bug500 (missed optimization)

Percentage Accurate: 69.2% → 98.8%

Time: 6.3s

Alternatives: 9

Speedup: 6.5×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 69.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 2.7557319223985893 \cdot 10^{-6}, -0.0001984126984126984\right), x \cdot x, 0.008333333333333333\right)\\ \left(\frac{\mathsf{fma}\left({t\_0}^{2}, {x}^{4}, -0.027777777777777776\right) \cdot x}{\mathsf{fma}\left(t\_0, x \cdot x, 0.16666666666666666\right)} \cdot x\right) \cdot x \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0
         (fma
          (fma (* x x) 2.7557319223985893e-6 -0.0001984126984126984)
          (* x x)
          0.008333333333333333)))
   (*
    (*
     (/
      (* (fma (pow t_0 2.0) (pow x 4.0) -0.027777777777777776) x)
      (fma t_0 (* x x) 0.16666666666666666))
     x)
    x)))

C

double code(double x) {
	double t_0 = fma(fma((x * x), 2.7557319223985893e-6, -0.0001984126984126984), (x * x), 0.008333333333333333);
	return (((fma(pow(t_0, 2.0), pow(x, 4.0), -0.027777777777777776) * x) / fma(t_0, (x * x), 0.16666666666666666)) * x) * x;
}

Julia

function code(x)
	t_0 = fma(fma(Float64(x * x), 2.7557319223985893e-6, -0.0001984126984126984), Float64(x * x), 0.008333333333333333)
	return Float64(Float64(Float64(Float64(fma((t_0 ^ 2.0), (x ^ 4.0), -0.027777777777777776) * x) / fma(t_0, Float64(x * x), 0.16666666666666666)) * x) * x)
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(N[(x * x), $MachinePrecision] * 2.7557319223985893e-6 + -0.0001984126984126984), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.008333333333333333), $MachinePrecision]}, N[(N[(N[(N[(N[(N[Power[t$95$0, 2.0], $MachinePrecision] * N[Power[x, 4.0], $MachinePrecision] + -0.027777777777777776), $MachinePrecision] * x), $MachinePrecision] / N[(t$95$0 * N[(x * x), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Initial program 73.7%

   \[\sin x - x \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \color{blue}{\sin x - x} \]

   2. flip-- N/A

      \[\leadsto \color{blue}{\frac{\sin x \cdot \sin x - x \cdot x}{\sin x + x}} \]

   3. div-sub N/A

      \[\leadsto \color{blue}{\frac{\sin x \cdot \sin x}{\sin x + x} - \frac{x \cdot x}{\sin x + x}} \]

   4. lift-sin.f64 N/A

      \[\leadsto \frac{\color{blue}{\sin x} \cdot \sin x}{\sin x + x} - \frac{x \cdot x}{\sin x + x} \]

   5. lift-sin.f64 N/A

      \[\leadsto \frac{\sin x \cdot \color{blue}{\sin x}}{\sin x + x} - \frac{x \cdot x}{\sin x + x} \]

   6. sin-mult N/A

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

   7. associate-/l/ N/A

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

   8. frac-sub N/A

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

   9. lower-/.f64 N/A

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

4. Applied rewrites 21.9%

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

5. Taylor expanded in x around inf

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

   1. *-commutative N/A

      \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \frac{-2 \cdot \sin x - -4 \cdot \sin x}{x} - 1\right) \cdot x} \]

   2. distribute-rgt-out-- N/A

      \[\leadsto \left(\frac{1}{2} \cdot \frac{\color{blue}{\sin x \cdot \left(-2 - -4\right)}}{x} - 1\right) \cdot x \]

   3. metadata-eval N/A

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

   4. *-commutative N/A

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

   5. associate-*r/ N/A

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

   6. associate-*r* N/A

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

   7. metadata-eval N/A

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

   8. *-lft-identity N/A

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

   9. lower-*.f64 N/A

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

   10. lower--.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lower-sin.f64 73.8

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

7. Applied rewrites 73.8%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 99.5%

    \[\leadsto \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(2.7557319223985893 \cdot 10^{-6}, x \cdot x, -0.0001984126984126984\right), x \cdot x, 0.008333333333333333\right), x \cdot x, -0.16666666666666666\right) \cdot x\right) \cdot x\right) \cdot x \]
11. Step-by-step derivation

12. Applied rewrites 99.5%

    \[\leadsto \left(\frac{\mathsf{fma}\left({\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 2.7557319223985893 \cdot 10^{-6}, -0.0001984126984126984\right), x \cdot x, 0.008333333333333333\right)\right)}^{2}, {x}^{4}, -0.027777777777777776\right) \cdot x}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 2.7557319223985893 \cdot 10^{-6}, -0.0001984126984126984\right), x \cdot x, 0.008333333333333333\right), x \cdot x, 0.16666666666666666\right)} \cdot x\right) \cdot x \]
13. Add Preprocessing

Alternative 2: 98.8% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 73.7%

   \[\sin x - x \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \color{blue}{\sin x - x} \]

   2. flip-- N/A

      \[\leadsto \color{blue}{\frac{\sin x \cdot \sin x - x \cdot x}{\sin x + x}} \]

   3. div-sub N/A

      \[\leadsto \color{blue}{\frac{\sin x \cdot \sin x}{\sin x + x} - \frac{x \cdot x}{\sin x + x}} \]

   4. lift-sin.f64 N/A

      \[\leadsto \frac{\color{blue}{\sin x} \cdot \sin x}{\sin x + x} - \frac{x \cdot x}{\sin x + x} \]

   5. lift-sin.f64 N/A

      \[\leadsto \frac{\sin x \cdot \color{blue}{\sin x}}{\sin x + x} - \frac{x \cdot x}{\sin x + x} \]

   6. sin-mult N/A

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

   7. associate-/l/ N/A

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

   8. frac-sub N/A

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

   9. lower-/.f64 N/A

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

4. Applied rewrites 21.9%

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

5. Taylor expanded in x around inf

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

   1. *-commutative N/A

      \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \frac{-2 \cdot \sin x - -4 \cdot \sin x}{x} - 1\right) \cdot x} \]

   2. distribute-rgt-out-- N/A

      \[\leadsto \left(\frac{1}{2} \cdot \frac{\color{blue}{\sin x \cdot \left(-2 - -4\right)}}{x} - 1\right) \cdot x \]

   3. metadata-eval N/A

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

   4. *-commutative N/A

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

   5. associate-*r/ N/A

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

   6. associate-*r* N/A

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

   7. metadata-eval N/A

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

   8. *-lft-identity N/A

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

   9. lower-*.f64 N/A

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

   10. lower--.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lower-sin.f64 73.8

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

7. Applied rewrites 73.8%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 99.5%

    \[\leadsto \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(2.7557319223985893 \cdot 10^{-6}, x \cdot x, -0.0001984126984126984\right), x \cdot x, 0.008333333333333333\right), x \cdot x, -0.16666666666666666\right) \cdot x\right) \cdot x\right) \cdot x \]
11. Add Preprocessing

Alternative 3: 98.8% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 73.7%

   \[\sin x - x \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \color{blue}{\sin x - x} \]

   2. flip-- N/A

      \[\leadsto \color{blue}{\frac{\sin x \cdot \sin x - x \cdot x}{\sin x + x}} \]

   3. div-sub N/A

      \[\leadsto \color{blue}{\frac{\sin x \cdot \sin x}{\sin x + x} - \frac{x \cdot x}{\sin x + x}} \]

   4. lift-sin.f64 N/A

      \[\leadsto \frac{\color{blue}{\sin x} \cdot \sin x}{\sin x + x} - \frac{x \cdot x}{\sin x + x} \]

   5. lift-sin.f64 N/A

      \[\leadsto \frac{\sin x \cdot \color{blue}{\sin x}}{\sin x + x} - \frac{x \cdot x}{\sin x + x} \]

   6. sin-mult N/A

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

   7. associate-/l/ N/A

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

   8. frac-sub N/A

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

   9. lower-/.f64 N/A

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

4. Applied rewrites 21.9%

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

5. Taylor expanded in x around inf

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

   1. *-commutative N/A

      \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \frac{-2 \cdot \sin x - -4 \cdot \sin x}{x} - 1\right) \cdot x} \]

   2. distribute-rgt-out-- N/A

      \[\leadsto \left(\frac{1}{2} \cdot \frac{\color{blue}{\sin x \cdot \left(-2 - -4\right)}}{x} - 1\right) \cdot x \]

   3. metadata-eval N/A

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

   4. *-commutative N/A

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

   5. associate-*r/ N/A

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

   6. associate-*r* N/A

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

   7. metadata-eval N/A

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

   8. *-lft-identity N/A

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

   9. lower-*.f64 N/A

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

   10. lower--.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lower-sin.f64 73.8

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

7. Applied rewrites 73.8%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 99.4%

    \[\leadsto \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, x \cdot x, 0.008333333333333333\right), x \cdot x, -0.16666666666666666\right) \cdot x\right) \cdot x\right) \cdot x \]
11. Add Preprocessing

Alternative 4: 98.5% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 73.7%

   \[\sin x - x \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \color{blue}{\sin x - x} \]

   2. flip-- N/A

      \[\leadsto \color{blue}{\frac{\sin x \cdot \sin x - x \cdot x}{\sin x + x}} \]

   3. div-sub N/A

      \[\leadsto \color{blue}{\frac{\sin x \cdot \sin x}{\sin x + x} - \frac{x \cdot x}{\sin x + x}} \]

   4. lift-sin.f64 N/A

      \[\leadsto \frac{\color{blue}{\sin x} \cdot \sin x}{\sin x + x} - \frac{x \cdot x}{\sin x + x} \]

   5. lift-sin.f64 N/A

      \[\leadsto \frac{\sin x \cdot \color{blue}{\sin x}}{\sin x + x} - \frac{x \cdot x}{\sin x + x} \]

   6. sin-mult N/A

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

   7. associate-/l/ N/A

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

   8. frac-sub N/A

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

   9. lower-/.f64 N/A

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

4. Applied rewrites 21.9%

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

5. Taylor expanded in x around inf

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

   1. *-commutative N/A

      \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \frac{-2 \cdot \sin x - -4 \cdot \sin x}{x} - 1\right) \cdot x} \]

   2. distribute-rgt-out-- N/A

      \[\leadsto \left(\frac{1}{2} \cdot \frac{\color{blue}{\sin x \cdot \left(-2 - -4\right)}}{x} - 1\right) \cdot x \]

   3. metadata-eval N/A

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

   4. *-commutative N/A

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

   5. associate-*r/ N/A

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

   6. associate-*r* N/A

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

   7. metadata-eval N/A

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

   8. *-lft-identity N/A

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

   9. lower-*.f64 N/A

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

   10. lower--.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lower-sin.f64 73.8

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

7. Applied rewrites 73.8%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 99.3%

    \[\leadsto \left(\left(\mathsf{fma}\left(x \cdot x, 0.008333333333333333, -0.16666666666666666\right) \cdot x\right) \cdot x\right) \cdot x \]
11. Add Preprocessing

Alternative 5: 98.1% accurate, 6.5× speedup

Math

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

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.7%

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

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

      \[\leadsto \color{blue}{{x}^{3} \cdot \frac{-1}{6}} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{{x}^{3} \cdot \frac{-1}{6}} \]

   3. lower-pow.f64 99.3

      \[\leadsto \color{blue}{{x}^{3}} \cdot -0.16666666666666666 \]

5. Applied rewrites 99.3%

   \[\leadsto \color{blue}{{x}^{3} \cdot -0.16666666666666666} \]
6. Step-by-step derivation

7. Applied rewrites 99.2%

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

Alternative 6: 98.1% accurate, 6.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.7%

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

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

      \[\leadsto \color{blue}{{x}^{3} \cdot \frac{-1}{6}} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{{x}^{3} \cdot \frac{-1}{6}} \]

   3. lower-pow.f64 99.3

      \[\leadsto \color{blue}{{x}^{3}} \cdot -0.16666666666666666 \]

5. Applied rewrites 99.3%

   \[\leadsto \color{blue}{{x}^{3} \cdot -0.16666666666666666} \]
6. Step-by-step derivation

7. Applied rewrites 99.2%

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

9. Applied rewrites 99.2%

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

Alternative 7: 98.1% accurate, 6.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.7%

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

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

      \[\leadsto \color{blue}{{x}^{3} \cdot \frac{-1}{6}} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{{x}^{3} \cdot \frac{-1}{6}} \]

   3. lower-pow.f64 99.3

      \[\leadsto \color{blue}{{x}^{3}} \cdot -0.16666666666666666 \]

5. Applied rewrites 99.3%

   \[\leadsto \color{blue}{{x}^{3} \cdot -0.16666666666666666} \]
6. Step-by-step derivation

7. Applied rewrites 99.2%

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

Alternative 8: 66.9% accurate, 11.6× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (* (- 1.0 1.0) x))

C

double code(double x) {
	return (1.0 - 1.0) * x;
}

Fortran

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

Java

public static double code(double x) {
	return (1.0 - 1.0) * x;
}

Python

def code(x):
	return (1.0 - 1.0) * x

Julia

function code(x)
	return Float64(Float64(1.0 - 1.0) * x)
end

MATLAB

function tmp = code(x)
	tmp = (1.0 - 1.0) * x;
end

Wolfram

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

Derivation

1. Initial program 73.7%

   \[\sin x - x \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \color{blue}{\sin x - x} \]

   2. flip-- N/A

      \[\leadsto \color{blue}{\frac{\sin x \cdot \sin x - x \cdot x}{\sin x + x}} \]

   3. div-sub N/A

      \[\leadsto \color{blue}{\frac{\sin x \cdot \sin x}{\sin x + x} - \frac{x \cdot x}{\sin x + x}} \]

   4. lift-sin.f64 N/A

      \[\leadsto \frac{\color{blue}{\sin x} \cdot \sin x}{\sin x + x} - \frac{x \cdot x}{\sin x + x} \]

   5. lift-sin.f64 N/A

      \[\leadsto \frac{\sin x \cdot \color{blue}{\sin x}}{\sin x + x} - \frac{x \cdot x}{\sin x + x} \]

   6. sin-mult N/A

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

   7. associate-/l/ N/A

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

   8. frac-sub N/A

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

   9. lower-/.f64 N/A

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

4. Applied rewrites 21.9%

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

5. Taylor expanded in x around inf

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

   1. *-commutative N/A

      \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \frac{-2 \cdot \sin x - -4 \cdot \sin x}{x} - 1\right) \cdot x} \]

   2. distribute-rgt-out-- N/A

      \[\leadsto \left(\frac{1}{2} \cdot \frac{\color{blue}{\sin x \cdot \left(-2 - -4\right)}}{x} - 1\right) \cdot x \]

   3. metadata-eval N/A

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

   4. *-commutative N/A

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

   5. associate-*r/ N/A

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

   6. associate-*r* N/A

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

   7. metadata-eval N/A

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

   8. *-lft-identity N/A

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

   9. lower-*.f64 N/A

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

   10. lower--.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lower-sin.f64 73.8

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

7. Applied rewrites 73.8%

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

8. Taylor expanded in x around 0

   \[\leadsto \left(1 - 1\right) \cdot x \]
9. Step-by-step derivation

10. Applied rewrites 72.9%

    \[\leadsto \left(1 - 1\right) \cdot x \]
11. Add Preprocessing

Alternative 9: 6.5% accurate, 34.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return -x

Julia

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

MATLAB

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

Wolfram

code[x_] := (-x)

Derivation

1. Initial program 73.7%

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

3. Taylor expanded in x around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 6.2

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

5. Applied rewrites 6.2%

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

Developer Target 1: 99.8% accurate, 0.3× 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 2024282 
(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))