bug500 (missed optimization)

Percentage Accurate: 70.0% → 98.9%

Time: 9.6s

Alternatives: 13

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: 70.0% 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: 98.9% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 70.9%

   \[\sin x - x \]
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

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

   2. unpow2 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. sub-neg N/A

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

   11. metadata-eval N/A

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

   12. lower-fma.f64 N/A

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

5. Applied rewrites 99.0%

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

7. Applied rewrites 99.1%

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

Alternative 2: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot x\right)\\ \frac{t\_0 \cdot \mathsf{fma}\left(x \cdot t\_0, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 8.529646426471823 \cdot 10^{-8}, -3.306878306878307 \cdot 10^{-6}\right), 6.944444444444444 \cdot 10^{-5}\right), -0.027777777777777776\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 2.7557319223985893 \cdot 10^{-6}, -0.0001984126984126984\right), 0.008333333333333333\right), 0.16666666666666666\right)} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* x (* x x))))
   (/
    (*
     t_0
     (fma
      (* x t_0)
      (fma
       (* x x)
       (fma x (* x 8.529646426471823e-8) -3.306878306878307e-6)
       6.944444444444444e-5)
      -0.027777777777777776))
    (fma
     (* x x)
     (fma
      x
      (* x (fma x (* x 2.7557319223985893e-6) -0.0001984126984126984))
      0.008333333333333333)
     0.16666666666666666))))

C

double code(double x) {
	double t_0 = x * (x * x);
	return (t_0 * fma((x * t_0), fma((x * x), fma(x, (x * 8.529646426471823e-8), -3.306878306878307e-6), 6.944444444444444e-5), -0.027777777777777776)) / fma((x * x), fma(x, (x * fma(x, (x * 2.7557319223985893e-6), -0.0001984126984126984)), 0.008333333333333333), 0.16666666666666666);
}

Julia

function code(x)
	t_0 = Float64(x * Float64(x * x))
	return Float64(Float64(t_0 * fma(Float64(x * t_0), fma(Float64(x * x), fma(x, Float64(x * 8.529646426471823e-8), -3.306878306878307e-6), 6.944444444444444e-5), -0.027777777777777776)) / fma(Float64(x * x), fma(x, Float64(x * fma(x, Float64(x * 2.7557319223985893e-6), -0.0001984126984126984)), 0.008333333333333333), 0.16666666666666666))
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]}, N[(N[(t$95$0 * N[(N[(x * t$95$0), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * 8.529646426471823e-8), $MachinePrecision] + -3.306878306878307e-6), $MachinePrecision] + 6.944444444444444e-5), $MachinePrecision] + -0.027777777777777776), $MachinePrecision]), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * N[(x * N[(x * 2.7557319223985893e-6), $MachinePrecision] + -0.0001984126984126984), $MachinePrecision]), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 70.9%

   \[\sin x - x \]
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

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

   2. unpow2 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. sub-neg N/A

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

   11. metadata-eval N/A

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

   12. lower-fma.f64 N/A

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

5. Applied rewrites 99.0%

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

7. Applied rewrites 99.1%

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

8. Taylor expanded in x around 0

   \[\leadsto \frac{{x}^{3} \cdot \left({x}^{4} \cdot \left(\frac{1}{14400} + {x}^{2} \cdot \left(\frac{13}{152409600} \cdot {x}^{2} - \frac{1}{302400}\right)\right) - \frac{1}{36}\right)}{\mathsf{fma}\left(\color{blue}{x \cdot x}, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot \frac{1}{362880}, \frac{-1}{5040}\right), \frac{1}{120}\right), \frac{1}{6}\right)} \]
9. Step-by-step derivation

10. Applied rewrites 99.1%

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

Alternative 3: 98.9% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 70.9%

   \[\sin x - x \]
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

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

   2. unpow2 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. sub-neg N/A

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

   11. metadata-eval N/A

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

   12. lower-fma.f64 N/A

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

5. Applied rewrites 99.0%

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

7. Applied rewrites 99.1%

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

Alternative 4: 98.9% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 70.9%

   \[\sin x - x \]
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

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

   2. unpow2 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. sub-neg N/A

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

   11. metadata-eval N/A

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

   12. lower-fma.f64 N/A

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

5. Applied rewrites 99.0%

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

7. Applied rewrites 99.1%

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

9. Applied rewrites 99.1%

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

10. Final simplification 99.1%

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

Alternative 5: 98.9% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 70.9%

   \[\sin x - x \]
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

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

   2. unpow2 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. sub-neg N/A

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

   11. metadata-eval N/A

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

   12. lower-fma.f64 N/A

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

5. Applied rewrites 99.0%

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

7. Applied rewrites 99.1%

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

Alternative 6: 98.9% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 70.9%

   \[\sin x - x \]
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

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

   2. unpow2 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. sub-neg N/A

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

   11. metadata-eval N/A

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

   12. lower-fma.f64 N/A

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

5. Applied rewrites 99.0%

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

7. Applied rewrites 99.1%

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

Alternative 7: 98.9% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 70.9%

   \[\sin x - x \]
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

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

   2. unpow2 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. sub-neg N/A

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

   11. metadata-eval N/A

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

   12. lower-fma.f64 N/A

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

5. Applied rewrites 99.0%

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

7. Applied rewrites 99.1%

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

9. Applied rewrites 99.0%

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

Alternative 8: 98.9% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 70.9%

   \[\sin x - x \]
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

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

   2. unpow2 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. sub-neg N/A

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

   11. metadata-eval N/A

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

   12. lower-fma.f64 N/A

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

5. Applied rewrites 99.0%

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

Alternative 9: 98.9% 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 70.9%

   \[\sin x - x \]
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 10: 98.7% 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 70.9%

   \[\sin x - x \]
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 98.5

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

5. Applied rewrites 98.5%

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

Alternative 11: 98.2% accurate, 6.5× 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 70.9%

   \[\sin x - x \]
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

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

   2. unpow2 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. sub-neg N/A

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

   11. metadata-eval N/A

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

   12. lower-fma.f64 N/A

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

5. Applied rewrites 99.0%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 97.9%

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

9. Final simplification 97.9%

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

Alternative 12: 98.2% accurate, 6.5× 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 70.9%

   \[\sin x - x \]
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 97.9

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

5. Applied rewrites 97.9%

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

6. Final simplification 97.9%

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

Alternative 13: 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 70.9%

   \[\sin x - x \]
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 6.3

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

5. Applied rewrites 6.3%

   \[\leadsto \color{blue}{-x} \]
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 2024221 
(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))