bug500 (missed optimization)

Percentage Accurate: 69.9% → 98.7%

Time: 8.0s

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: 69.9% 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: 98.7% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 72.6%

   \[\sin x - x \]
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. cube-mult N/A

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

   2. unpow2 N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. associate-*l* N/A

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

   7. lower-*.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. sub-neg N/A

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

   10. metadata-eval N/A

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

   11. lower-fma.f64 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f64 N/A

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

   14. +-commutative N/A

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

   15. *-commutative N/A

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

   16. unpow2 N/A

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

   17. associate-*l* N/A

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

   18. lower-fma.f64 N/A

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

   19. lower-*.f64 98.9

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

5. Applied rewrites 98.9%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-fma.f64 N/A

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

   4. lift-fma.f64 N/A

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

   5. lift-fma.f64 N/A

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

   6. +-commutative N/A

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

   7. distribute-lft-in N/A

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

   8. *-commutative N/A

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

   9. distribute-rgt-in N/A

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

   10. lower-fma.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-*.f64 N/A

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

7. Applied rewrites 98.9%

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

8. Final simplification 98.9%

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

Alternative 2: 98.7% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 72.6%

   \[\sin x - x \]
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. cube-mult N/A

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

   2. unpow2 N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. associate-*l* N/A

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

   7. lower-*.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. sub-neg N/A

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

   10. metadata-eval N/A

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

   11. lower-fma.f64 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f64 N/A

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

   14. +-commutative N/A

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

   15. *-commutative N/A

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

   16. unpow2 N/A

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

   17. associate-*l* N/A

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

   18. lower-fma.f64 N/A

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

   19. lower-*.f64 98.9

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

5. Applied rewrites 98.9%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-fma.f64 N/A

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

   4. lift-fma.f64 N/A

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

   5. lift-fma.f64 N/A

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

   6. +-commutative N/A

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

   7. distribute-lft-in N/A

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

   8. *-commutative N/A

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

   9. distribute-rgt-in N/A

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

   10. lower-fma.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-*.f64 N/A

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

7. Applied rewrites 98.9%

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

8. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. sub-neg N/A

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

   4. metadata-eval N/A

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

   5. lower-fma.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 N/A

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

   8. +-commutative N/A

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

   9. *-commutative N/A

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

   10. unpow2 N/A

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

   11. associate-*l* N/A

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

   12. lower-fma.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. cube-mult N/A

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

   15. unpow2 N/A

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

   16. lower-*.f64 N/A

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

   17. unpow2 N/A

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

   18. lower-*.f64 98.9

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

10. Applied rewrites 98.9%

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

Alternative 3: 98.7% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 72.6%

   \[\sin x - x \]
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. cube-mult N/A

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

   2. unpow2 N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. associate-*l* N/A

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

   7. lower-*.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. sub-neg N/A

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

   10. metadata-eval N/A

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

   11. lower-fma.f64 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f64 N/A

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

   14. +-commutative N/A

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

   15. *-commutative N/A

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

   16. unpow2 N/A

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

   17. associate-*l* N/A

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

   18. lower-fma.f64 N/A

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

   19. lower-*.f64 98.9

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

5. Applied rewrites 98.9%

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

Alternative 4: 98.4% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 72.6%

   \[\sin x - x \]
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

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

   2. unpow2 N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. associate-*l* N/A

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

   7. lower-*.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. sub-neg N/A

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

   10. *-commutative N/A

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

   11. unpow2 N/A

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

   12. associate-*l* N/A

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

   13. metadata-eval N/A

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

   14. lower-fma.f64 N/A

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

   15. lower-*.f64 98.7

       \[\leadsto x \cdot \left(x \cdot \left(x \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot 0.008333333333333333}, -0.16666666666666666\right)\right)\right) \]

5. Applied rewrites 98.7%

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

   1. lift-*.f64 N/A

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

   2. distribute-lft-in N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*r* N/A

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

   7. lift-*.f64 N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-*.f64 98.7

      \[\leadsto x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot \left(x \cdot x\right), 0.008333333333333333, \color{blue}{x \cdot -0.16666666666666666}\right)\right) \]

7. Applied rewrites 98.7%

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

Alternative 5: 98.4% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 72.6%

   \[\sin x - x \]
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

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

   2. unpow2 N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. associate-*l* N/A

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

   7. lower-*.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. sub-neg N/A

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

   10. *-commutative N/A

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

   11. unpow2 N/A

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

   12. associate-*l* N/A

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

   13. metadata-eval N/A

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

   14. lower-fma.f64 N/A

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

   15. lower-*.f64 98.7

       \[\leadsto x \cdot \left(x \cdot \left(x \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot 0.008333333333333333}, -0.16666666666666666\right)\right)\right) \]

5. Applied rewrites 98.7%

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

Alternative 6: 98.0% accurate, 4.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.6%

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

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

   2. cube-mult N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 98.3

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

5. Applied rewrites 98.3%

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

   1. associate-*r* N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 98.3

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

7. Applied rewrites 98.3%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. --rgt-identity N/A

      \[\leadsto \left(\color{blue}{\left(x \cdot x - 0\right)} \cdot \frac{-1}{6}\right) \cdot x \]

   4. flip-- N/A

      \[\leadsto \left(\color{blue}{\frac{\left(x \cdot x\right) \cdot \left(x \cdot x\right) - 0 \cdot 0}{x \cdot x + 0}} \cdot \frac{-1}{6}\right) \cdot x \]

   5. lift-*.f64 N/A

      \[\leadsto \left(\frac{\color{blue}{\left(x \cdot x\right)} \cdot \left(x \cdot x\right) - 0 \cdot 0}{x \cdot x + 0} \cdot \frac{-1}{6}\right) \cdot x \]

   6. associate-*r* N/A

      \[\leadsto \left(\frac{\color{blue}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)} - 0 \cdot 0}{x \cdot x + 0} \cdot \frac{-1}{6}\right) \cdot x \]

   7. lift-*.f64 N/A

      \[\leadsto \left(\frac{x \cdot \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} - 0 \cdot 0}{x \cdot x + 0} \cdot \frac{-1}{6}\right) \cdot x \]

   8. lift-*.f64 N/A

      \[\leadsto \left(\frac{\color{blue}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)} - 0 \cdot 0}{x \cdot x + 0} \cdot \frac{-1}{6}\right) \cdot x \]

   9. metadata-eval N/A

      \[\leadsto \left(\frac{x \cdot \left(x \cdot \left(x \cdot x\right)\right) - \color{blue}{0}}{x \cdot x + 0} \cdot \frac{-1}{6}\right) \cdot x \]

   10. +-rgt-identity N/A

       \[\leadsto \left(\frac{x \cdot \left(x \cdot \left(x \cdot x\right)\right) - 0}{\color{blue}{x \cdot x}} \cdot \frac{-1}{6}\right) \cdot x \]

   11. --rgt-identity N/A

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

   12. associate-*l/ N/A

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

   13. associate-*r/ N/A

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

   14. clear-num N/A

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

   15. un-div-inv N/A

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

   16. lift-*.f64 N/A

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

   17. div-inv N/A

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

   18. *-commutative N/A

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

   19. times-frac N/A

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

9. Applied rewrites 98.3%

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

10. Final simplification 98.3%

    \[\leadsto x \cdot \left(x \cdot \frac{x}{-6}\right) \]
11. Add Preprocessing

Alternative 7: 97.9% accurate, 6.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.6%

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

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

   2. cube-mult N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 98.3

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

5. Applied rewrites 98.3%

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

   1. associate-*r* N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 98.3

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

7. Applied rewrites 98.3%

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

8. Final simplification 98.3%

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

Alternative 8: 97.9% accurate, 6.5× speedup

Math

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

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.6%

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

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

   2. cube-mult N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 98.3

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

5. Applied rewrites 98.3%

   \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left(x \cdot \left(x \cdot x\right)\right)} \]
6. 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 72.6%

   \[\sin x - x \]
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 6.5

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

5. Applied rewrites 6.5%

   \[\leadsto \color{blue}{-x} \]
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 2024219 
(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))