ENA, Section 1.4, Exercise 4a

Percentage Accurate: 53.4% → 99.6%

Time: 10.2s

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: 99.6% accurate, 8.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 57.2%

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

3. Taylor expanded in x around 0

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

   1. unpow2 N/A

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

   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 99.7%

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. *-commutative N/A

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

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

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

7. Applied egg-rr 99.7%

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

Alternative 2: 99.6% 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 57.2%

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

3. Taylor expanded in x around 0

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

   1. unpow2 N/A

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

   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 99.7%

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

Alternative 3: 99.5% accurate, 10.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 57.2%

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

3. Taylor expanded in x around 0

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

   1. unpow2 N/A

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

   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 99.7%

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. *-commutative N/A

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

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

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

7. Applied egg-rr 99.7%

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

8. Taylor expanded in x around 0

   \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\color{blue}{\left({x}^{3} \cdot \left(\frac{-11}{15120} \cdot {x}^{2} - \frac{23}{360}\right)\right)}, \mathsf{*.f64}\left(x, \frac{1}{6}\right)\right)\right) \]
9. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\left(\frac{-11}{15120} \cdot {x}^{2} - \frac{23}{360}\right) \cdot {x}^{3}\right), \mathsf{*.f64}\left(\color{blue}{x}, \frac{1}{6}\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\frac{-11}{15120} \cdot {x}^{2} - \frac{23}{360}\right), \left({x}^{3}\right)\right), \mathsf{*.f64}\left(\color{blue}{x}, \frac{1}{6}\right)\right)\right) \]

   3. sub-neg N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\frac{-11}{15120} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{23}{360}\right)\right)\right), \left({x}^{3}\right)\right), \mathsf{*.f64}\left(x, \frac{1}{6}\right)\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\frac{-11}{15120} \cdot {x}^{2} + \frac{-23}{360}\right), \left({x}^{3}\right)\right), \mathsf{*.f64}\left(x, \frac{1}{6}\right)\right)\right) \]

   5. +-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\frac{-23}{360} + \frac{-11}{15120} \cdot {x}^{2}\right), \left({x}^{3}\right)\right), \mathsf{*.f64}\left(x, \frac{1}{6}\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-23}{360}, \left(\frac{-11}{15120} \cdot {x}^{2}\right)\right), \left({x}^{3}\right)\right), \mathsf{*.f64}\left(x, \frac{1}{6}\right)\right)\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-23}{360}, \left(\frac{-11}{15120} \cdot \left(x \cdot x\right)\right)\right), \left({x}^{3}\right)\right), \mathsf{*.f64}\left(x, \frac{1}{6}\right)\right)\right) \]

   8. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-23}{360}, \left(\left(\frac{-11}{15120} \cdot x\right) \cdot x\right)\right), \left({x}^{3}\right)\right), \mathsf{*.f64}\left(x, \frac{1}{6}\right)\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-23}{360}, \left(x \cdot \left(\frac{-11}{15120} \cdot x\right)\right)\right), \left({x}^{3}\right)\right), \mathsf{*.f64}\left(x, \frac{1}{6}\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-23}{360}, \mathsf{*.f64}\left(x, \left(\frac{-11}{15120} \cdot x\right)\right)\right), \left({x}^{3}\right)\right), \mathsf{*.f64}\left(x, \frac{1}{6}\right)\right)\right) \]

   11. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-23}{360}, \mathsf{*.f64}\left(x, \left(x \cdot \frac{-11}{15120}\right)\right)\right), \left({x}^{3}\right)\right), \mathsf{*.f64}\left(x, \frac{1}{6}\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-23}{360}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{-11}{15120}\right)\right)\right), \left({x}^{3}\right)\right), \mathsf{*.f64}\left(x, \frac{1}{6}\right)\right)\right) \]

   13. cube-mult N/A

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

   14. unpow2 N/A

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

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

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

   16. unpow2 N/A

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

   17. *-lowering-*.f64 99.6%

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

10. Simplified 99.6%

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

11. Final simplification 99.6%

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

Alternative 4: 99.5% accurate, 12.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 57.2%

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

3. Taylor expanded in x around 0

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

   1. *-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 99.6%

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

5. Simplified 99.6%

   \[\leadsto \color{blue}{\left(x \cdot x\right) \cdot \left(0.16666666666666666 + x \cdot \left(x \cdot \left(-0.06388888888888888 + \left(x \cdot x\right) \cdot -0.0007275132275132275\right)\right)\right)} \]
6. 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 57.2%

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

3. Taylor expanded in x around 0

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

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

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

   2. unpow2 N/A

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

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

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

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

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

   5. *-commutative N/A

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

   6. unpow2 N/A

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

   7. associate-*l* N/A

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

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

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

   9. *-lowering-*.f64 99.5%

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

5. Simplified 99.5%

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

   1. associate-*l* N/A

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

   2. *-commutative N/A

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

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

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

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

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

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

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

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

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

   7. *-lowering-*.f64 99.5%

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

7. Applied egg-rr 99.5%

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

8. Final simplification 99.5%

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

Alternative 6: 99.3% accurate, 18.6× speedup

Math

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

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(Float64(x * 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[(N[(x * x), $MachinePrecision] * N[(0.16666666666666666 + N[(x * N[(x * -0.06388888888888888), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 57.2%

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

3. Taylor expanded in x around 0

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

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

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

   2. unpow2 N/A

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

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

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

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

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

   5. *-commutative N/A

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

   6. unpow2 N/A

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

   7. associate-*l* N/A

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

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

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

   9. *-lowering-*.f64 99.5%

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

5. Simplified 99.5%

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

Alternative 7: 98.7% 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 57.2%

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

3. Taylor expanded in x around 0

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

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

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

   2. unpow2 N/A

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

   3. *-lowering-*.f64 98.9%

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

5. Simplified 98.9%

   \[\leadsto \color{blue}{0.16666666666666666 \cdot \left(x \cdot x\right)} \]
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 98.9%

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

7. Applied egg-rr 98.9%

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

8. Final simplification 98.9%

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

Alternative 8: 98.7% 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 57.2%

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

3. Taylor expanded in x around 0

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

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

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

   2. unpow2 N/A

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

   3. *-lowering-*.f64 98.9%

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

5. Simplified 98.9%

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

6. Final simplification 98.9%

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

Developer Target 1: 98.7% 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 2024155 
(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)))