ENA, Section 1.4, Exercise 4a

Percentage Accurate: 53.4% → 99.5%

Time: 13.2s

Alternatives: 8

Speedup: 41.0×

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: 53.4% 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: 99.5% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -0.06388888888888888 + x \cdot \left(x \cdot \left(-0.0007275132275132275 + \left(x \cdot x\right) \cdot -0.00023644179894179894\right)\right)\\ t_1 := \left(x \cdot x\right) \cdot t\_0\\ x \cdot \frac{x \cdot \left(0.027777777777777776 - x \cdot \left(\left(x \cdot t\_0\right) \cdot t\_1\right)\right)}{0.16666666666666666 - t\_1} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0
         (+
          -0.06388888888888888
          (*
           x
           (*
            x
            (+ -0.0007275132275132275 (* (* x x) -0.00023644179894179894))))))
        (t_1 (* (* x x) t_0)))
   (*
    x
    (/
     (* x (- 0.027777777777777776 (* x (* (* x t_0) t_1))))
     (- 0.16666666666666666 t_1)))))

C

double code(double x) {
	double t_0 = -0.06388888888888888 + (x * (x * (-0.0007275132275132275 + ((x * x) * -0.00023644179894179894))));
	double t_1 = (x * x) * t_0;
	return x * ((x * (0.027777777777777776 - (x * ((x * t_0) * t_1)))) / (0.16666666666666666 - t_1));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    t_0 = (-0.06388888888888888d0) + (x * (x * ((-0.0007275132275132275d0) + ((x * x) * (-0.00023644179894179894d0)))))
    t_1 = (x * x) * t_0
    code = x * ((x * (0.027777777777777776d0 - (x * ((x * t_0) * t_1)))) / (0.16666666666666666d0 - t_1))
end function

Java

public static double code(double x) {
	double t_0 = -0.06388888888888888 + (x * (x * (-0.0007275132275132275 + ((x * x) * -0.00023644179894179894))));
	double t_1 = (x * x) * t_0;
	return x * ((x * (0.027777777777777776 - (x * ((x * t_0) * t_1)))) / (0.16666666666666666 - t_1));
}

Python

def code(x):
	t_0 = -0.06388888888888888 + (x * (x * (-0.0007275132275132275 + ((x * x) * -0.00023644179894179894))))
	t_1 = (x * x) * t_0
	return x * ((x * (0.027777777777777776 - (x * ((x * t_0) * t_1)))) / (0.16666666666666666 - t_1))

Julia

function code(x)
	t_0 = Float64(-0.06388888888888888 + Float64(x * Float64(x * Float64(-0.0007275132275132275 + Float64(Float64(x * x) * -0.00023644179894179894)))))
	t_1 = Float64(Float64(x * x) * t_0)
	return Float64(x * Float64(Float64(x * Float64(0.027777777777777776 - Float64(x * Float64(Float64(x * t_0) * t_1)))) / Float64(0.16666666666666666 - t_1)))
end

MATLAB

function tmp = code(x)
	t_0 = -0.06388888888888888 + (x * (x * (-0.0007275132275132275 + ((x * x) * -0.00023644179894179894))));
	t_1 = (x * x) * t_0;
	tmp = x * ((x * (0.027777777777777776 - (x * ((x * t_0) * t_1)))) / (0.16666666666666666 - t_1));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(-0.06388888888888888 + N[(x * N[(x * N[(-0.0007275132275132275 + N[(N[(x * x), $MachinePrecision] * -0.00023644179894179894), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * x), $MachinePrecision] * t$95$0), $MachinePrecision]}, N[(x * N[(N[(x * N[(0.027777777777777776 - N[(x * N[(N[(x * t$95$0), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.16666666666666666 - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 51.6%

   \[\frac{x - \sin x}{\tan x} \]
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

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

   2. associate-*l* N/A

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

   3. *-commutative N/A

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

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

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right) \cdot x\right)}\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right)}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right)}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right)}\right)\right)\right) \]

   8. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \left(\left(x \cdot x\right) \cdot \left(\color{blue}{{x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right)} - \frac{23}{360}\right)\right)\right)\right)\right) \]

   9. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \left(x \cdot \color{blue}{\left(x \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right)}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right)}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)}\right)\right)\right)\right)\right) \]

   12. sub-neg N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{23}{360}\right)\right)}\right)\right)\right)\right)\right)\right) \]

5. Simplified 99.5%

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

   1. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\frac{1}{6} + x \cdot \left(x \cdot \left(\frac{-23}{360} + \left(x \cdot x\right) \cdot \left(\frac{-11}{15120} + \left(x \cdot x\right) \cdot \frac{-143}{604800}\right)\right)\right)\right) \cdot \color{blue}{x}\right)\right) \]

   2. flip-+ N/A

      \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{\frac{1}{6} \cdot \frac{1}{6} - \left(x \cdot \left(x \cdot \left(\frac{-23}{360} + \left(x \cdot x\right) \cdot \left(\frac{-11}{15120} + \left(x \cdot x\right) \cdot \frac{-143}{604800}\right)\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(\frac{-23}{360} + \left(x \cdot x\right) \cdot \left(\frac{-11}{15120} + \left(x \cdot x\right) \cdot \frac{-143}{604800}\right)\right)\right)\right)}{\frac{1}{6} - x \cdot \left(x \cdot \left(\frac{-23}{360} + \left(x \cdot x\right) \cdot \left(\frac{-11}{15120} + \left(x \cdot x\right) \cdot \frac{-143}{604800}\right)\right)\right)} \cdot x\right)\right) \]

   3. associate-*l/ N/A

      \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{\left(\frac{1}{6} \cdot \frac{1}{6} - \left(x \cdot \left(x \cdot \left(\frac{-23}{360} + \left(x \cdot x\right) \cdot \left(\frac{-11}{15120} + \left(x \cdot x\right) \cdot \frac{-143}{604800}\right)\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(\frac{-23}{360} + \left(x \cdot x\right) \cdot \left(\frac{-11}{15120} + \left(x \cdot x\right) \cdot \frac{-143}{604800}\right)\right)\right)\right)\right) \cdot x}{\color{blue}{\frac{1}{6} - x \cdot \left(x \cdot \left(\frac{-23}{360} + \left(x \cdot x\right) \cdot \left(\frac{-11}{15120} + \left(x \cdot x\right) \cdot \frac{-143}{604800}\right)\right)\right)}}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\left(\left(\frac{1}{6} \cdot \frac{1}{6} - \left(x \cdot \left(x \cdot \left(\frac{-23}{360} + \left(x \cdot x\right) \cdot \left(\frac{-11}{15120} + \left(x \cdot x\right) \cdot \frac{-143}{604800}\right)\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(\frac{-23}{360} + \left(x \cdot x\right) \cdot \left(\frac{-11}{15120} + \left(x \cdot x\right) \cdot \frac{-143}{604800}\right)\right)\right)\right)\right) \cdot x\right), \color{blue}{\left(\frac{1}{6} - x \cdot \left(x \cdot \left(\frac{-23}{360} + \left(x \cdot x\right) \cdot \left(\frac{-11}{15120} + \left(x \cdot x\right) \cdot \frac{-143}{604800}\right)\right)\right)\right)}\right)\right) \]

7. Applied egg-rr 99.6%

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

8. Final simplification 99.6%

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

Alternative 2: 99.5% accurate, 8.9× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * (x * (0.16666666666666666d0 + (x * (x * ((-0.06388888888888888d0) + ((x * x) * ((-0.0007275132275132275d0) + ((x * x) * (-0.00023644179894179894d0)))))))))
end function

Java

public static double code(double x) {
	return x * (x * (0.16666666666666666 + (x * (x * (-0.06388888888888888 + ((x * x) * (-0.0007275132275132275 + ((x * x) * -0.00023644179894179894))))))));
}

Python

def code(x):
	return x * (x * (0.16666666666666666 + (x * (x * (-0.06388888888888888 + ((x * x) * (-0.0007275132275132275 + ((x * x) * -0.00023644179894179894))))))))

Julia

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

MATLAB

function tmp = code(x)
	tmp = x * (x * (0.16666666666666666 + (x * (x * (-0.06388888888888888 + ((x * x) * (-0.0007275132275132275 + ((x * x) * -0.00023644179894179894))))))));
end

Wolfram

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

Derivation

1. Initial program 51.6%

   \[\frac{x - \sin x}{\tan x} \]
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

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

   2. associate-*l* N/A

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

   3. *-commutative N/A

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

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

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right) \cdot x\right)}\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right)}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right)}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right)}\right)\right)\right) \]

   8. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \left(\left(x \cdot x\right) \cdot \left(\color{blue}{{x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right)} - \frac{23}{360}\right)\right)\right)\right)\right) \]

   9. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \left(x \cdot \color{blue}{\left(x \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right)}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right)}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)}\right)\right)\right)\right)\right) \]

   12. sub-neg N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{23}{360}\right)\right)}\right)\right)\right)\right)\right)\right) \]

5. Simplified 99.5%

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

Alternative 3: 99.5% accurate, 12.1× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * (x * (0.16666666666666666d0 + (x * (x * ((-0.06388888888888888d0) + ((-0.0007275132275132275d0) * (x * x)))))))
end function

Java

public static double code(double x) {
	return x * (x * (0.16666666666666666 + (x * (x * (-0.06388888888888888 + (-0.0007275132275132275 * (x * x)))))));
}

Python

def code(x):
	return x * (x * (0.16666666666666666 + (x * (x * (-0.06388888888888888 + (-0.0007275132275132275 * (x * x)))))))

Julia

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

MATLAB

function tmp = code(x)
	tmp = x * (x * (0.16666666666666666 + (x * (x * (-0.06388888888888888 + (-0.0007275132275132275 * (x * x)))))));
end

Wolfram

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

Derivation

1. Initial program 51.6%

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

      \[\leadsto \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{-11}{15120} \cdot {x}^{2} - \frac{23}{360}\right)\right)}\right) \]

   2. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{1}{6}} + {x}^{2} \cdot \left(\frac{-11}{15120} \cdot {x}^{2} - \frac{23}{360}\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{1}{6}} + {x}^{2} \cdot \left(\frac{-11}{15120} \cdot {x}^{2} - \frac{23}{360}\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{6}, \color{blue}{\left({x}^{2} \cdot \left(\frac{-11}{15120} \cdot {x}^{2} - \frac{23}{360}\right)\right)}\right)\right) \]

   5. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{6}, \left(\left(x \cdot x\right) \cdot \left(\color{blue}{\frac{-11}{15120} \cdot {x}^{2}} - \frac{23}{360}\right)\right)\right)\right) \]

   6. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{6}, \left(x \cdot \color{blue}{\left(x \cdot \left(\frac{-11}{15120} \cdot {x}^{2} - \frac{23}{360}\right)\right)}\right)\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{6}, \left(x \cdot \left(\left(\frac{-11}{15120} \cdot {x}^{2} - \frac{23}{360}\right) \cdot \color{blue}{x}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \color{blue}{\left(\left(\frac{-11}{15120} \cdot {x}^{2} - \frac{23}{360}\right) \cdot x\right)}\right)\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left(\frac{-11}{15120} \cdot {x}^{2} - \frac{23}{360}\right)}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{-11}{15120} \cdot {x}^{2} - \frac{23}{360}\right)}\right)\right)\right)\right) \]

   11. sub-neg N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{-11}{15120} \cdot {x}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{23}{360}\right)\right)}\right)\right)\right)\right)\right) \]

   12. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{-11}{15120} \cdot {x}^{2} + \frac{-23}{360}\right)\right)\right)\right)\right) \]

   13. +-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{-23}{360} + \color{blue}{\frac{-11}{15120} \cdot {x}^{2}}\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-23}{360}, \color{blue}{\left(\frac{-11}{15120} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right) \]

   15. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-23}{360}, \left({x}^{2} \cdot \color{blue}{\frac{-11}{15120}}\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-23}{360}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{-11}{15120}}\right)\right)\right)\right)\right)\right) \]

   17. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-23}{360}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{-11}{15120}\right)\right)\right)\right)\right)\right) \]

   18. *-lowering-*.f64 99.4%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-23}{360}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-11}{15120}\right)\right)\right)\right)\right)\right) \]

5. Simplified 99.4%

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

   1. associate-*l* N/A

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

   2. *-commutative N/A

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

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

      \[\leadsto \mathsf{*.f64}\left(\left(x \cdot \left(\frac{1}{6} + x \cdot \left(x \cdot \left(\frac{-23}{360} + \left(x \cdot x\right) \cdot \frac{-11}{15120}\right)\right)\right)\right), \color{blue}{x}\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{1}{6} + x \cdot \left(x \cdot \left(\frac{-23}{360} + \left(x \cdot x\right) \cdot \frac{-11}{15120}\right)\right)\right)\right), x\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \left(x \cdot \left(x \cdot \left(\frac{-23}{360} + \left(x \cdot x\right) \cdot \frac{-11}{15120}\right)\right)\right)\right)\right), x\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{-23}{360} + \left(x \cdot x\right) \cdot \frac{-11}{15120}\right)\right)\right)\right)\right), x\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{-23}{360} + \left(x \cdot x\right) \cdot \frac{-11}{15120}\right)\right)\right)\right)\right), x\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-23}{360}, \left(\left(x \cdot x\right) \cdot \frac{-11}{15120}\right)\right)\right)\right)\right)\right), x\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-23}{360}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{-11}{15120}\right)\right)\right)\right)\right)\right), x\right) \]

   10. *-lowering-*.f64 99.5%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-23}{360}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-11}{15120}\right)\right)\right)\right)\right)\right), x\right) \]

7. Applied egg-rr 99.5%

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

8. Final simplification 99.5%

   \[\leadsto x \cdot \left(x \cdot \left(0.16666666666666666 + x \cdot \left(x \cdot \left(-0.06388888888888888 + -0.0007275132275132275 \cdot \left(x \cdot x\right)\right)\right)\right)\right) \]
9. Add Preprocessing

Alternative 4: 99.4% accurate, 12.1× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (x * x) * (0.16666666666666666d0 + (x * (x * ((-0.06388888888888888d0) + ((-0.0007275132275132275d0) * (x * x))))))
end function

Java

public static double code(double x) {
	return (x * x) * (0.16666666666666666 + (x * (x * (-0.06388888888888888 + (-0.0007275132275132275 * (x * x))))));
}

Python

def code(x):
	return (x * x) * (0.16666666666666666 + (x * (x * (-0.06388888888888888 + (-0.0007275132275132275 * (x * x))))))

Julia

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

MATLAB

function tmp = code(x)
	tmp = (x * x) * (0.16666666666666666 + (x * (x * (-0.06388888888888888 + (-0.0007275132275132275 * (x * x))))));
end

Wolfram

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

Derivation

1. Initial program 51.6%

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

      \[\leadsto \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{-11}{15120} \cdot {x}^{2} - \frac{23}{360}\right)\right)}\right) \]

   2. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{1}{6}} + {x}^{2} \cdot \left(\frac{-11}{15120} \cdot {x}^{2} - \frac{23}{360}\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{1}{6}} + {x}^{2} \cdot \left(\frac{-11}{15120} \cdot {x}^{2} - \frac{23}{360}\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{6}, \color{blue}{\left({x}^{2} \cdot \left(\frac{-11}{15120} \cdot {x}^{2} - \frac{23}{360}\right)\right)}\right)\right) \]

   5. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{6}, \left(\left(x \cdot x\right) \cdot \left(\color{blue}{\frac{-11}{15120} \cdot {x}^{2}} - \frac{23}{360}\right)\right)\right)\right) \]

   6. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{6}, \left(x \cdot \color{blue}{\left(x \cdot \left(\frac{-11}{15120} \cdot {x}^{2} - \frac{23}{360}\right)\right)}\right)\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{6}, \left(x \cdot \left(\left(\frac{-11}{15120} \cdot {x}^{2} - \frac{23}{360}\right) \cdot \color{blue}{x}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \color{blue}{\left(\left(\frac{-11}{15120} \cdot {x}^{2} - \frac{23}{360}\right) \cdot x\right)}\right)\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left(\frac{-11}{15120} \cdot {x}^{2} - \frac{23}{360}\right)}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{-11}{15120} \cdot {x}^{2} - \frac{23}{360}\right)}\right)\right)\right)\right) \]

   11. sub-neg N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{-11}{15120} \cdot {x}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{23}{360}\right)\right)}\right)\right)\right)\right)\right) \]

   12. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{-11}{15120} \cdot {x}^{2} + \frac{-23}{360}\right)\right)\right)\right)\right) \]

   13. +-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{-23}{360} + \color{blue}{\frac{-11}{15120} \cdot {x}^{2}}\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-23}{360}, \color{blue}{\left(\frac{-11}{15120} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right) \]

   15. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-23}{360}, \left({x}^{2} \cdot \color{blue}{\frac{-11}{15120}}\right)\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-23}{360}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{-11}{15120}}\right)\right)\right)\right)\right)\right) \]

   17. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-23}{360}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{-11}{15120}\right)\right)\right)\right)\right)\right) \]

   18. *-lowering-*.f64 99.4%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-23}{360}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-11}{15120}\right)\right)\right)\right)\right)\right) \]

5. Simplified 99.4%

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

6. Final simplification 99.4%

   \[\leadsto \left(x \cdot x\right) \cdot \left(0.16666666666666666 + x \cdot \left(x \cdot \left(-0.06388888888888888 + -0.0007275132275132275 \cdot \left(x \cdot x\right)\right)\right)\right) \]
7. Add Preprocessing

Alternative 5: 99.3% accurate, 18.6× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * (x * (0.16666666666666666d0 + (x * (x * (-0.06388888888888888d0)))))
end function

Java

public static double code(double x) {
	return x * (x * (0.16666666666666666 + (x * (x * -0.06388888888888888))));
}

Python

def code(x):
	return x * (x * (0.16666666666666666 + (x * (x * -0.06388888888888888))))

Julia

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

MATLAB

function tmp = code(x)
	tmp = x * (x * (0.16666666666666666 + (x * (x * -0.06388888888888888))));
end

Wolfram

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

Derivation

1. Initial program 51.6%

   \[\frac{x - \sin x}{\tan x} \]
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

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

   2. associate-*l* N/A

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

   3. *-commutative N/A

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

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

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right) \cdot x\right)}\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right)}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right)}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right)}\right)\right)\right) \]

   8. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \left(\left(x \cdot x\right) \cdot \left(\color{blue}{{x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right)} - \frac{23}{360}\right)\right)\right)\right)\right) \]

   9. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \left(x \cdot \color{blue}{\left(x \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right)}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right)}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)}\right)\right)\right)\right)\right) \]

   12. sub-neg N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{23}{360}\right)\right)}\right)\right)\right)\right)\right)\right) \]

5. Simplified 99.5%

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

6. Taylor expanded in x around 0

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

   1. unpow2 N/A

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

   2. associate-*l* N/A

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

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

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left(\frac{1}{6} + \frac{-23}{360} \cdot {x}^{2}\right)\right)}\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{6} + \frac{-23}{360} \cdot {x}^{2}\right)}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \color{blue}{\left(\frac{-23}{360} \cdot {x}^{2}\right)}\right)\right)\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \left({x}^{2} \cdot \color{blue}{\frac{-23}{360}}\right)\right)\right)\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \left(\left(x \cdot x\right) \cdot \frac{-23}{360}\right)\right)\right)\right) \]

   8. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \left(x \cdot \color{blue}{\left(x \cdot \frac{-23}{360}\right)}\right)\right)\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \left(x \cdot \left(\frac{-23}{360} \cdot \color{blue}{x}\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{-23}{360} \cdot x\right)}\right)\right)\right)\right) \]

   11. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\frac{-23}{360}}\right)\right)\right)\right)\right) \]

   12. *-lowering-*.f64 99.4%

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\frac{-23}{360}}\right)\right)\right)\right)\right) \]

8. Simplified 99.4%

   \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(0.16666666666666666 + x \cdot \left(x \cdot -0.06388888888888888\right)\right)\right)} \]
9. Add Preprocessing

Alternative 6: 98.8% accurate, 41.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.6%

   \[\frac{x - \sin x}{\tan x} \]
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

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

   2. associate-*l* N/A

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

   3. *-commutative N/A

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

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

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right) \cdot x\right)}\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right)}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{6} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right)}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right)}\right)\right)\right) \]

   8. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \left(\left(x \cdot x\right) \cdot \left(\color{blue}{{x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right)} - \frac{23}{360}\right)\right)\right)\right)\right) \]

   9. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \left(x \cdot \color{blue}{\left(x \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right)}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)\right)}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) - \frac{23}{360}\right)}\right)\right)\right)\right)\right) \]

   12. sub-neg N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \left(\frac{-143}{604800} \cdot {x}^{2} - \frac{11}{15120}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{23}{360}\right)\right)}\right)\right)\right)\right)\right)\right) \]

5. Simplified 99.5%

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

   1. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\frac{1}{6} + x \cdot \left(x \cdot \left(\frac{-23}{360} + \left(x \cdot x\right) \cdot \left(\frac{-11}{15120} + \left(x \cdot x\right) \cdot \frac{-143}{604800}\right)\right)\right)\right) \cdot \color{blue}{x}\right)\right) \]

   2. flip-+ N/A

      \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{\frac{1}{6} \cdot \frac{1}{6} - \left(x \cdot \left(x \cdot \left(\frac{-23}{360} + \left(x \cdot x\right) \cdot \left(\frac{-11}{15120} + \left(x \cdot x\right) \cdot \frac{-143}{604800}\right)\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(\frac{-23}{360} + \left(x \cdot x\right) \cdot \left(\frac{-11}{15120} + \left(x \cdot x\right) \cdot \frac{-143}{604800}\right)\right)\right)\right)}{\frac{1}{6} - x \cdot \left(x \cdot \left(\frac{-23}{360} + \left(x \cdot x\right) \cdot \left(\frac{-11}{15120} + \left(x \cdot x\right) \cdot \frac{-143}{604800}\right)\right)\right)} \cdot x\right)\right) \]

   3. associate-*l/ N/A

      \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{\left(\frac{1}{6} \cdot \frac{1}{6} - \left(x \cdot \left(x \cdot \left(\frac{-23}{360} + \left(x \cdot x\right) \cdot \left(\frac{-11}{15120} + \left(x \cdot x\right) \cdot \frac{-143}{604800}\right)\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(\frac{-23}{360} + \left(x \cdot x\right) \cdot \left(\frac{-11}{15120} + \left(x \cdot x\right) \cdot \frac{-143}{604800}\right)\right)\right)\right)\right) \cdot x}{\color{blue}{\frac{1}{6} - x \cdot \left(x \cdot \left(\frac{-23}{360} + \left(x \cdot x\right) \cdot \left(\frac{-11}{15120} + \left(x \cdot x\right) \cdot \frac{-143}{604800}\right)\right)\right)}}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\left(\left(\frac{1}{6} \cdot \frac{1}{6} - \left(x \cdot \left(x \cdot \left(\frac{-23}{360} + \left(x \cdot x\right) \cdot \left(\frac{-11}{15120} + \left(x \cdot x\right) \cdot \frac{-143}{604800}\right)\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(\frac{-23}{360} + \left(x \cdot x\right) \cdot \left(\frac{-11}{15120} + \left(x \cdot x\right) \cdot \frac{-143}{604800}\right)\right)\right)\right)\right) \cdot x\right), \color{blue}{\left(\frac{1}{6} - x \cdot \left(x \cdot \left(\frac{-23}{360} + \left(x \cdot x\right) \cdot \left(\frac{-11}{15120} + \left(x \cdot x\right) \cdot \frac{-143}{604800}\right)\right)\right)\right)}\right)\right) \]

7. Applied egg-rr 99.6%

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

   1. clear-num N/A

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

   2. un-div-inv N/A

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

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

      \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(\frac{\frac{1}{6} - \left(x \cdot x\right) \cdot \left(\frac{-23}{360} + x \cdot \left(x \cdot \left(\frac{-11}{15120} + \left(x \cdot x\right) \cdot \frac{-143}{604800}\right)\right)\right)}{\left(\frac{1}{36} - x \cdot \left(\left(x \cdot \left(\frac{-23}{360} + x \cdot \left(x \cdot \left(\frac{-11}{15120} + \left(x \cdot x\right) \cdot \frac{-143}{604800}\right)\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{-23}{360} + x \cdot \left(x \cdot \left(\frac{-11}{15120} + \left(x \cdot x\right) \cdot \frac{-143}{604800}\right)\right)\right)\right)\right)\right) \cdot x}\right)}\right) \]

9. Applied egg-rr 99.5%

   \[\leadsto \color{blue}{\frac{x}{\frac{1}{x \cdot \left(x \cdot \left(x \cdot \left(-0.06388888888888888 + x \cdot \left(x \cdot \left(-0.0007275132275132275 + \left(x \cdot x\right) \cdot -0.00023644179894179894\right)\right)\right)\right) + 0.16666666666666666\right)}}} \]

10. Taylor expanded in x around 0

    \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(\frac{6}{x}\right)}\right) \]
11. Step-by-step derivation

    1. /-lowering-/.f64 99.2%

       \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(6, \color{blue}{x}\right)\right) \]

12. Simplified 99.2%

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

Alternative 7: 98.8% accurate, 41.0× 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 51.6%

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

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

   2. unpow2 N/A

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

   3. *-lowering-*.f64 99.1%

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

5. Simplified 99.1%

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

   1. associate-*r* N/A

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

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

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

   3. *-commutative N/A

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

   4. *-lowering-*.f64 99.1%

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

7. Applied egg-rr 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 8: 98.8% accurate, 41.0× 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 51.6%

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

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

   2. unpow2 N/A

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

   3. *-lowering-*.f64 99.1%

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

5. Simplified 99.1%

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

6. Final simplification 99.1%

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

Developer Target 1: 98.8% accurate, 41.0× 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 2024161 
(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)))