ENA, Section 1.4, Exercise 4a

Percentage Accurate: 53.7% → 99.6%

Time: 15.6s

Alternatives: 13

Speedup: 19.5×

Specification

\[-1 \leq x \land x \leq 1\]

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 53.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 1.3× speedup

Math

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

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)))
   (tan x))))

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))) / tan(x));
}

Julia

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

Wolfram

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

Derivation

1. Initial program 54.2%

   \[\frac{x - \sin x}{\tan x} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

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

   1. cube-mult N/A

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

   2. unpow2 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

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

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

   6. *-commutative N/A

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

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

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

   8. unpow2 N/A

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

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

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

   10. +-commutative N/A

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

   11. accelerator-lowering-fma.f64 N/A

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

5. Simplified 87.6%

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

   1. associate-/l* N/A

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

   2. *-commutative N/A

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

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

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

7. Applied egg-rr 99.6%

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

   1. associate-*l* N/A

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

   2. *-commutative N/A

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

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

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

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

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

   5. associate-*r* N/A

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

   6. accelerator-lowering-fma.f64 N/A

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

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

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

   8. accelerator-lowering-fma.f64 N/A

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

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

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

   10. associate-*r* N/A

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

   11. accelerator-lowering-fma.f64 N/A

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

   12. *-lowering-*.f64 99.6

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

9. Applied egg-rr 99.6%

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

10. Final simplification 99.6%

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

Alternative 2: 99.6% accurate, 1.3× speedup

Math

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

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))
   (tan x))))

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)) / tan(x));
}

Julia

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

Wolfram

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

Derivation

1. Initial program 54.2%

   \[\frac{x - \sin x}{\tan x} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

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

   1. cube-mult N/A

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

   2. unpow2 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

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

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

   6. *-commutative N/A

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

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

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

   8. unpow2 N/A

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

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

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

   10. +-commutative N/A

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

   11. accelerator-lowering-fma.f64 N/A

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

5. Simplified 87.6%

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

   1. associate-/l* N/A

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

   2. *-commutative N/A

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

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

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

7. Applied egg-rr 99.6%

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

8. Final simplification 99.6%

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

Alternative 3: 99.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot -0.00023644179894179894, -0.0007275132275132275\right), -0.06388888888888888\right)\\ \left(x \cdot \mathsf{fma}\left(t\_0, t\_0 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right), -0.027777777777777776\right)\right) \cdot \frac{x}{\mathsf{fma}\left(x, x \cdot t\_0, -0.16666666666666666\right)} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0
         (fma
          (* x x)
          (fma x (* x -0.00023644179894179894) -0.0007275132275132275)
          -0.06388888888888888)))
   (*
    (* x (fma t_0 (* t_0 (* x (* x (* x x)))) -0.027777777777777776))
    (/ x (fma x (* x t_0) -0.16666666666666666)))))

C

double code(double x) {
	double t_0 = fma((x * x), fma(x, (x * -0.00023644179894179894), -0.0007275132275132275), -0.06388888888888888);
	return (x * fma(t_0, (t_0 * (x * (x * (x * x)))), -0.027777777777777776)) * (x / fma(x, (x * t_0), -0.16666666666666666));
}

Julia

function code(x)
	t_0 = fma(Float64(x * x), fma(x, Float64(x * -0.00023644179894179894), -0.0007275132275132275), -0.06388888888888888)
	return Float64(Float64(x * fma(t_0, Float64(t_0 * Float64(x * Float64(x * Float64(x * x)))), -0.027777777777777776)) * Float64(x / fma(x, Float64(x * t_0), -0.16666666666666666)))
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * -0.00023644179894179894), $MachinePrecision] + -0.0007275132275132275), $MachinePrecision] + -0.06388888888888888), $MachinePrecision]}, N[(N[(x * N[(t$95$0 * N[(t$95$0 * N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -0.027777777777777776), $MachinePrecision]), $MachinePrecision] * N[(x / N[(x * N[(x * t$95$0), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 54.2%

   \[\frac{x - \sin x}{\tan x} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right)} \]
4. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right) \]

   2. associate-*l* N/A

      \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right)\right)} \]

   3. *-commutative N/A

      \[\leadsto x \cdot \color{blue}{\left(\left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right) \cdot x\right)} \]

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

      \[\leadsto \color{blue}{x \cdot \left(\left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right) \cdot x\right)} \]

   5. *-commutative N/A

      \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right)\right)} \]

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

      \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right)\right)} \]

   7. +-commutative N/A

      \[\leadsto x \cdot \left(x \cdot \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right) + \frac{1}{6}\right)}\right) \]

   8. unpow2 N/A

      \[\leadsto x \cdot \left(x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right) + \frac{1}{6}\right)\right) \]

   9. associate-*l* N/A

      \[\leadsto x \cdot \left(x \cdot \left(\color{blue}{x \cdot \left(x \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right)} + \frac{1}{6}\right)\right) \]

   10. accelerator-lowering-fma.f64 N/A

       \[\leadsto x \cdot \left(x \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right), \frac{1}{6}\right)}\right) \]

5. Simplified 99.5%

   \[\leadsto \color{blue}{x \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, -0.00023644179894179894, -0.0007275132275132275\right), -0.06388888888888888\right), 0.16666666666666666\right)\right)} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto x \cdot \color{blue}{\left(\left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot \frac{-143}{604800} + \frac{-11}{15120}\right)\right) + \frac{-23}{360}\right)\right) + \frac{1}{6}\right) \cdot x\right)} \]

   2. flip-+ N/A

      \[\leadsto x \cdot \left(\color{blue}{\frac{\left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot \frac{-143}{604800} + \frac{-11}{15120}\right)\right) + \frac{-23}{360}\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot \frac{-143}{604800} + \frac{-11}{15120}\right)\right) + \frac{-23}{360}\right)\right)\right) - \frac{1}{6} \cdot \frac{1}{6}}{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot \frac{-143}{604800} + \frac{-11}{15120}\right)\right) + \frac{-23}{360}\right)\right) - \frac{1}{6}}} \cdot x\right) \]

   3. associate-*l/ N/A

      \[\leadsto x \cdot \color{blue}{\frac{\left(\left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot \frac{-143}{604800} + \frac{-11}{15120}\right)\right) + \frac{-23}{360}\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot \frac{-143}{604800} + \frac{-11}{15120}\right)\right) + \frac{-23}{360}\right)\right)\right) - \frac{1}{6} \cdot \frac{1}{6}\right) \cdot x}{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot \frac{-143}{604800} + \frac{-11}{15120}\right)\right) + \frac{-23}{360}\right)\right) - \frac{1}{6}}} \]

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

      \[\leadsto x \cdot \color{blue}{\frac{\left(\left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot \frac{-143}{604800} + \frac{-11}{15120}\right)\right) + \frac{-23}{360}\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot \frac{-143}{604800} + \frac{-11}{15120}\right)\right) + \frac{-23}{360}\right)\right)\right) - \frac{1}{6} \cdot \frac{1}{6}\right) \cdot x}{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot \frac{-143}{604800} + \frac{-11}{15120}\right)\right) + \frac{-23}{360}\right)\right) - \frac{1}{6}}} \]

7. Applied egg-rr 99.5%

   \[\leadsto x \cdot \color{blue}{\frac{\mathsf{fma}\left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot -0.00023644179894179894, -0.0007275132275132275\right), -0.06388888888888888\right), \left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot -0.00023644179894179894, -0.0007275132275132275\right), -0.06388888888888888\right), -0.027777777777777776\right) \cdot x}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot -0.00023644179894179894, -0.0007275132275132275\right), -0.06388888888888888\right), -0.16666666666666666\right)}} \]
8. Step-by-step derivation

   1. associate-/l* N/A

      \[\leadsto x \cdot \color{blue}{\left(\left(\left(x \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \frac{-143}{604800}\right) + \frac{-11}{15120}\right) + \frac{-23}{360}\right)\right) \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \frac{-143}{604800}\right) + \frac{-11}{15120}\right) + \frac{-23}{360}\right)\right) + \frac{-1}{36}\right) \cdot \frac{x}{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \frac{-143}{604800}\right) + \frac{-11}{15120}\right) + \frac{-23}{360}\right) + \frac{-1}{6}}\right)} \]

   2. associate-*r* N/A

      \[\leadsto \color{blue}{\left(x \cdot \left(\left(x \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \frac{-143}{604800}\right) + \frac{-11}{15120}\right) + \frac{-23}{360}\right)\right) \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \frac{-143}{604800}\right) + \frac{-11}{15120}\right) + \frac{-23}{360}\right)\right) + \frac{-1}{36}\right)\right) \cdot \frac{x}{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \frac{-143}{604800}\right) + \frac{-11}{15120}\right) + \frac{-23}{360}\right) + \frac{-1}{6}}} \]

   3. *-commutative N/A

      \[\leadsto \color{blue}{\left(\left(\left(x \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \frac{-143}{604800}\right) + \frac{-11}{15120}\right) + \frac{-23}{360}\right)\right) \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \frac{-143}{604800}\right) + \frac{-11}{15120}\right) + \frac{-23}{360}\right)\right) + \frac{-1}{36}\right) \cdot x\right)} \cdot \frac{x}{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \frac{-143}{604800}\right) + \frac{-11}{15120}\right) + \frac{-23}{360}\right) + \frac{-1}{6}} \]

9. Applied egg-rr 99.5%

   \[\leadsto \color{blue}{\left(x \cdot \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot -0.00023644179894179894, -0.0007275132275132275\right), -0.06388888888888888\right), \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot -0.00023644179894179894, -0.0007275132275132275\right), -0.06388888888888888\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right), -0.027777777777777776\right)\right) \cdot \frac{x}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot -0.00023644179894179894, -0.0007275132275132275\right), -0.06388888888888888\right), -0.16666666666666666\right)}} \]
10. Add Preprocessing

Alternative 4: 99.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot -0.00023644179894179894, -0.0007275132275132275\right), -0.06388888888888888\right)\\ x \cdot \frac{x \cdot \mathsf{fma}\left(x \cdot t\_0, t\_0 \cdot \left(x \cdot \left(x \cdot x\right)\right), -0.027777777777777776\right)}{\mathsf{fma}\left(x \cdot x, t\_0, -0.16666666666666666\right)} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0
         (fma
          (* x x)
          (fma x (* x -0.00023644179894179894) -0.0007275132275132275)
          -0.06388888888888888)))
   (*
    x
    (/
     (* x (fma (* x t_0) (* t_0 (* x (* x x))) -0.027777777777777776))
     (fma (* x x) t_0 -0.16666666666666666)))))

C

double code(double x) {
	double t_0 = fma((x * x), fma(x, (x * -0.00023644179894179894), -0.0007275132275132275), -0.06388888888888888);
	return x * ((x * fma((x * t_0), (t_0 * (x * (x * x))), -0.027777777777777776)) / fma((x * x), t_0, -0.16666666666666666));
}

Julia

function code(x)
	t_0 = fma(Float64(x * x), fma(x, Float64(x * -0.00023644179894179894), -0.0007275132275132275), -0.06388888888888888)
	return Float64(x * Float64(Float64(x * fma(Float64(x * t_0), Float64(t_0 * Float64(x * Float64(x * x))), -0.027777777777777776)) / fma(Float64(x * x), t_0, -0.16666666666666666)))
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * -0.00023644179894179894), $MachinePrecision] + -0.0007275132275132275), $MachinePrecision] + -0.06388888888888888), $MachinePrecision]}, N[(x * N[(N[(x * N[(N[(x * t$95$0), $MachinePrecision] * N[(t$95$0 * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -0.027777777777777776), $MachinePrecision]), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] * t$95$0 + -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 54.2%

   \[\frac{x - \sin x}{\tan x} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right)} \]
4. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right) \]

   2. associate-*l* N/A

      \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right)\right)} \]

   3. *-commutative N/A

      \[\leadsto x \cdot \color{blue}{\left(\left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right) \cdot x\right)} \]

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

      \[\leadsto \color{blue}{x \cdot \left(\left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right) \cdot x\right)} \]

   5. *-commutative N/A

      \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right)\right)} \]

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

      \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right)\right)} \]

   7. +-commutative N/A

      \[\leadsto x \cdot \left(x \cdot \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right) + \frac{1}{6}\right)}\right) \]

   8. unpow2 N/A

      \[\leadsto x \cdot \left(x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right) + \frac{1}{6}\right)\right) \]

   9. associate-*l* N/A

      \[\leadsto x \cdot \left(x \cdot \left(\color{blue}{x \cdot \left(x \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right)} + \frac{1}{6}\right)\right) \]

   10. accelerator-lowering-fma.f64 N/A

       \[\leadsto x \cdot \left(x \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right), \frac{1}{6}\right)}\right) \]

5. Simplified 99.5%

   \[\leadsto \color{blue}{x \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, -0.00023644179894179894, -0.0007275132275132275\right), -0.06388888888888888\right), 0.16666666666666666\right)\right)} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto x \cdot \color{blue}{\left(\left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot \frac{-143}{604800} + \frac{-11}{15120}\right)\right) + \frac{-23}{360}\right)\right) + \frac{1}{6}\right) \cdot x\right)} \]

   2. flip-+ N/A

      \[\leadsto x \cdot \left(\color{blue}{\frac{\left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot \frac{-143}{604800} + \frac{-11}{15120}\right)\right) + \frac{-23}{360}\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot \frac{-143}{604800} + \frac{-11}{15120}\right)\right) + \frac{-23}{360}\right)\right)\right) - \frac{1}{6} \cdot \frac{1}{6}}{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot \frac{-143}{604800} + \frac{-11}{15120}\right)\right) + \frac{-23}{360}\right)\right) - \frac{1}{6}}} \cdot x\right) \]

   3. associate-*l/ N/A

      \[\leadsto x \cdot \color{blue}{\frac{\left(\left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot \frac{-143}{604800} + \frac{-11}{15120}\right)\right) + \frac{-23}{360}\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot \frac{-143}{604800} + \frac{-11}{15120}\right)\right) + \frac{-23}{360}\right)\right)\right) - \frac{1}{6} \cdot \frac{1}{6}\right) \cdot x}{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot \frac{-143}{604800} + \frac{-11}{15120}\right)\right) + \frac{-23}{360}\right)\right) - \frac{1}{6}}} \]

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

      \[\leadsto x \cdot \color{blue}{\frac{\left(\left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot \frac{-143}{604800} + \frac{-11}{15120}\right)\right) + \frac{-23}{360}\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot \frac{-143}{604800} + \frac{-11}{15120}\right)\right) + \frac{-23}{360}\right)\right)\right) - \frac{1}{6} \cdot \frac{1}{6}\right) \cdot x}{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot \frac{-143}{604800} + \frac{-11}{15120}\right)\right) + \frac{-23}{360}\right)\right) - \frac{1}{6}}} \]

7. Applied egg-rr 99.5%

   \[\leadsto x \cdot \color{blue}{\frac{\mathsf{fma}\left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot -0.00023644179894179894, -0.0007275132275132275\right), -0.06388888888888888\right), \left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot -0.00023644179894179894, -0.0007275132275132275\right), -0.06388888888888888\right), -0.027777777777777776\right) \cdot x}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot -0.00023644179894179894, -0.0007275132275132275\right), -0.06388888888888888\right), -0.16666666666666666\right)}} \]

8. Final simplification 99.5%

   \[\leadsto x \cdot \frac{x \cdot \mathsf{fma}\left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot -0.00023644179894179894, -0.0007275132275132275\right), -0.06388888888888888\right), \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot -0.00023644179894179894, -0.0007275132275132275\right), -0.06388888888888888\right) \cdot \left(x \cdot \left(x \cdot x\right)\right), -0.027777777777777776\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot -0.00023644179894179894, -0.0007275132275132275\right), -0.06388888888888888\right), -0.16666666666666666\right)} \]
9. Add Preprocessing

Alternative 5: 99.6% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x \cdot 0.16666666666666666, x, \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot -0.00023644179894179894, -0.0007275132275132275\right), -0.06388888888888888\right)\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fma
  (* x 0.16666666666666666)
  x
  (*
   (* x x)
   (*
    (* x x)
    (fma
     (* x x)
     (fma x (* x -0.00023644179894179894) -0.0007275132275132275)
     -0.06388888888888888)))))

C

double code(double x) {
	return fma((x * 0.16666666666666666), x, ((x * x) * ((x * x) * fma((x * x), fma(x, (x * -0.00023644179894179894), -0.0007275132275132275), -0.06388888888888888))));
}

Julia

function code(x)
	return fma(Float64(x * 0.16666666666666666), x, Float64(Float64(x * x) * Float64(Float64(x * x) * fma(Float64(x * x), fma(x, Float64(x * -0.00023644179894179894), -0.0007275132275132275), -0.06388888888888888))))
end

Wolfram

code[x_] := N[(N[(x * 0.16666666666666666), $MachinePrecision] * x + N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * -0.00023644179894179894), $MachinePrecision] + -0.0007275132275132275), $MachinePrecision] + -0.06388888888888888), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.2%

   \[\frac{x - \sin x}{\tan x} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right)} \]
4. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right) \]

   2. associate-*l* N/A

      \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right)\right)} \]

   3. *-commutative N/A

      \[\leadsto x \cdot \color{blue}{\left(\left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right) \cdot x\right)} \]

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

      \[\leadsto \color{blue}{x \cdot \left(\left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right) \cdot x\right)} \]

   5. *-commutative N/A

      \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right)\right)} \]

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

      \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right)\right)} \]

   7. +-commutative N/A

      \[\leadsto x \cdot \left(x \cdot \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right) + \frac{1}{6}\right)}\right) \]

   8. unpow2 N/A

      \[\leadsto x \cdot \left(x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right) + \frac{1}{6}\right)\right) \]

   9. associate-*l* N/A

      \[\leadsto x \cdot \left(x \cdot \left(\color{blue}{x \cdot \left(x \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right)} + \frac{1}{6}\right)\right) \]

   10. accelerator-lowering-fma.f64 N/A

       \[\leadsto x \cdot \left(x \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right), \frac{1}{6}\right)}\right) \]

5. Simplified 99.5%

   \[\leadsto \color{blue}{x \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, -0.00023644179894179894, -0.0007275132275132275\right), -0.06388888888888888\right), 0.16666666666666666\right)\right)} \]
6. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \color{blue}{\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot \frac{-143}{604800} + \frac{-11}{15120}\right)\right) + \frac{-23}{360}\right)\right) + \frac{1}{6}\right)} \]

   2. +-commutative N/A

      \[\leadsto \left(x \cdot x\right) \cdot \color{blue}{\left(\frac{1}{6} + x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot \frac{-143}{604800} + \frac{-11}{15120}\right)\right) + \frac{-23}{360}\right)\right)\right)} \]

   3. distribute-rgt-in N/A

      \[\leadsto \color{blue}{\frac{1}{6} \cdot \left(x \cdot x\right) + \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot \frac{-143}{604800} + \frac{-11}{15120}\right)\right) + \frac{-23}{360}\right)\right)\right) \cdot \left(x \cdot x\right)} \]

   4. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\frac{1}{6} \cdot x\right) \cdot x} + \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot \frac{-143}{604800} + \frac{-11}{15120}\right)\right) + \frac{-23}{360}\right)\right)\right) \cdot \left(x \cdot x\right) \]

   5. *-commutative N/A

      \[\leadsto \color{blue}{\left(x \cdot \frac{1}{6}\right)} \cdot x + \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot \frac{-143}{604800} + \frac{-11}{15120}\right)\right) + \frac{-23}{360}\right)\right)\right) \cdot \left(x \cdot x\right) \]

   6. accelerator-lowering-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot \frac{1}{6}, x, \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot \frac{-143}{604800} + \frac{-11}{15120}\right)\right) + \frac{-23}{360}\right)\right)\right) \cdot \left(x \cdot x\right)\right)} \]

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

      \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot \frac{1}{6}}, x, \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot \frac{-143}{604800} + \frac{-11}{15120}\right)\right) + \frac{-23}{360}\right)\right)\right) \cdot \left(x \cdot x\right)\right) \]

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

      \[\leadsto \mathsf{fma}\left(x \cdot \frac{1}{6}, x, \color{blue}{\left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot \frac{-143}{604800} + \frac{-11}{15120}\right)\right) + \frac{-23}{360}\right)\right)\right) \cdot \left(x \cdot x\right)}\right) \]

7. Applied egg-rr 99.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot 0.16666666666666666, x, \left(\left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot -0.00023644179894179894, -0.0007275132275132275\right), -0.06388888888888888\right)\right) \cdot \left(x \cdot x\right)\right)} \]

8. Final simplification 99.5%

   \[\leadsto \mathsf{fma}\left(x \cdot 0.16666666666666666, x, \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot -0.00023644179894179894, -0.0007275132275132275\right), -0.06388888888888888\right)\right)\right) \]
9. Add Preprocessing

Alternative 6: 99.6% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ x \cdot \mathsf{fma}\left(\left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot -0.00023644179894179894, -0.0007275132275132275\right), -0.06388888888888888\right), x, x \cdot 0.16666666666666666\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  x
  (fma
   (*
    (* x x)
    (fma
     (* x x)
     (fma x (* x -0.00023644179894179894) -0.0007275132275132275)
     -0.06388888888888888))
   x
   (* x 0.16666666666666666))))

C

double code(double x) {
	return x * fma(((x * x) * fma((x * x), fma(x, (x * -0.00023644179894179894), -0.0007275132275132275), -0.06388888888888888)), x, (x * 0.16666666666666666));
}

Julia

function code(x)
	return Float64(x * fma(Float64(Float64(x * x) * fma(Float64(x * x), fma(x, Float64(x * -0.00023644179894179894), -0.0007275132275132275), -0.06388888888888888)), x, Float64(x * 0.16666666666666666)))
end

Wolfram

code[x_] := N[(x * N[(N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * -0.00023644179894179894), $MachinePrecision] + -0.0007275132275132275), $MachinePrecision] + -0.06388888888888888), $MachinePrecision]), $MachinePrecision] * x + N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.2%

   \[\frac{x - \sin x}{\tan x} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right)} \]
4. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right) \]

   2. associate-*l* N/A

      \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right)\right)} \]

   3. *-commutative N/A

      \[\leadsto x \cdot \color{blue}{\left(\left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right) \cdot x\right)} \]

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

      \[\leadsto \color{blue}{x \cdot \left(\left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right) \cdot x\right)} \]

   5. *-commutative N/A

      \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right)\right)} \]

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

      \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right)\right)} \]

   7. +-commutative N/A

      \[\leadsto x \cdot \left(x \cdot \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right) + \frac{1}{6}\right)}\right) \]

   8. unpow2 N/A

      \[\leadsto x \cdot \left(x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right) + \frac{1}{6}\right)\right) \]

   9. associate-*l* N/A

      \[\leadsto x \cdot \left(x \cdot \left(\color{blue}{x \cdot \left(x \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right)} + \frac{1}{6}\right)\right) \]

   10. accelerator-lowering-fma.f64 N/A

       \[\leadsto x \cdot \left(x \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right), \frac{1}{6}\right)}\right) \]

5. Simplified 99.5%

   \[\leadsto \color{blue}{x \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, -0.00023644179894179894, -0.0007275132275132275\right), -0.06388888888888888\right), 0.16666666666666666\right)\right)} \]
6. Step-by-step derivation

   1. distribute-rgt-in N/A

      \[\leadsto x \cdot \color{blue}{\left(\left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot \frac{-143}{604800} + \frac{-11}{15120}\right)\right) + \frac{-23}{360}\right)\right)\right) \cdot x + \frac{1}{6} \cdot x\right)} \]

   2. *-commutative N/A

      \[\leadsto x \cdot \left(\left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot \frac{-143}{604800} + \frac{-11}{15120}\right)\right) + \frac{-23}{360}\right)\right)\right) \cdot x + \color{blue}{x \cdot \frac{1}{6}}\right) \]

   3. accelerator-lowering-fma.f64 N/A

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot \frac{-143}{604800} + \frac{-11}{15120}\right)\right) + \frac{-23}{360}\right)\right), x, x \cdot \frac{1}{6}\right)} \]

   4. associate-*r* N/A

      \[\leadsto x \cdot \mathsf{fma}\left(\color{blue}{\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot \frac{-143}{604800} + \frac{-11}{15120}\right)\right) + \frac{-23}{360}\right)}, x, x \cdot \frac{1}{6}\right) \]

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

      \[\leadsto x \cdot \mathsf{fma}\left(\color{blue}{\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot \frac{-143}{604800} + \frac{-11}{15120}\right)\right) + \frac{-23}{360}\right)}, x, x \cdot \frac{1}{6}\right) \]

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

      \[\leadsto x \cdot \mathsf{fma}\left(\color{blue}{\left(x \cdot x\right)} \cdot \left(x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot \frac{-143}{604800} + \frac{-11}{15120}\right)\right) + \frac{-23}{360}\right), x, x \cdot \frac{1}{6}\right) \]

   7. associate-*r* N/A

      \[\leadsto x \cdot \mathsf{fma}\left(\left(x \cdot x\right) \cdot \left(\color{blue}{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{-143}{604800} + \frac{-11}{15120}\right)} + \frac{-23}{360}\right), x, x \cdot \frac{1}{6}\right) \]

   8. accelerator-lowering-fma.f64 N/A

      \[\leadsto x \cdot \mathsf{fma}\left(\left(x \cdot x\right) \cdot \color{blue}{\mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot \frac{-143}{604800} + \frac{-11}{15120}, \frac{-23}{360}\right)}, x, x \cdot \frac{1}{6}\right) \]

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

      \[\leadsto x \cdot \mathsf{fma}\left(\left(x \cdot x\right) \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \left(x \cdot x\right) \cdot \frac{-143}{604800} + \frac{-11}{15120}, \frac{-23}{360}\right), x, x \cdot \frac{1}{6}\right) \]

   10. associate-*l* N/A

       \[\leadsto x \cdot \mathsf{fma}\left(\left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(x \cdot \frac{-143}{604800}\right)} + \frac{-11}{15120}, \frac{-23}{360}\right), x, x \cdot \frac{1}{6}\right) \]

   11. accelerator-lowering-fma.f64 N/A

       \[\leadsto x \cdot \mathsf{fma}\left(\left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{-143}{604800}, \frac{-11}{15120}\right)}, \frac{-23}{360}\right), x, x \cdot \frac{1}{6}\right) \]

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

       \[\leadsto x \cdot \mathsf{fma}\left(\left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{-143}{604800}}, \frac{-11}{15120}\right), \frac{-23}{360}\right), x, x \cdot \frac{1}{6}\right) \]

   13. *-lowering-*.f64 99.5

       \[\leadsto x \cdot \mathsf{fma}\left(\left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot -0.00023644179894179894, -0.0007275132275132275\right), -0.06388888888888888\right), x, \color{blue}{x \cdot 0.16666666666666666}\right) \]

7. Applied egg-rr 99.5%

   \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(\left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot -0.00023644179894179894, -0.0007275132275132275\right), -0.06388888888888888\right), x, x \cdot 0.16666666666666666\right)} \]
8. Add Preprocessing

Alternative 7: 99.6% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, -0.00023644179894179894, -0.0007275132275132275\right), -0.06388888888888888\right), 0.16666666666666666\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  x
  (*
   x
   (fma
    x
    (*
     x
     (fma
      x
      (* x (fma (* x x) -0.00023644179894179894 -0.0007275132275132275))
      -0.06388888888888888))
    0.16666666666666666))))

C

double code(double x) {
	return x * (x * fma(x, (x * fma(x, (x * fma((x * x), -0.00023644179894179894, -0.0007275132275132275)), -0.06388888888888888)), 0.16666666666666666));
}

Julia

function code(x)
	return Float64(x * Float64(x * fma(x, Float64(x * fma(x, Float64(x * fma(Float64(x * x), -0.00023644179894179894, -0.0007275132275132275)), -0.06388888888888888)), 0.16666666666666666)))
end

Wolfram

code[x_] := N[(x * N[(x * N[(x * N[(x * N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * -0.00023644179894179894 + -0.0007275132275132275), $MachinePrecision]), $MachinePrecision] + -0.06388888888888888), $MachinePrecision]), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.2%

   \[\frac{x - \sin x}{\tan x} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right)} \]
4. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right) \]

   2. associate-*l* N/A

      \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right)\right)} \]

   3. *-commutative N/A

      \[\leadsto x \cdot \color{blue}{\left(\left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right) \cdot x\right)} \]

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

      \[\leadsto \color{blue}{x \cdot \left(\left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right) \cdot x\right)} \]

   5. *-commutative N/A

      \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right)\right)} \]

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

      \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right)\right)} \]

   7. +-commutative N/A

      \[\leadsto x \cdot \left(x \cdot \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right) + \frac{1}{6}\right)}\right) \]

   8. unpow2 N/A

      \[\leadsto x \cdot \left(x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right) + \frac{1}{6}\right)\right) \]

   9. associate-*l* N/A

      \[\leadsto x \cdot \left(x \cdot \left(\color{blue}{x \cdot \left(x \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right)} + \frac{1}{6}\right)\right) \]

   10. accelerator-lowering-fma.f64 N/A

       \[\leadsto x \cdot \left(x \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right), \frac{1}{6}\right)}\right) \]

5. Simplified 99.5%

   \[\leadsto \color{blue}{x \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, -0.00023644179894179894, -0.0007275132275132275\right), -0.06388888888888888\right), 0.16666666666666666\right)\right)} \]
6. Add Preprocessing

Alternative 8: 99.5% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.0007275132275132275, -0.06388888888888888\right), 0.16666666666666666\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  x
  (*
   x
   (fma
    (* x x)
    (fma (* x x) -0.0007275132275132275 -0.06388888888888888)
    0.16666666666666666))))

C

double code(double x) {
	return x * (x * fma((x * x), fma((x * x), -0.0007275132275132275, -0.06388888888888888), 0.16666666666666666));
}

Julia

function code(x)
	return Float64(x * Float64(x * fma(Float64(x * x), fma(Float64(x * x), -0.0007275132275132275, -0.06388888888888888), 0.16666666666666666)))
end

Wolfram

code[x_] := N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.0007275132275132275 + -0.06388888888888888), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.2%

   \[\frac{x - \sin x}{\tan x} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{-11}{15120} \cdot {x}^{2} - \frac{23}{360}\right)\right)} \]
4. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{-11}{15120} \cdot {x}^{2} - \frac{23}{360}\right)\right) \]

   2. associate-*l* N/A

      \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{-11}{15120} \cdot {x}^{2} - \frac{23}{360}\right)\right)\right)} \]

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

      \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{-11}{15120} \cdot {x}^{2} - \frac{23}{360}\right)\right)\right)} \]

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

      \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{-11}{15120} \cdot {x}^{2} - \frac{23}{360}\right)\right)\right)} \]

   5. +-commutative N/A

      \[\leadsto x \cdot \left(x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{-11}{15120} \cdot {x}^{2} - \frac{23}{360}\right) + \frac{1}{6}\right)}\right) \]

   6. accelerator-lowering-fma.f64 N/A

      \[\leadsto x \cdot \left(x \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{-11}{15120} \cdot {x}^{2} - \frac{23}{360}, \frac{1}{6}\right)}\right) \]

   7. unpow2 N/A

      \[\leadsto x \cdot \left(x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-11}{15120} \cdot {x}^{2} - \frac{23}{360}, \frac{1}{6}\right)\right) \]

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

      \[\leadsto x \cdot \left(x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-11}{15120} \cdot {x}^{2} - \frac{23}{360}, \frac{1}{6}\right)\right) \]

   9. sub-neg N/A

      \[\leadsto x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{-11}{15120} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{23}{360}\right)\right)}, \frac{1}{6}\right)\right) \]

   10. *-commutative N/A

       \[\leadsto x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{-11}{15120}} + \left(\mathsf{neg}\left(\frac{23}{360}\right)\right), \frac{1}{6}\right)\right) \]

   11. metadata-eval N/A

       \[\leadsto x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, {x}^{2} \cdot \frac{-11}{15120} + \color{blue}{\frac{-23}{360}}, \frac{1}{6}\right)\right) \]

   12. accelerator-lowering-fma.f64 N/A

       \[\leadsto x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{-11}{15120}, \frac{-23}{360}\right)}, \frac{1}{6}\right)\right) \]

   13. unpow2 N/A

       \[\leadsto x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-11}{15120}, \frac{-23}{360}\right), \frac{1}{6}\right)\right) \]

   14. *-lowering-*.f64 99.3

       \[\leadsto x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, -0.0007275132275132275, -0.06388888888888888\right), 0.16666666666666666\right)\right) \]

5. Simplified 99.3%

   \[\leadsto \color{blue}{x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.0007275132275132275, -0.06388888888888888\right), 0.16666666666666666\right)\right)} \]
6. Add Preprocessing

Alternative 9: 99.4% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x \cdot 0.16666666666666666, x, \left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot -0.06388888888888888\right)\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fma
  (* x 0.16666666666666666)
  x
  (* (* x x) (* x (* x -0.06388888888888888)))))

C

double code(double x) {
	return fma((x * 0.16666666666666666), x, ((x * x) * (x * (x * -0.06388888888888888))));
}

Julia

function code(x)
	return fma(Float64(x * 0.16666666666666666), x, Float64(Float64(x * x) * Float64(x * Float64(x * -0.06388888888888888))))
end

Wolfram

code[x_] := N[(N[(x * 0.16666666666666666), $MachinePrecision] * x + N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * -0.06388888888888888), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.2%

   \[\frac{x - \sin x}{\tan x} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{6} + \frac{-23}{360} \cdot {x}^{2}\right)} \]
4. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{1}{6} + \frac{-23}{360} \cdot {x}^{2}\right) \]

   2. associate-*l* N/A

      \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(\frac{1}{6} + \frac{-23}{360} \cdot {x}^{2}\right)\right)} \]

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

      \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(\frac{1}{6} + \frac{-23}{360} \cdot {x}^{2}\right)\right)} \]

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

      \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{6} + \frac{-23}{360} \cdot {x}^{2}\right)\right)} \]

   5. +-commutative N/A

      \[\leadsto x \cdot \left(x \cdot \color{blue}{\left(\frac{-23}{360} \cdot {x}^{2} + \frac{1}{6}\right)}\right) \]

   6. *-commutative N/A

      \[\leadsto x \cdot \left(x \cdot \left(\color{blue}{{x}^{2} \cdot \frac{-23}{360}} + \frac{1}{6}\right)\right) \]

   7. accelerator-lowering-fma.f64 N/A

      \[\leadsto x \cdot \left(x \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{-23}{360}, \frac{1}{6}\right)}\right) \]

   8. unpow2 N/A

      \[\leadsto x \cdot \left(x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-23}{360}, \frac{1}{6}\right)\right) \]

   9. *-lowering-*.f64 99.1

      \[\leadsto x \cdot \left(x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, -0.06388888888888888, 0.16666666666666666\right)\right) \]

5. Simplified 99.1%

   \[\leadsto \color{blue}{x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, -0.06388888888888888, 0.16666666666666666\right)\right)} \]
6. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \color{blue}{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{-23}{360} + \frac{1}{6}\right)} \]

   2. +-commutative N/A

      \[\leadsto \left(x \cdot x\right) \cdot \color{blue}{\left(\frac{1}{6} + \left(x \cdot x\right) \cdot \frac{-23}{360}\right)} \]

   3. distribute-rgt-in N/A

      \[\leadsto \color{blue}{\frac{1}{6} \cdot \left(x \cdot x\right) + \left(\left(x \cdot x\right) \cdot \frac{-23}{360}\right) \cdot \left(x \cdot x\right)} \]

   4. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\frac{1}{6} \cdot x\right) \cdot x} + \left(\left(x \cdot x\right) \cdot \frac{-23}{360}\right) \cdot \left(x \cdot x\right) \]

   5. *-commutative N/A

      \[\leadsto \color{blue}{\left(x \cdot \frac{1}{6}\right)} \cdot x + \left(\left(x \cdot x\right) \cdot \frac{-23}{360}\right) \cdot \left(x \cdot x\right) \]

   6. accelerator-lowering-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot \frac{1}{6}, x, \left(\left(x \cdot x\right) \cdot \frac{-23}{360}\right) \cdot \left(x \cdot x\right)\right)} \]

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

      \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot \frac{1}{6}}, x, \left(\left(x \cdot x\right) \cdot \frac{-23}{360}\right) \cdot \left(x \cdot x\right)\right) \]

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

      \[\leadsto \mathsf{fma}\left(x \cdot \frac{1}{6}, x, \color{blue}{\left(\left(x \cdot x\right) \cdot \frac{-23}{360}\right) \cdot \left(x \cdot x\right)}\right) \]

   9. associate-*l* N/A

      \[\leadsto \mathsf{fma}\left(x \cdot \frac{1}{6}, x, \color{blue}{\left(x \cdot \left(x \cdot \frac{-23}{360}\right)\right)} \cdot \left(x \cdot x\right)\right) \]

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

       \[\leadsto \mathsf{fma}\left(x \cdot \frac{1}{6}, x, \color{blue}{\left(x \cdot \left(x \cdot \frac{-23}{360}\right)\right)} \cdot \left(x \cdot x\right)\right) \]

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

       \[\leadsto \mathsf{fma}\left(x \cdot \frac{1}{6}, x, \left(x \cdot \color{blue}{\left(x \cdot \frac{-23}{360}\right)}\right) \cdot \left(x \cdot x\right)\right) \]

   12. *-lowering-*.f64 99.2

       \[\leadsto \mathsf{fma}\left(x \cdot 0.16666666666666666, x, \left(x \cdot \left(x \cdot -0.06388888888888888\right)\right) \cdot \color{blue}{\left(x \cdot x\right)}\right) \]

7. Applied egg-rr 99.2%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot 0.16666666666666666, x, \left(x \cdot \left(x \cdot -0.06388888888888888\right)\right) \cdot \left(x \cdot x\right)\right)} \]

8. Final simplification 99.2%

   \[\leadsto \mathsf{fma}\left(x \cdot 0.16666666666666666, x, \left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot -0.06388888888888888\right)\right)\right) \]
9. Add Preprocessing

Alternative 10: 99.4% accurate, 8.0× speedup

Math

\[\begin{array}{l} \\ x \cdot \mathsf{fma}\left(x \cdot \left(x \cdot x\right), -0.06388888888888888, x \cdot 0.16666666666666666\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* x (fma (* x (* x x)) -0.06388888888888888 (* x 0.16666666666666666))))

C

double code(double x) {
	return x * fma((x * (x * x)), -0.06388888888888888, (x * 0.16666666666666666));
}

Julia

function code(x)
	return Float64(x * fma(Float64(x * Float64(x * x)), -0.06388888888888888, Float64(x * 0.16666666666666666)))
end

Wolfram

code[x_] := N[(x * N[(N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] * -0.06388888888888888 + N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.2%

   \[\frac{x - \sin x}{\tan x} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{6} + \frac{-23}{360} \cdot {x}^{2}\right)} \]
4. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{1}{6} + \frac{-23}{360} \cdot {x}^{2}\right) \]

   2. associate-*l* N/A

      \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(\frac{1}{6} + \frac{-23}{360} \cdot {x}^{2}\right)\right)} \]

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

      \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(\frac{1}{6} + \frac{-23}{360} \cdot {x}^{2}\right)\right)} \]

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

      \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{6} + \frac{-23}{360} \cdot {x}^{2}\right)\right)} \]

   5. +-commutative N/A

      \[\leadsto x \cdot \left(x \cdot \color{blue}{\left(\frac{-23}{360} \cdot {x}^{2} + \frac{1}{6}\right)}\right) \]

   6. *-commutative N/A

      \[\leadsto x \cdot \left(x \cdot \left(\color{blue}{{x}^{2} \cdot \frac{-23}{360}} + \frac{1}{6}\right)\right) \]

   7. accelerator-lowering-fma.f64 N/A

      \[\leadsto x \cdot \left(x \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{-23}{360}, \frac{1}{6}\right)}\right) \]

   8. unpow2 N/A

      \[\leadsto x \cdot \left(x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-23}{360}, \frac{1}{6}\right)\right) \]

   9. *-lowering-*.f64 99.1

      \[\leadsto x \cdot \left(x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, -0.06388888888888888, 0.16666666666666666\right)\right) \]

5. Simplified 99.1%

   \[\leadsto \color{blue}{x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, -0.06388888888888888, 0.16666666666666666\right)\right)} \]
6. Step-by-step derivation

   1. distribute-lft-in N/A

      \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(\left(x \cdot x\right) \cdot \frac{-23}{360}\right) + x \cdot \frac{1}{6}\right)} \]

   2. associate-*r* N/A

      \[\leadsto x \cdot \left(\color{blue}{\left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{-23}{360}} + x \cdot \frac{1}{6}\right) \]

   3. accelerator-lowering-fma.f64 N/A

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), \frac{-23}{360}, x \cdot \frac{1}{6}\right)} \]

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

      \[\leadsto x \cdot \mathsf{fma}\left(\color{blue}{x \cdot \left(x \cdot x\right)}, \frac{-23}{360}, x \cdot \frac{1}{6}\right) \]

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

      \[\leadsto x \cdot \mathsf{fma}\left(x \cdot \color{blue}{\left(x \cdot x\right)}, \frac{-23}{360}, x \cdot \frac{1}{6}\right) \]

   6. *-lowering-*.f64 99.1

      \[\leadsto x \cdot \mathsf{fma}\left(x \cdot \left(x \cdot x\right), -0.06388888888888888, \color{blue}{x \cdot 0.16666666666666666}\right) \]

7. Applied egg-rr 99.1%

   \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), -0.06388888888888888, x \cdot 0.16666666666666666\right)} \]
8. Add Preprocessing

Alternative 11: 99.4% accurate, 9.8× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, -0.06388888888888888, 0.16666666666666666\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* x (* x (fma (* x x) -0.06388888888888888 0.16666666666666666))))

C

double code(double x) {
	return x * (x * fma((x * x), -0.06388888888888888, 0.16666666666666666));
}

Julia

function code(x)
	return Float64(x * Float64(x * fma(Float64(x * x), -0.06388888888888888, 0.16666666666666666)))
end

Wolfram

code[x_] := N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * -0.06388888888888888 + 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.2%

   \[\frac{x - \sin x}{\tan x} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{6} + \frac{-23}{360} \cdot {x}^{2}\right)} \]
4. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{1}{6} + \frac{-23}{360} \cdot {x}^{2}\right) \]

   2. associate-*l* N/A

      \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(\frac{1}{6} + \frac{-23}{360} \cdot {x}^{2}\right)\right)} \]

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

      \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(\frac{1}{6} + \frac{-23}{360} \cdot {x}^{2}\right)\right)} \]

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

      \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(\frac{1}{6} + \frac{-23}{360} \cdot {x}^{2}\right)\right)} \]

   5. +-commutative N/A

      \[\leadsto x \cdot \left(x \cdot \color{blue}{\left(\frac{-23}{360} \cdot {x}^{2} + \frac{1}{6}\right)}\right) \]

   6. *-commutative N/A

      \[\leadsto x \cdot \left(x \cdot \left(\color{blue}{{x}^{2} \cdot \frac{-23}{360}} + \frac{1}{6}\right)\right) \]

   7. accelerator-lowering-fma.f64 N/A

      \[\leadsto x \cdot \left(x \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{-23}{360}, \frac{1}{6}\right)}\right) \]

   8. unpow2 N/A

      \[\leadsto x \cdot \left(x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-23}{360}, \frac{1}{6}\right)\right) \]

   9. *-lowering-*.f64 99.1

      \[\leadsto x \cdot \left(x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, -0.06388888888888888, 0.16666666666666666\right)\right) \]

5. Simplified 99.1%

   \[\leadsto \color{blue}{x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, -0.06388888888888888, 0.16666666666666666\right)\right)} \]
6. Add Preprocessing

Alternative 12: 98.8% accurate, 19.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.2%

   \[\frac{x - \sin x}{\tan x} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{1}{6} \cdot {x}^{2}} \]
4. Step-by-step derivation

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

      \[\leadsto \color{blue}{\frac{1}{6} \cdot {x}^{2}} \]

   2. unpow2 N/A

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

   3. *-lowering-*.f64 98.4

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

5. Simplified 98.4%

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

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

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

   4. *-lowering-*.f64 98.4

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

7. Applied egg-rr 98.4%

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

8. Final simplification 98.4%

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

Alternative 13: 98.8% accurate, 19.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.2%

   \[\frac{x - \sin x}{\tan x} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{1}{6} \cdot {x}^{2}} \]
4. Step-by-step derivation

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

      \[\leadsto \color{blue}{\frac{1}{6} \cdot {x}^{2}} \]

   2. unpow2 N/A

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

   3. *-lowering-*.f64 98.4

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

5. Simplified 98.4%

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

6. Final simplification 98.4%

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

Developer Target 1: 98.8% accurate, 19.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024205 
(FPCore (x)
  :name "ENA, Section 1.4, Exercise 4a"
  :precision binary64
  :pre (and (<= -1.0 x) (<= x 1.0))

  :alt
  (! :herbie-platform default (* 1/6 (* x x)))

  (/ (- x (sin x)) (tan x)))