left(x \cdot x\right) \cdot \left(-0.16666666666666666 + x \cdot \left(x \cdot \left(0.008333333333333333 + -0.0001984126984126984 \cdot \left(x \cdot x\right)\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  x
  (*
   (* x x)
   (+
    -0.16666666666666666
    (* x (* x (+ 0.008333333333333333 (* -0.0001984126984126984 (* x x)))))))))

C

double code(double x) {
	return x * ((x * x) * (-0.16666666666666666 + (x * (x * (0.008333333333333333 + (-0.0001984126984126984 * (x * x)))))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * ((x * x) * ((-0.16666666666666666d0) + (x * (x * (0.008333333333333333d0 + ((-0.0001984126984126984d0) * (x * x)))))))
end function

Java

public static double code(double x) {
	return x * ((x * x) * (-0.16666666666666666 + (x * (x * (0.008333333333333333 + (-0.0001984126984126984 * (x * x)))))));
}

Python

def code(x):
	return x * ((x * x) * (-0.16666666666666666 + (x * (x * (0.008333333333333333 + (-0.0001984126984126984 * (x * x)))))))

Julia

function code(x)
	return Float64(x * Float64(Float64(x * x) * Float64(-0.16666666666666666 + Float64(x * Float64(x * Float64(0.008333333333333333 + Float64(-0.0001984126984126984 * Float64(x * x))))))))
end

MATLAB

function tmp = code(x)
	tmp = x * ((x * x) * (-0.16666666666666666 + (x * (x * (0.008333333333333333 + (-0.0001984126984126984 * (x * x)))))));
end

Wolfram

code[x_] := N[(x * N[(N[(x * x), $MachinePrecision] * N[(-0.16666666666666666 + N[(x * N[(x * N[(0.008333333333333333 + N[(-0.0001984126984126984 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 66.1%

   \[\sin x - x \]
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%

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

6. Taylor expanded in x around 0

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

   1. unpow2 N/A

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

   2. associate-*l* N/A

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

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

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

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

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

   5. sub-neg N/A

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

   6. metadata-eval N/A

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

   7. +-commutative N/A

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

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

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

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

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

   10. unpow2 N/A

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

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

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

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

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

   13. *-commutative N/A

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

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

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

   15. unpow2 N/A

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

   16. *-lowering-*.f64 99.3%

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

8. Simplified 99.3%

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

   1. *-commutative N/A

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

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

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

   3. associate-*r* N/A

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

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

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

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

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

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

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

   7. associate-*l* N/A

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

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

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

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

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

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

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

   11. *-commutative N/A

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

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

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

   13. *-lowering-*.f64 99.3%

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

10. Applied egg-rr 99.3%

    \[\leadsto \color{blue}{\left(\left(x \cdot x\right) \cdot \left(-0.16666666666666666 + x \cdot \left(x \cdot \left(0.008333333333333333 + -0.0001984126984126984 \cdot \left(x \cdot x\right)\right)\right)\right)\right) \cdot x} \]

11. Final simplification 99.3%

    \[\leadsto x \cdot \left(\left(x \cdot x\right) \cdot \left(-0.16666666666666666 + x \cdot \left(x \cdot \left(0.008333333333333333 + -0.0001984126984126984 \cdot \left(x \cdot x\right)\right)\right)\right)\right) \]
12. Add Preprocessing

Alternative 7: 99.0% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(-0.16666666666666666 + x \cdot \left(x \cdot \left(0.008333333333333333 + -0.0001984126984126984 \cdot \left(x \cdot x\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (* x (* x x))
  (+
   -0.16666666666666666
   (* x (* x (+ 0.008333333333333333 (* -0.0001984126984126984 (* x x))))))))

C

double code(double x) {
	return (x * (x * x)) * (-0.16666666666666666 + (x * (x * (0.008333333333333333 + (-0.0001984126984126984 * (x * x))))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (x * (x * x)) * ((-0.16666666666666666d0) + (x * (x * (0.008333333333333333d0 + ((-0.0001984126984126984d0) * (x * x))))))
end function

Java

public static double code(double x) {
	return (x * (x * x)) * (-0.16666666666666666 + (x * (x * (0.008333333333333333 + (-0.0001984126984126984 * (x * x))))));
}

Python

def code(x):
	return (x * (x * x)) * (-0.16666666666666666 + (x * (x * (0.008333333333333333 + (-0.0001984126984126984 * (x * x))))))

Julia

function code(x)
	return Float64(Float64(x * Float64(x * x)) * Float64(-0.16666666666666666 + Float64(x * Float64(x * Float64(0.008333333333333333 + Float64(-0.0001984126984126984 * Float64(x * x)))))))
end

MATLAB

function tmp = code(x)
	tmp = (x * (x * x)) * (-0.16666666666666666 + (x * (x * (0.008333333333333333 + (-0.0001984126984126984 * (x * x))))));
end

Wolfram

code[x_] := N[(N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(-0.16666666666666666 + N[(x * N[(x * N[(0.008333333333333333 + N[(-0.0001984126984126984 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 66.1%

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

3. Taylor expanded in x around 0

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

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

      \[\leadsto \mathsf{*.f64}\left(\left({x}^{3}\right), \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right)}\right) \]

   2. cube-mult N/A

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

   3. unpow2 N/A

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

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

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

   5. unpow2 N/A

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

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

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

   7. sub-neg N/A

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

   8. metadata-eval N/A

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

   9. +-commutative N/A

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

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

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

   11. unpow2 N/A

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

   12. associate-*l* N/A

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

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

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

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

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

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

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

   16. *-commutative N/A

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

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

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

   18. unpow2 N/A

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

   19. *-lowering-*.f64 99.3%

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

5. Simplified 99.3%

   \[\leadsto \color{blue}{\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(-0.16666666666666666 + x \cdot \left(x \cdot \left(0.008333333333333333 + \left(x \cdot x\right) \cdot -0.0001984126984126984\right)\right)\right)} \]

6. Final simplification 99.3%

   \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(-0.16666666666666666 + x \cdot \left(x \cdot \left(0.008333333333333333 + -0.0001984126984126984 \cdot \left(x \cdot x\right)\right)\right)\right) \]
7. Add Preprocessing

Alternative 8: 99.0% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(x \cdot \left(x \cdot \left(-0.16666666666666666 + \left(x \cdot x\right) \cdot \left(0.008333333333333333 + -0.0001984126984126984 \cdot \left(x \cdot x\right)\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  x
  (*
   x
   (*
    x
    (+
     -0.16666666666666666
     (*
      (* x x)
      (+ 0.008333333333333333 (* -0.0001984126984126984 (* x x)))))))))

C

double code(double x) {
	return x * (x * (x * (-0.16666666666666666 + ((x * x) * (0.008333333333333333 + (-0.0001984126984126984 * (x * x)))))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * (x * (x * ((-0.16666666666666666d0) + ((x * x) * (0.008333333333333333d0 + ((-0.0001984126984126984d0) * (x * x)))))))
end function

Java

public static double code(double x) {
	return x * (x * (x * (-0.16666666666666666 + ((x * x) * (0.008333333333333333 + (-0.0001984126984126984 * (x * x)))))));
}

Python

def code(x):
	return x * (x * (x * (-0.16666666666666666 + ((x * x) * (0.008333333333333333 + (-0.0001984126984126984 * (x * x)))))))

Julia

function code(x)
	return Float64(x * Float64(x * Float64(x * Float64(-0.16666666666666666 + Float64(Float64(x * x) * Float64(0.008333333333333333 + Float64(-0.0001984126984126984 * Float64(x * x))))))))
end

MATLAB

function tmp = code(x)
	tmp = x * (x * (x * (-0.16666666666666666 + ((x * x) * (0.008333333333333333 + (-0.0001984126984126984 * (x * x)))))));
end

Wolfram

code[x_] := N[(x * N[(x * N[(x * N[(-0.16666666666666666 + N[(N[(x * x), $MachinePrecision] * N[(0.008333333333333333 + N[(-0.0001984126984126984 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 66.1%

   \[\sin x - x \]
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%

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

6. Taylor expanded in x around 0

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

   1. unpow2 N/A

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

   2. associate-*l* N/A

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

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

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

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

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

   5. sub-neg N/A

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

   6. metadata-eval N/A

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

   7. +-commutative N/A

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

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

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

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

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

   10. unpow2 N/A

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

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

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

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

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

   13. *-commutative N/A

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

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

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

   15. unpow2 N/A

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

   16. *-lowering-*.f64 99.3%

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

8. Simplified 99.3%

   \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(x \cdot \left(-0.16666666666666666 + \left(x \cdot x\right) \cdot \left(0.008333333333333333 + \left(x \cdot x\right) \cdot -0.0001984126984126984\right)\right)\right)\right)} \]

9. Final simplification 99.3%

   \[\leadsto x \cdot \left(x \cdot \left(x \cdot \left(-0.16666666666666666 + \left(x \cdot x\right) \cdot \left(0.008333333333333333 + -0.0001984126984126984 \cdot \left(x \cdot x\right)\right)\right)\right)\right) \]
10. Add Preprocessing

Alternative 9: 98.9% accurate, 6.9× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(x \cdot \frac{x \cdot -0.004629629629629629}{0.027777777777777776 + \left(x \cdot x\right) \cdot 0.001388888888888889}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  x
  (*
   x
   (/
    (* x -0.004629629629629629)
    (+ 0.027777777777777776 (* (* x x) 0.001388888888888889))))))

C

double code(double x) {
	return x * (x * ((x * -0.004629629629629629) / (0.027777777777777776 + ((x * x) * 0.001388888888888889))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * (x * ((x * (-0.004629629629629629d0)) / (0.027777777777777776d0 + ((x * x) * 0.001388888888888889d0))))
end function

Java

public static double code(double x) {
	return x * (x * ((x * -0.004629629629629629) / (0.027777777777777776 + ((x * x) * 0.001388888888888889))));
}

Python

def code(x):
	return x * (x * ((x * -0.004629629629629629) / (0.027777777777777776 + ((x * x) * 0.001388888888888889))))

Julia

function code(x)
	return Float64(x * Float64(x * Float64(Float64(x * -0.004629629629629629) / Float64(0.027777777777777776 + Float64(Float64(x * x) * 0.001388888888888889)))))
end

MATLAB

function tmp = code(x)
	tmp = x * (x * ((x * -0.004629629629629629) / (0.027777777777777776 + ((x * x) * 0.001388888888888889))));
end

Wolfram

code[x_] := N[(x * N[(x * N[(N[(x * -0.004629629629629629), $MachinePrecision] / N[(0.027777777777777776 + N[(N[(x * x), $MachinePrecision] * 0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 66.1%

   \[\sin x - x \]
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%

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

   1. *-commutative N/A

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

   2. flip3-+ N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{{\frac{-1}{6}}^{3} + {\left(\left(x \cdot x\right) \cdot \left(\frac{1}{120} + \left(x \cdot x\right) \cdot \left(\frac{-1}{5040} + x \cdot \left(x \cdot \frac{1}{362880}\right)\right)\right)\right)}^{3}}{\frac{-1}{6} \cdot \frac{-1}{6} + \left(\left(\left(x \cdot x\right) \cdot \left(\frac{1}{120} + \left(x \cdot x\right) \cdot \left(\frac{-1}{5040} + x \cdot \left(x \cdot \frac{1}{362880}\right)\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{120} + \left(x \cdot x\right) \cdot \left(\frac{-1}{5040} + x \cdot \left(x \cdot \frac{1}{362880}\right)\right)\right)\right) - \frac{-1}{6} \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{120} + \left(x \cdot x\right) \cdot \left(\frac{-1}{5040} + x \cdot \left(x \cdot \frac{1}{362880}\right)\right)\right)\right)\right)} \cdot x\right)\right)\right) \]

   3. associate-*l/ N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{\left({\frac{-1}{6}}^{3} + {\left(\left(x \cdot x\right) \cdot \left(\frac{1}{120} + \left(x \cdot x\right) \cdot \left(\frac{-1}{5040} + x \cdot \left(x \cdot \frac{1}{362880}\right)\right)\right)\right)}^{3}\right) \cdot x}{\color{blue}{\frac{-1}{6} \cdot \frac{-1}{6} + \left(\left(\left(x \cdot x\right) \cdot \left(\frac{1}{120} + \left(x \cdot x\right) \cdot \left(\frac{-1}{5040} + x \cdot \left(x \cdot \frac{1}{362880}\right)\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{120} + \left(x \cdot x\right) \cdot \left(\frac{-1}{5040} + x \cdot \left(x \cdot \frac{1}{362880}\right)\right)\right)\right) - \frac{-1}{6} \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{120} + \left(x \cdot x\right) \cdot \left(\frac{-1}{5040} + x \cdot \left(x \cdot \frac{1}{362880}\right)\right)\right)\right)\right)}}\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\left(\left({\frac{-1}{6}}^{3} + {\left(\left(x \cdot x\right) \cdot \left(\frac{1}{120} + \left(x \cdot x\right) \cdot \left(\frac{-1}{5040} + x \cdot \left(x \cdot \frac{1}{362880}\right)\right)\right)\right)}^{3}\right) \cdot x\right), \color{blue}{\left(\frac{-1}{6} \cdot \frac{-1}{6} + \left(\left(\left(x \cdot x\right) \cdot \left(\frac{1}{120} + \left(x \cdot x\right) \cdot \left(\frac{-1}{5040} + x \cdot \left(x \cdot \frac{1}{362880}\right)\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{120} + \left(x \cdot x\right) \cdot \left(\frac{-1}{5040} + x \cdot \left(x \cdot \frac{1}{362880}\right)\right)\right)\right) - \frac{-1}{6} \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{120} + \left(x \cdot x\right) \cdot \left(\frac{-1}{5040} + x \cdot \left(x \cdot \frac{1}{362880}\right)\right)\right)\right)\right)\right)}\right)\right)\right) \]

7. Applied egg-rr 99.4%

   \[\leadsto x \cdot \left(x \cdot \color{blue}{\frac{\left(-0.004629629629629629 + \left(x \cdot \left(x \cdot \left(0.008333333333333333 + x \cdot \left(x \cdot \left(-0.0001984126984126984 + x \cdot \left(x \cdot 2.7557319223985893 \cdot 10^{-6}\right)\right)\right)\right)\right)\right) \cdot \left(\left(x \cdot \left(x \cdot \left(0.008333333333333333 + x \cdot \left(x \cdot \left(-0.0001984126984126984 + x \cdot \left(x \cdot 2.7557319223985893 \cdot 10^{-6}\right)\right)\right)\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(0.008333333333333333 + x \cdot \left(x \cdot \left(-0.0001984126984126984 + x \cdot \left(x \cdot 2.7557319223985893 \cdot 10^{-6}\right)\right)\right)\right)\right)\right)\right)\right) \cdot x}{0.027777777777777776 + \left(x \cdot \left(x \cdot \left(0.008333333333333333 + x \cdot \left(x \cdot \left(-0.0001984126984126984 + x \cdot \left(x \cdot 2.7557319223985893 \cdot 10^{-6}\right)\right)\right)\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(0.008333333333333333 + x \cdot \left(x \cdot \left(-0.0001984126984126984 + x \cdot \left(x \cdot 2.7557319223985893 \cdot 10^{-6}\right)\right)\right)\right)\right) - -0.16666666666666666\right)}}\right) \]

8. Taylor expanded in x around 0

   \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\color{blue}{\left(\frac{-1}{216} \cdot x\right)}, \mathsf{+.f64}\left(\frac{1}{36}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{5040}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{362880}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{5040}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{362880}\right)\right)\right)\right)\right)\right)\right)\right), \frac{-1}{6}\right)\right)\right)\right)\right)\right) \]
9. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\left(x \cdot \frac{-1}{216}\right), \mathsf{+.f64}\left(\color{blue}{\frac{1}{36}}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{5040}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{362880}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{5040}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{362880}\right)\right)\right)\right)\right)\right)\right)\right), \frac{-1}{6}\right)\right)\right)\right)\right)\right) \]

   2. *-lowering-*.f64 99.3%

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \frac{-1}{216}\right), \mathsf{+.f64}\left(\color{blue}{\frac{1}{36}}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{5040}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{362880}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{5040}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{362880}\right)\right)\right)\right)\right)\right)\right)\right), \frac{-1}{6}\right)\right)\right)\right)\right)\right) \]

10. Simplified 99.3%

    \[\leadsto x \cdot \left(x \cdot \frac{\color{blue}{x \cdot -0.004629629629629629}}{0.027777777777777776 + \left(x \cdot \left(x \cdot \left(0.008333333333333333 + x \cdot \left(x \cdot \left(-0.0001984126984126984 + x \cdot \left(x \cdot 2.7557319223985893 \cdot 10^{-6}\right)\right)\right)\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(0.008333333333333333 + x \cdot \left(x \cdot \left(-0.0001984126984126984 + x \cdot \left(x \cdot 2.7557319223985893 \cdot 10^{-6}\right)\right)\right)\right)\right) - -0.16666666666666666\right)}\right) \]

11. Taylor expanded in x around 0

    \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \frac{-1}{216}\right), \mathsf{+.f64}\left(\frac{1}{36}, \color{blue}{\left(\frac{1}{720} \cdot {x}^{2}\right)}\right)\right)\right)\right) \]
12. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \frac{-1}{216}\right), \mathsf{+.f64}\left(\frac{1}{36}, \left({x}^{2} \cdot \color{blue}{\frac{1}{720}}\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \frac{-1}{216}\right), \mathsf{+.f64}\left(\frac{1}{36}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{1}{720}}\right)\right)\right)\right)\right) \]

    3. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \frac{-1}{216}\right), \mathsf{+.f64}\left(\frac{1}{36}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{720}\right)\right)\right)\right)\right) \]

    4. *-lowering-*.f64 99.1%

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \frac{-1}{216}\right), \mathsf{+.f64}\left(\frac{1}{36}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{720}\right)\right)\right)\right)\right) \]

13. Simplified 99.1%

    \[\leadsto x \cdot \left(x \cdot \frac{x \cdot -0.004629629629629629}{0.027777777777777776 + \color{blue}{\left(x \cdot x\right) \cdot 0.001388888888888889}}\right) \]
14. Add Preprocessing

Alternative 10: 98.8% accurate, 7.9× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(\left(x \cdot x\right) \cdot \left(-0.16666666666666666 + 0.008333333333333333 \cdot \left(x \cdot x\right)\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* x (* (* x x) (+ -0.16666666666666666 (* 0.008333333333333333 (* x x))))))

C

double code(double x) {
	return x * ((x * x) * (-0.16666666666666666 + (0.008333333333333333 * (x * x))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * ((x * x) * ((-0.16666666666666666d0) + (0.008333333333333333d0 * (x * x))))
end function

Java

public static double code(double x) {
	return x * ((x * x) * (-0.16666666666666666 + (0.008333333333333333 * (x * x))));
}

Python

def code(x):
	return x * ((x * x) * (-0.16666666666666666 + (0.008333333333333333 * (x * x))))

Julia

function code(x)
	return Float64(x * Float64(Float64(x * x) * Float64(-0.16666666666666666 + Float64(0.008333333333333333 * Float64(x * x)))))
end

MATLAB

function tmp = code(x)
	tmp = x * ((x * x) * (-0.16666666666666666 + (0.008333333333333333 * (x * x))));
end

Wolfram

code[x_] := N[(x * N[(N[(x * x), $MachinePrecision] * N[(-0.16666666666666666 + N[(0.008333333333333333 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 66.1%

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

3. Taylor expanded in x around 0

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

   1. cube-mult N/A

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

   2. unpow2 N/A

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

   3. associate-*r* N/A

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

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

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

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

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

   6. unpow2 N/A

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

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

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

   8. sub-neg N/A

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

   9. metadata-eval N/A

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

   10. +-commutative N/A

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

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

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

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

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

   13. unpow2 N/A

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

   14. *-lowering-*.f64 99.1%

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

5. Simplified 99.1%

   \[\leadsto \color{blue}{x \cdot \left(\left(x \cdot x\right) \cdot \left(-0.16666666666666666 + 0.008333333333333333 \cdot \left(x \cdot x\right)\right)\right)} \]
6. Add Preprocessing

Alternative 11: 98.3% 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 66.1%

   \[\sin x - x \]
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. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{-1}{6}, \color{blue}{\left({x}^{3}\right)}\right) \]

   2. cube-mult N/A

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

   3. unpow2 N/A

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

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

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

   5. unpow2 N/A

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

   6. *-lowering-*.f64 98.9%

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

5. Simplified 98.9%

   \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left(x \cdot \left(x \cdot x\right)\right)} \]
6. Step-by-step derivation

   1. associate-*r* N/A

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

   2. associate-*r* N/A

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

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

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

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

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

   5. *-lowering-*.f64 98.9%

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

7. Applied egg-rr 98.9%

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

8. Final simplification 98.9%

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

Alternative 12: 98.3% accurate, 14.7× speedup

Math

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

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(Float64(x * Float64(x * x)) * -0.16666666666666666)
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 66.1%

   \[\sin x - x \]
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. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{-1}{6}, \color{blue}{\left({x}^{3}\right)}\right) \]

   2. cube-mult N/A

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

   3. unpow2 N/A

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

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

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

   5. unpow2 N/A

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

   6. *-lowering-*.f64 98.9%

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

5. Simplified 98.9%

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

6. Final simplification 98.9%

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

Alternative 13: 67.5% 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 66.1%

   \[\sin x - x \]
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 64.9%

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

   1. +-inverses 64.9%

      \[\leadsto 0 \]

7. Applied egg-rr 64.9%

   \[\leadsto \color{blue}{0} \]
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 2024148 
(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))