ENA, Section 1.4, Exercise 4a

Percentage Accurate: 54.0% → 99.6%

Time: 13.9s

Alternatives: 9

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: 54.0% 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, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 50.6%

   \[\frac{x - \sin x}{\tan x} \]
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. lower-*.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. lower-*.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. lower-*.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. lower-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. Applied rewrites 81.3%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-fma.f64 N/A

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

   6. lift-fma.f64 N/A

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

   7. lift-fma.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lift-tan.f64 N/A

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

7. Applied rewrites 99.6%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-fma.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-fma.f64 N/A

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

   6. flip-+ N/A

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

   7. lower-/.f64 N/A

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

9. Applied rewrites 99.6%

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

Alternative 2: 99.6% accurate, 1.3× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (*
  (/ x (tan 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 / tan(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(Float64(x / tan(x)) * Float64(Float64(x * x) * fma(Float64(x * x), fma(x, Float64(x * fma(Float64(x * x), -2.7557319223985893e-6, 0.0001984126984126984)), -0.008333333333333333), 0.16666666666666666)))
end

Wolfram

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

Derivation

1. Initial program 50.6%

   \[\frac{x - \sin x}{\tan x} \]
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. lower-*.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. lower-*.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. lower-*.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. lower-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. Applied rewrites 81.3%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-fma.f64 N/A

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

   6. lift-fma.f64 N/A

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

   7. lift-fma.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lift-tan.f64 N/A

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

7. Applied rewrites 99.6%

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

8. Final simplification 99.6%

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

Alternative 3: 99.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, -0.00023644179894179894, -0.0007275132275132275\right), -0.06388888888888888\right)\\ t_1 := 0.16666666666666666 - x \cdot \left(x \cdot t\_0\right)\\ x \cdot \left(\mathsf{fma}\left(x \cdot x, t\_0, 0.16666666666666666\right) \cdot \frac{x \cdot t\_1}{t\_1}\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0
         (fma
          x
          (* x (fma (* x x) -0.00023644179894179894 -0.0007275132275132275))
          -0.06388888888888888))
        (t_1 (- 0.16666666666666666 (* x (* x t_0)))))
   (* x (* (fma (* x x) t_0 0.16666666666666666) (/ (* x t_1) t_1)))))

C

double code(double x) {
	double t_0 = fma(x, (x * fma((x * x), -0.00023644179894179894, -0.0007275132275132275)), -0.06388888888888888);
	double t_1 = 0.16666666666666666 - (x * (x * t_0));
	return x * (fma((x * x), t_0, 0.16666666666666666) * ((x * t_1) / t_1));
}

Julia

function code(x)
	t_0 = fma(x, Float64(x * fma(Float64(x * x), -0.00023644179894179894, -0.0007275132275132275)), -0.06388888888888888)
	t_1 = Float64(0.16666666666666666 - Float64(x * Float64(x * t_0)))
	return Float64(x * Float64(fma(Float64(x * x), t_0, 0.16666666666666666) * Float64(Float64(x * t_1) / t_1)))
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * -0.00023644179894179894 + -0.0007275132275132275), $MachinePrecision]), $MachinePrecision] + -0.06388888888888888), $MachinePrecision]}, Block[{t$95$1 = N[(0.16666666666666666 - N[(x * N[(x * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x * N[(N[(N[(x * x), $MachinePrecision] * t$95$0 + 0.16666666666666666), $MachinePrecision] * N[(N[(x * t$95$1), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 50.6%

   \[\frac{x - \sin x}{\tan x} \]
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. lower-*.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. lower-*.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. lower-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. Applied rewrites 99.6%

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

   1. lift-*.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-fma.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. distribute-rgt-in N/A

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

   7. +-commutative N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. *-commutative N/A

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

   11. associate-*r* N/A

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

7. Applied rewrites 99.5%

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

9. Applied rewrites 99.5%

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

10. Taylor expanded in x around 0

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

    1. +-commutative N/A

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

    2. lower-fma.f64 N/A

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

    3. unpow2 N/A

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

    4. lower-*.f64 N/A

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

    5. sub-neg N/A

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

    6. unpow2 N/A

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

    7. associate-*l* N/A

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

    8. metadata-eval N/A

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

    9. lower-fma.f64 N/A

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

    10. lower-*.f64 N/A

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

    11. sub-neg N/A

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

    12. *-commutative N/A

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

    13. metadata-eval N/A

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

    14. lower-fma.f64 N/A

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

    15. unpow2 N/A

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

    16. lower-*.f64 99.6

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

12. Applied rewrites 99.6%

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

Alternative 4: 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 50.6%

   \[\frac{x - \sin x}{\tan x} \]
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. lower-*.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. lower-*.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. lower-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. Applied rewrites 99.6%

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

Alternative 5: 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 50.6%

   \[\frac{x - \sin x}{\tan x} \]
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. lower-*.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. lower-*.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. lower-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. lower-*.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. lower-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. lower-*.f64 99.4

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

5. Applied rewrites 99.4%

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

Alternative 6: 99.3% 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 50.6%

   \[\frac{x - \sin x}{\tan x} \]
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. lower-*.f64 N/A

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

   4. lower-*.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. lower-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. lower-*.f64 99.3

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

5. Applied rewrites 99.3%

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

Alternative 7: 98.7% accurate, 12.6× speedup

Math

\[\begin{array}{l} \\ x \cdot \frac{x}{6} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.6%

   \[\frac{x - \sin x}{\tan x} \]
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. lower-*.f64 N/A

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

   4. lower-*.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. lower-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. lower-*.f64 99.3

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

5. Applied rewrites 99.3%

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

   1. lift-*.f64 N/A

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

   2. flip3-+ N/A

      \[\leadsto x \cdot \left(x \cdot \color{blue}{\frac{{\left(\left(x \cdot x\right) \cdot \frac{-23}{360}\right)}^{3} + {\frac{1}{6}}^{3}}{\left(\left(x \cdot x\right) \cdot \frac{-23}{360}\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{-23}{360}\right) + \left(\frac{1}{6} \cdot \frac{1}{6} - \left(\left(x \cdot x\right) \cdot \frac{-23}{360}\right) \cdot \frac{1}{6}\right)}}\right) \]

   3. clear-num N/A

      \[\leadsto x \cdot \left(x \cdot \color{blue}{\frac{1}{\frac{\left(\left(x \cdot x\right) \cdot \frac{-23}{360}\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{-23}{360}\right) + \left(\frac{1}{6} \cdot \frac{1}{6} - \left(\left(x \cdot x\right) \cdot \frac{-23}{360}\right) \cdot \frac{1}{6}\right)}{{\left(\left(x \cdot x\right) \cdot \frac{-23}{360}\right)}^{3} + {\frac{1}{6}}^{3}}}}\right) \]

   4. un-div-inv N/A

      \[\leadsto x \cdot \color{blue}{\frac{x}{\frac{\left(\left(x \cdot x\right) \cdot \frac{-23}{360}\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{-23}{360}\right) + \left(\frac{1}{6} \cdot \frac{1}{6} - \left(\left(x \cdot x\right) \cdot \frac{-23}{360}\right) \cdot \frac{1}{6}\right)}{{\left(\left(x \cdot x\right) \cdot \frac{-23}{360}\right)}^{3} + {\frac{1}{6}}^{3}}}} \]

   5. lower-/.f64 N/A

      \[\leadsto x \cdot \color{blue}{\frac{x}{\frac{\left(\left(x \cdot x\right) \cdot \frac{-23}{360}\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{-23}{360}\right) + \left(\frac{1}{6} \cdot \frac{1}{6} - \left(\left(x \cdot x\right) \cdot \frac{-23}{360}\right) \cdot \frac{1}{6}\right)}{{\left(\left(x \cdot x\right) \cdot \frac{-23}{360}\right)}^{3} + {\frac{1}{6}}^{3}}}} \]

   6. clear-num N/A

      \[\leadsto x \cdot \frac{x}{\color{blue}{\frac{1}{\frac{{\left(\left(x \cdot x\right) \cdot \frac{-23}{360}\right)}^{3} + {\frac{1}{6}}^{3}}{\left(\left(x \cdot x\right) \cdot \frac{-23}{360}\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{-23}{360}\right) + \left(\frac{1}{6} \cdot \frac{1}{6} - \left(\left(x \cdot x\right) \cdot \frac{-23}{360}\right) \cdot \frac{1}{6}\right)}}}} \]

   7. flip3-+ N/A

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

   8. lift-fma.f64 N/A

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

   9. lower-/.f64 99.4

      \[\leadsto x \cdot \frac{x}{\color{blue}{\frac{1}{\mathsf{fma}\left(x \cdot x, -0.06388888888888888, 0.16666666666666666\right)}}} \]

   10. lift-fma.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. associate-*l* N/A

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

7. Applied rewrites 99.4%

   \[\leadsto x \cdot \color{blue}{\frac{x}{\frac{1}{\mathsf{fma}\left(x, x \cdot -0.06388888888888888, 0.16666666666666666\right)}}} \]

8. Taylor expanded in x around 0

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

10. Applied rewrites 99.1%

    \[\leadsto x \cdot \frac{x}{\color{blue}{6}} \]
11. Add Preprocessing

Alternative 8: 98.7% 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 50.6%

   \[\frac{x - \sin x}{\tan x} \]
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. lower-*.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. lower-*.f64 99.0

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

5. Applied rewrites 99.0%

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

   1. associate-*r* N/A

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

   2. lower-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 99.1

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

7. Applied rewrites 99.1%

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

8. Final simplification 99.1%

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

Alternative 9: 98.6% 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 50.6%

   \[\frac{x - \sin x}{\tan x} \]
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. lower-*.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. lower-*.f64 99.0

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

5. Applied rewrites 99.0%

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

6. Final simplification 99.0%

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

Developer Target 1: 98.6% 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 2024219 
(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)))