bug500 (missed optimization)

Percentage Accurate: 69.1% → 98.8%

Time: 8.3s

Alternatives: 9

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: 69.1% 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, 4.1× speedup

Math

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

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(Float64(x * Float64(x * x)) * Float64(-0.16666666666666666 + Float64(Float64(x * 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[(N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(-0.16666666666666666 + N[(N[(x * x), $MachinePrecision] * 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]

Derivation

1. Initial program 71.2%

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

3. Taylor expanded in x around 0

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

   1. cube-mult N/A

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

   2. unpow2 N/A

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

   3. associate-*l* N/A

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

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

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

   5. unpow2 N/A

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

   6. associate-*l* N/A

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

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

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

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

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

   9. sub-neg N/A

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

   10. metadata-eval N/A

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

   11. +-commutative N/A

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

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

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

5. Simplified 99.3%

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

   1. associate-*r* N/A

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

   2. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

   14. *-lowering-*.f64 99.3%

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

7. Applied egg-rr 99.3%

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

Alternative 2: 98.8% accurate, 4.1× speedup

Math

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

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(Float64(x * 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[(N[(x * x), $MachinePrecision] * 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]

Derivation

1. Initial program 71.2%

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

3. Taylor expanded in x around 0

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

   1. cube-mult N/A

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

   2. unpow2 N/A

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

   3. associate-*l* N/A

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

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

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

   5. unpow2 N/A

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

   6. associate-*l* N/A

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

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

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

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

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

   9. sub-neg N/A

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

   10. metadata-eval N/A

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

   11. +-commutative N/A

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

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

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

5. Simplified 99.3%

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

   13. *-lowering-*.f64 99.3%

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

7. Applied egg-rr 99.3%

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

8. Final simplification 99.3%

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

Alternative 3: 98.8% accurate, 4.1× speedup

Math

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

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(x * Float64(x * Float64(-0.16666666666666666 + Float64(Float64(x * 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[(x * N[(x * N[(-0.16666666666666666 + N[(N[(x * x), $MachinePrecision] * 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 71.2%

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

3. Taylor expanded in x around 0

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

   1. cube-mult N/A

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

   2. unpow2 N/A

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

   3. associate-*l* N/A

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

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

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

   5. unpow2 N/A

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

   6. associate-*l* N/A

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

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

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

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

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

   9. sub-neg N/A

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

   10. metadata-eval N/A

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

   11. +-commutative N/A

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

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

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

5. Simplified 99.3%

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

Alternative 4: 98.8% accurate, 5.4× speedup

Math

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

FPCore

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

C

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

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)))))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.2%

   \[\sin x - x \]
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. unpow2 N/A

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

   18. associate-*l* N/A

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

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

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

   20. *-lowering-*.f64 99.2%

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

5. Simplified 99.2%

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

Alternative 5: 98.7% accurate, 6.9× speedup

Math

\[\begin{array}{l} \\ \frac{\left(x \cdot \left(x \cdot x\right)\right) \cdot 0.027777777777777776}{-0.16666666666666666 - x \cdot \left(x \cdot 0.008333333333333333\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  (* (* x (* x x)) 0.027777777777777776)
  (- -0.16666666666666666 (* x (* x 0.008333333333333333)))))

C

double code(double x) {
	return ((x * (x * x)) * 0.027777777777777776) / (-0.16666666666666666 - (x * (x * 0.008333333333333333)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((x * (x * x)) * 0.027777777777777776d0) / ((-0.16666666666666666d0) - (x * (x * 0.008333333333333333d0)))
end function

Java

public static double code(double x) {
	return ((x * (x * x)) * 0.027777777777777776) / (-0.16666666666666666 - (x * (x * 0.008333333333333333)));
}

Python

def code(x):
	return ((x * (x * x)) * 0.027777777777777776) / (-0.16666666666666666 - (x * (x * 0.008333333333333333)))

Julia

function code(x)
	return Float64(Float64(Float64(x * Float64(x * x)) * 0.027777777777777776) / Float64(-0.16666666666666666 - Float64(x * Float64(x * 0.008333333333333333))))
end

MATLAB

function tmp = code(x)
	tmp = ((x * (x * x)) * 0.027777777777777776) / (-0.16666666666666666 - (x * (x * 0.008333333333333333)));
end

Wolfram

code[x_] := N[(N[(N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] * 0.027777777777777776), $MachinePrecision] / N[(-0.16666666666666666 - N[(x * N[(x * 0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 71.2%

   \[\sin x - x \]
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. unpow2 N/A

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

   13. associate-*r* N/A

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

   14. *-commutative N/A

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

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

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

   16. *-commutative N/A

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

   17. *-lowering-*.f64 98.8%

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

5. Simplified 98.8%

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

   1. associate-*r* N/A

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

   2. flip-+ N/A

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

   3. associate-*r/ N/A

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

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

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

7. Applied egg-rr 98.8%

   \[\leadsto \color{blue}{\frac{\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(0.027777777777777776 - 6.944444444444444 \cdot 10^{-5} \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot x\right)\right)}{-0.16666666666666666 - x \cdot \left(x \cdot 0.008333333333333333\right)}} \]

8. Taylor expanded in x around 0

   \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{1}{36} \cdot {x}^{3}\right)}, \mathsf{\_.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{120}\right)\right)\right)\right) \]
9. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left({x}^{3} \cdot \frac{1}{36}\right), \mathsf{\_.f64}\left(\color{blue}{\frac{-1}{6}}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{120}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({x}^{3}\right), \frac{1}{36}\right), \mathsf{\_.f64}\left(\color{blue}{\frac{-1}{6}}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{120}\right)\right)\right)\right) \]

   3. cube-mult N/A

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

   4. unpow2 N/A

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

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left({x}^{2}\right)\right), \frac{1}{36}\right), \mathsf{\_.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{120}\right)\right)\right)\right) \]

   6. unpow2 N/A

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

   7. *-lowering-*.f64 98.8%

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

10. Simplified 98.8%

    \[\leadsto \frac{\color{blue}{\left(x \cdot \left(x \cdot x\right)\right) \cdot 0.027777777777777776}}{-0.16666666666666666 - x \cdot \left(x \cdot 0.008333333333333333\right)} \]
11. Add Preprocessing

Alternative 6: 98.6% accurate, 7.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.2%

   \[\sin x - x \]
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. unpow2 N/A

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

   13. associate-*r* N/A

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

   14. *-commutative N/A

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

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

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

   16. *-commutative N/A

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

   17. *-lowering-*.f64 98.8%

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

5. Simplified 98.8%

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

Alternative 7: 98.1% 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 71.2%

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

3. Taylor expanded in x around 0

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

   1. unpow3 N/A

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

   2. unpow2 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

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

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

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

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

   7. unpow2 N/A

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

   8. *-lowering-*.f64 98.1%

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

5. Simplified 98.1%

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

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

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

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

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

   5. *-lowering-*.f64 98.1%

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

7. Applied egg-rr 98.1%

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

Alternative 8: 98.1% 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 71.2%

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

3. Taylor expanded in x around 0

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

   1. unpow3 N/A

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

   2. unpow2 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

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

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

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

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

   7. unpow2 N/A

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

   8. *-lowering-*.f64 98.1%

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

5. Simplified 98.1%

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

6. Final simplification 98.1%

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

Alternative 9: 66.9% 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 71.2%

   \[\sin x - x \]
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 68.9%

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

   1. +-inverses 68.9%

      \[\leadsto 0 \]

7. Applied egg-rr 68.9%

   \[\leadsto \color{blue}{0} \]
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 2024158 
(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))