cos2 (problem 3.4.1)

Percentage Accurate: 50.4% → 99.8%

Time: 7.8s

Alternatives: 11

Speedup: 120.0×

Specification

Math

\[\begin{array}{l} \\ \frac{1 - \cos x}{x \cdot x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (- 1.0 (cos x)) (* x x)))

C

double code(double x) {
	return (1.0 - cos(x)) / (x * x);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    code = (1.0d0 - cos(x)) / (x * x)
end function

Java

public static double code(double x) {
	return (1.0 - Math.cos(x)) / (x * x);
}

Python

def code(x):
	return (1.0 - math.cos(x)) / (x * x)

Julia

function code(x)
	return Float64(Float64(1.0 - cos(x)) / Float64(x * x))
end

MATLAB

function tmp = code(x)
	tmp = (1.0 - cos(x)) / (x * x);
end

Wolfram

code[x_] := N[(N[(1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]

Initial Program: 50.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{1 - \cos x}{x \cdot x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (- 1.0 (cos x)) (* x x)))

C

double code(double x) {
	return (1.0 - cos(x)) / (x * x);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    code = (1.0d0 - cos(x)) / (x * x)
end function

Java

public static double code(double x) {
	return (1.0 - Math.cos(x)) / (x * x);
}

Python

def code(x):
	return (1.0 - math.cos(x)) / (x * x)

Julia

function code(x)
	return Float64(Float64(1.0 - cos(x)) / Float64(x * x))
end

MATLAB

function tmp = code(x)
	tmp = (1.0 - cos(x)) / (x * x);
end

Wolfram

code[x_] := N[(N[(1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\tan \left(x \cdot 0.5\right)}{x} \cdot \sin x}{x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (* (/ (tan (* x 0.5)) x) (sin x)) x))

C

double code(double x) {
	return ((tan((x * 0.5)) / x) * sin(x)) / x;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    code = ((tan((x * 0.5d0)) / x) * sin(x)) / x
end function

Java

public static double code(double x) {
	return ((Math.tan((x * 0.5)) / x) * Math.sin(x)) / x;
}

Python

def code(x):
	return ((math.tan((x * 0.5)) / x) * math.sin(x)) / x

Julia

function code(x)
	return Float64(Float64(Float64(tan(Float64(x * 0.5)) / x) * sin(x)) / x)
end

MATLAB

function tmp = code(x)
	tmp = ((tan((x * 0.5)) / x) * sin(x)) / x;
end

Wolfram

code[x_] := N[(N[(N[(N[Tan[N[(x * 0.5), $MachinePrecision]], $MachinePrecision] / x), $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]

Derivation

1. Initial program 43.7%

   \[\frac{1 - \cos x}{x \cdot x} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{1 - \cos x}{x \cdot x}} \]

   2. lift--.f64 N/A

      \[\leadsto \frac{\color{blue}{1 - \cos x}}{x \cdot x} \]

   3. flip-- N/A

      \[\leadsto \frac{\color{blue}{\frac{1 \cdot 1 - \cos x \cdot \cos x}{1 + \cos x}}}{x \cdot x} \]

   4. associate-/l/ N/A

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

   5. metadata-eval N/A

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

   6. lift-cos.f64 N/A

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

   7. lift-cos.f64 N/A

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

   8. 1-sub-cos N/A

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

   9. associate-/l* N/A

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

   10. lower-*.f64 N/A

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

   11. lower-sin.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lower-sin.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. associate-*r* N/A

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

   16. lower-*.f64 N/A

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

   17. lower-*.f64 N/A

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

   18. +-commutative N/A

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

   19. lower-+.f64 71.4

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

4. Applied rewrites 71.4%

   \[\leadsto \color{blue}{\sin x \cdot \frac{\sin x}{\left(\left(\cos x + 1\right) \cdot x\right) \cdot x}} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-/r* N/A

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

   6. associate-*l/ N/A

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

   7. lower-/.f64 N/A

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

6. Applied rewrites 99.8%

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

   1. lift-/.f64 N/A

      \[\leadsto \frac{\frac{\tan \color{blue}{\left(\frac{x}{2}\right)}}{x} \cdot \sin x}{x} \]

   2. frac-2neg N/A

      \[\leadsto \frac{\frac{\tan \color{blue}{\left(\frac{\mathsf{neg}\left(x\right)}{\mathsf{neg}\left(2\right)}\right)}}{x} \cdot \sin x}{x} \]

   3. mul-1-neg N/A

      \[\leadsto \frac{\frac{\tan \left(\frac{\color{blue}{-1 \cdot x}}{\mathsf{neg}\left(2\right)}\right)}{x} \cdot \sin x}{x} \]

   4. *-commutative N/A

      \[\leadsto \frac{\frac{\tan \left(\frac{\color{blue}{x \cdot -1}}{\mathsf{neg}\left(2\right)}\right)}{x} \cdot \sin x}{x} \]

   5. associate-/l* N/A

      \[\leadsto \frac{\frac{\tan \color{blue}{\left(x \cdot \frac{-1}{\mathsf{neg}\left(2\right)}\right)}}{x} \cdot \sin x}{x} \]

   6. metadata-eval N/A

      \[\leadsto \frac{\frac{\tan \left(x \cdot \frac{-1}{\color{blue}{-2}}\right)}{x} \cdot \sin x}{x} \]

   7. metadata-eval N/A

      \[\leadsto \frac{\frac{\tan \left(x \cdot \color{blue}{\frac{1}{2}}\right)}{x} \cdot \sin x}{x} \]

   8. lower-*.f64 99.8

      \[\leadsto \frac{\frac{\tan \color{blue}{\left(x \cdot 0.5\right)}}{x} \cdot \sin x}{x} \]

8. Applied rewrites 99.8%

   \[\leadsto \frac{\color{blue}{\frac{\tan \left(x \cdot 0.5\right)}{x} \cdot \sin x}}{x} \]
9. Add Preprocessing

Alternative 2: 75.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.055:\\ \;\;\;\;\mathsf{fma}\left(x \cdot \mathsf{fma}\left({x}^{4}, -2.48015873015873 \cdot 10^{-5}, \mathsf{fma}\left(0.001388888888888889 \cdot x, x, -0.041666666666666664\right)\right), x, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin x \cdot \tan \left(0.5 \cdot x\right)}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.055)
   (fma
    (*
     x
     (fma
      (pow x 4.0)
      -2.48015873015873e-5
      (fma (* 0.001388888888888889 x) x -0.041666666666666664)))
    x
    0.5)
   (/ (* (sin x) (tan (* 0.5 x))) (* x x))))

C

double code(double x) {
	double tmp;
	if (x <= 0.055) {
		tmp = fma((x * fma(pow(x, 4.0), -2.48015873015873e-5, fma((0.001388888888888889 * x), x, -0.041666666666666664))), x, 0.5);
	} else {
		tmp = (sin(x) * tan((0.5 * x))) / (x * x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.055)
		tmp = fma(Float64(x * fma((x ^ 4.0), -2.48015873015873e-5, fma(Float64(0.001388888888888889 * x), x, -0.041666666666666664))), x, 0.5);
	else
		tmp = Float64(Float64(sin(x) * tan(Float64(0.5 * x))) / Float64(x * x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 0.055], N[(N[(x * N[(N[Power[x, 4.0], $MachinePrecision] * -2.48015873015873e-5 + N[(N[(0.001388888888888889 * x), $MachinePrecision] * x + -0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x + 0.5), $MachinePrecision], N[(N[(N[Sin[x], $MachinePrecision] * N[Tan[N[(0.5 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0550000000000000003

   1. Initial program 26.8%

      \[\frac{1 - \cos x}{x \cdot x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{40320} \cdot {x}^{2}\right) - \frac{1}{24}\right)} \]

   5. Applied rewrites 75.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot \mathsf{fma}\left({x}^{4}, -2.48015873015873 \cdot 10^{-5}, \mathsf{fma}\left(0.001388888888888889 \cdot x, x, -0.041666666666666664\right)\right), x, 0.5\right)} \]

   if 0.0550000000000000003 < x

   1. Initial program 98.9%

      \[\frac{1 - \cos x}{x \cdot x} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1 - \cos x}{x \cdot x}} \]

      2. lift--.f64 N/A

         \[\leadsto \frac{\color{blue}{1 - \cos x}}{x \cdot x} \]

      3. flip-- N/A

         \[\leadsto \frac{\color{blue}{\frac{1 \cdot 1 - \cos x \cdot \cos x}{1 + \cos x}}}{x \cdot x} \]

      4. associate-/l/ N/A

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

      5. metadata-eval N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. 1-sub-cos N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sin.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*r* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 99.1

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

   4. Applied rewrites 99.1%

      \[\leadsto \color{blue}{\sin x \cdot \frac{\sin x}{\left(\left(\cos x + 1\right) \cdot x\right) \cdot x}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

   6. Applied rewrites 99.6%

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

      1. lift-/.f64 N/A

         \[\leadsto \frac{\frac{\tan \color{blue}{\left(\frac{x}{2}\right)}}{x} \cdot \sin x}{x} \]

      2. frac-2neg N/A

         \[\leadsto \frac{\frac{\tan \color{blue}{\left(\frac{\mathsf{neg}\left(x\right)}{\mathsf{neg}\left(2\right)}\right)}}{x} \cdot \sin x}{x} \]

      3. mul-1-neg N/A

         \[\leadsto \frac{\frac{\tan \left(\frac{\color{blue}{-1 \cdot x}}{\mathsf{neg}\left(2\right)}\right)}{x} \cdot \sin x}{x} \]

      4. *-commutative N/A

         \[\leadsto \frac{\frac{\tan \left(\frac{\color{blue}{x \cdot -1}}{\mathsf{neg}\left(2\right)}\right)}{x} \cdot \sin x}{x} \]

      5. associate-/l* N/A

         \[\leadsto \frac{\frac{\tan \color{blue}{\left(x \cdot \frac{-1}{\mathsf{neg}\left(2\right)}\right)}}{x} \cdot \sin x}{x} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\frac{\tan \left(x \cdot \frac{-1}{\color{blue}{-2}}\right)}{x} \cdot \sin x}{x} \]

      7. metadata-eval N/A

         \[\leadsto \frac{\frac{\tan \left(x \cdot \color{blue}{\frac{1}{2}}\right)}{x} \cdot \sin x}{x} \]

      8. lower-*.f64 99.6

         \[\leadsto \frac{\frac{\tan \color{blue}{\left(x \cdot 0.5\right)}}{x} \cdot \sin x}{x} \]

   8. Applied rewrites 99.6%

      \[\leadsto \frac{\color{blue}{\frac{\tan \left(x \cdot 0.5\right)}{x} \cdot \sin x}}{x} \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\tan \left(x \cdot \frac{1}{2}\right)}{x} \cdot \sin x}{x}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\tan \left(x \cdot \frac{1}{2}\right)}{x} \cdot \sin x}}{x} \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{\frac{\tan \left(x \cdot \frac{1}{2}\right)}{x} \cdot \frac{\sin x}{x}} \]

      4. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\tan \left(x \cdot \frac{1}{2}\right)}{x}} \cdot \frac{\sin x}{x} \]

      5. frac-times N/A

         \[\leadsto \color{blue}{\frac{\tan \left(x \cdot \frac{1}{2}\right) \cdot \sin x}{x \cdot x}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\tan \left(x \cdot \frac{1}{2}\right) \cdot \sin x}{\color{blue}{x \cdot x}} \]

      7. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\tan \left(x \cdot \frac{1}{2}\right) \cdot \sin x}{x \cdot x}} \]

      8. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sin x \cdot \tan \left(x \cdot \frac{1}{2}\right)}}{x \cdot x} \]

      9. lower-*.f64 99.5

         \[\leadsto \frac{\color{blue}{\sin x \cdot \tan \left(x \cdot 0.5\right)}}{x \cdot x} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\sin x \cdot \tan \color{blue}{\left(x \cdot \frac{1}{2}\right)}}{x \cdot x} \]

      11. *-commutative N/A

          \[\leadsto \frac{\sin x \cdot \tan \color{blue}{\left(\frac{1}{2} \cdot x\right)}}{x \cdot x} \]

      12. lower-*.f64 99.5

          \[\leadsto \frac{\sin x \cdot \tan \color{blue}{\left(0.5 \cdot x\right)}}{x \cdot x} \]

   10. Applied rewrites 99.5%

       \[\leadsto \color{blue}{\frac{\sin x \cdot \tan \left(0.5 \cdot x\right)}{x \cdot x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \sin x \cdot \frac{\frac{\tan \left(x \cdot 0.5\right)}{x}}{x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (sin x) (/ (/ (tan (* x 0.5)) x) x)))

C

double code(double x) {
	return sin(x) * ((tan((x * 0.5)) / x) / x);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    code = sin(x) * ((tan((x * 0.5d0)) / x) / x)
end function

Java

public static double code(double x) {
	return Math.sin(x) * ((Math.tan((x * 0.5)) / x) / x);
}

Python

def code(x):
	return math.sin(x) * ((math.tan((x * 0.5)) / x) / x)

Julia

function code(x)
	return Float64(sin(x) * Float64(Float64(tan(Float64(x * 0.5)) / x) / x))
end

MATLAB

function tmp = code(x)
	tmp = sin(x) * ((tan((x * 0.5)) / x) / x);
end

Wolfram

code[x_] := N[(N[Sin[x], $MachinePrecision] * N[(N[(N[Tan[N[(x * 0.5), $MachinePrecision]], $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 43.7%

   \[\frac{1 - \cos x}{x \cdot x} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{1 - \cos x}{x \cdot x}} \]

   2. lift--.f64 N/A

      \[\leadsto \frac{\color{blue}{1 - \cos x}}{x \cdot x} \]

   3. flip-- N/A

      \[\leadsto \frac{\color{blue}{\frac{1 \cdot 1 - \cos x \cdot \cos x}{1 + \cos x}}}{x \cdot x} \]

   4. associate-/l/ N/A

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

   5. metadata-eval N/A

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

   6. lift-cos.f64 N/A

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

   7. lift-cos.f64 N/A

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

   8. 1-sub-cos N/A

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

   9. associate-/l* N/A

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

   10. lower-*.f64 N/A

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

   11. lower-sin.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lower-sin.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. associate-*r* N/A

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

   16. lower-*.f64 N/A

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

   17. lower-*.f64 N/A

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

   18. +-commutative N/A

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

   19. lower-+.f64 71.4

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

4. Applied rewrites 71.4%

   \[\leadsto \color{blue}{\sin x \cdot \frac{\sin x}{\left(\left(\cos x + 1\right) \cdot x\right) \cdot x}} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-/r* N/A

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

   6. associate-*l/ N/A

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

   7. lower-/.f64 N/A

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

6. Applied rewrites 99.8%

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

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{\tan \left(\frac{x}{2}\right)}{x} \cdot \sin x}{x}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{\tan \left(\frac{x}{2}\right)}{x} \cdot \sin x}}{x} \]

   3. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\sin x \cdot \frac{\tan \left(\frac{x}{2}\right)}{x}}}{x} \]

   4. associate-/l* N/A

      \[\leadsto \color{blue}{\sin x \cdot \frac{\frac{\tan \left(\frac{x}{2}\right)}{x}}{x}} \]

   5. lower-*.f64 N/A

      \[\leadsto \color{blue}{\sin x \cdot \frac{\frac{\tan \left(\frac{x}{2}\right)}{x}}{x}} \]

   6. lower-/.f64 99.6

      \[\leadsto \sin x \cdot \color{blue}{\frac{\frac{\tan \left(\frac{x}{2}\right)}{x}}{x}} \]

   7. lift-/.f64 N/A

      \[\leadsto \sin x \cdot \frac{\frac{\tan \color{blue}{\left(\frac{x}{2}\right)}}{x}}{x} \]

   8. frac-2neg N/A

      \[\leadsto \sin x \cdot \frac{\frac{\tan \color{blue}{\left(\frac{\mathsf{neg}\left(x\right)}{\mathsf{neg}\left(2\right)}\right)}}{x}}{x} \]

   9. mul-1-neg N/A

      \[\leadsto \sin x \cdot \frac{\frac{\tan \left(\frac{\color{blue}{-1 \cdot x}}{\mathsf{neg}\left(2\right)}\right)}{x}}{x} \]

   10. *-commutative N/A

       \[\leadsto \sin x \cdot \frac{\frac{\tan \left(\frac{\color{blue}{x \cdot -1}}{\mathsf{neg}\left(2\right)}\right)}{x}}{x} \]

   11. associate-/l* N/A

       \[\leadsto \sin x \cdot \frac{\frac{\tan \color{blue}{\left(x \cdot \frac{-1}{\mathsf{neg}\left(2\right)}\right)}}{x}}{x} \]

   12. metadata-eval N/A

       \[\leadsto \sin x \cdot \frac{\frac{\tan \left(x \cdot \frac{-1}{\color{blue}{-2}}\right)}{x}}{x} \]

   13. metadata-eval N/A

       \[\leadsto \sin x \cdot \frac{\frac{\tan \left(x \cdot \color{blue}{\frac{1}{2}}\right)}{x}}{x} \]

   14. lower-*.f64 99.6

       \[\leadsto \sin x \cdot \frac{\frac{\tan \color{blue}{\left(x \cdot 0.5\right)}}{x}}{x} \]

8. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\sin x \cdot \frac{\frac{\tan \left(x \cdot 0.5\right)}{x}}{x}} \]
9. Add Preprocessing

Alternative 4: 64.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.8 \cdot 10^{+25}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.001388888888888889 \cdot x, x, -0.041666666666666664\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\left(-x\right) \cdot \mathsf{fma}\left({x}^{-1}, x, -1\right)}{x}}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.8e+25)
   (fma (* x x) (fma (* 0.001388888888888889 x) x -0.041666666666666664) 0.5)
   (/ (/ (/ (* (- x) (fma (pow x -1.0) x -1.0)) x) x) x)))

C

double code(double x) {
	double tmp;
	if (x <= 1.8e+25) {
		tmp = fma((x * x), fma((0.001388888888888889 * x), x, -0.041666666666666664), 0.5);
	} else {
		tmp = (((-x * fma(pow(x, -1.0), x, -1.0)) / x) / x) / x;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.8e+25)
		tmp = fma(Float64(x * x), fma(Float64(0.001388888888888889 * x), x, -0.041666666666666664), 0.5);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(-x) * fma((x ^ -1.0), x, -1.0)) / x) / x) / x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.8e+25], N[(N[(x * x), $MachinePrecision] * N[(N[(0.001388888888888889 * x), $MachinePrecision] * x + -0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision], N[(N[(N[(N[((-x) * N[(N[Power[x, -1.0], $MachinePrecision] * x + -1.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.80000000000000008e25

   1. Initial program 30.3%

      \[\frac{1 - \cos x}{x \cdot x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{720} \cdot {x}^{2} - \frac{1}{24}\right)} \]

   5. Applied rewrites 72.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.001388888888888889 \cdot x, x, -0.041666666666666664\right), 0.5\right)} \]

   if 1.80000000000000008e25 < x

   1. Initial program 99.2%

      \[\frac{1 - \cos x}{x \cdot x} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1 - \cos x}{x \cdot x}} \]

      2. lift--.f64 N/A

         \[\leadsto \frac{\color{blue}{1 - \cos x}}{x \cdot x} \]

      3. flip-- N/A

         \[\leadsto \frac{\color{blue}{\frac{1 \cdot 1 - \cos x \cdot \cos x}{1 + \cos x}}}{x \cdot x} \]

      4. associate-/l/ N/A

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

      5. metadata-eval N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. 1-sub-cos N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sin.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*r* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 99.2

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

   4. Applied rewrites 99.2%

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

   6. Applied rewrites 47.0%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(\frac{\cos x}{x}, x \cdot x, -x\right)}{-x}}{x \cdot x}} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\frac{\mathsf{fma}\left(\frac{\cos x}{x}, x \cdot x, -x\right)}{-x}\right)}{\mathsf{neg}\left(x \cdot x\right)}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\frac{\mathsf{fma}\left(\frac{\cos x}{x}, x \cdot x, -x\right)}{-x}\right)}{\mathsf{neg}\left(\color{blue}{x \cdot x}\right)} \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto \frac{\mathsf{neg}\left(\frac{\mathsf{fma}\left(\frac{\cos x}{x}, x \cdot x, -x\right)}{-x}\right)}{\color{blue}{x \cdot \left(\mathsf{neg}\left(x\right)\right)}} \]

      5. lift-neg.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\frac{\mathsf{fma}\left(\frac{\cos x}{x}, x \cdot x, -x\right)}{-x}\right)}{x \cdot \color{blue}{\left(-x\right)}} \]

      6. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{\mathsf{neg}\left(\frac{\mathsf{fma}\left(\frac{\cos x}{x}, x \cdot x, -x\right)}{-x}\right)}{x}}{-x}} \]

      7. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{\mathsf{neg}\left(\frac{\mathsf{fma}\left(\frac{\cos x}{x}, x \cdot x, -x\right)}{-x}\right)}{x}}{-x}} \]

   8. Applied rewrites 99.1%

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

   9. Taylor expanded in x around 0

      \[\leadsto \frac{\frac{\frac{x \cdot \mathsf{fma}\left(\color{blue}{\frac{1}{x}}, x, -1\right)}{x}}{x}}{-x} \]
   10. Step-by-step derivation

       1. lower-/.f64 55.7

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

   11. Applied rewrites 55.7%

       \[\leadsto \frac{\frac{\frac{x \cdot \mathsf{fma}\left(\color{blue}{\frac{1}{x}}, x, -1\right)}{x}}{x}}{-x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.8 \cdot 10^{+25}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.001388888888888889 \cdot x, x, -0.041666666666666664\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\left(-x\right) \cdot \mathsf{fma}\left({x}^{-1}, x, -1\right)}{x}}{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.1:\\ \;\;\;\;\mathsf{fma}\left(x \cdot \mathsf{fma}\left({x}^{4}, -2.48015873015873 \cdot 10^{-5}, \mathsf{fma}\left(0.001388888888888889 \cdot x, x, -0.041666666666666664\right)\right), x, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.1)
   (fma
    (*
     x
     (fma
      (pow x 4.0)
      -2.48015873015873e-5
      (fma (* 0.001388888888888889 x) x -0.041666666666666664)))
    x
    0.5)
   (/ (/ (- 1.0 (cos x)) x) x)))

C

double code(double x) {
	double tmp;
	if (x <= 0.1) {
		tmp = fma((x * fma(pow(x, 4.0), -2.48015873015873e-5, fma((0.001388888888888889 * x), x, -0.041666666666666664))), x, 0.5);
	} else {
		tmp = ((1.0 - cos(x)) / x) / x;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.1)
		tmp = fma(Float64(x * fma((x ^ 4.0), -2.48015873015873e-5, fma(Float64(0.001388888888888889 * x), x, -0.041666666666666664))), x, 0.5);
	else
		tmp = Float64(Float64(Float64(1.0 - cos(x)) / x) / x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 0.1], N[(N[(x * N[(N[Power[x, 4.0], $MachinePrecision] * -2.48015873015873e-5 + N[(N[(0.001388888888888889 * x), $MachinePrecision] * x + -0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x + 0.5), $MachinePrecision], N[(N[(N[(1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.10000000000000001

   1. Initial program 26.8%

      \[\frac{1 - \cos x}{x \cdot x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{40320} \cdot {x}^{2}\right) - \frac{1}{24}\right)} \]

   5. Applied rewrites 75.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot \mathsf{fma}\left({x}^{4}, -2.48015873015873 \cdot 10^{-5}, \mathsf{fma}\left(0.001388888888888889 \cdot x, x, -0.041666666666666664\right)\right), x, 0.5\right)} \]

   if 0.10000000000000001 < x

   1. Initial program 98.9%

      \[\frac{1 - \cos x}{x \cdot x} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1 - \cos x}{x \cdot x}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{1 - \cos x}{\color{blue}{x \cdot x}} \]

      3. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{1 - \cos x}{x}}{x}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{1 - \cos x}{x}}{x}} \]

      5. lower-/.f64 98.9

         \[\leadsto \frac{\color{blue}{\frac{1 - \cos x}{x}}}{x} \]

   4. Applied rewrites 98.9%

      \[\leadsto \color{blue}{\frac{\frac{1 - \cos x}{x}}{x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 76.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.031:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.001388888888888889 \cdot x, x, -0.041666666666666664\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.031)
   (fma (* x x) (fma (* 0.001388888888888889 x) x -0.041666666666666664) 0.5)
   (/ (/ (- 1.0 (cos x)) x) x)))

C

double code(double x) {
	double tmp;
	if (x <= 0.031) {
		tmp = fma((x * x), fma((0.001388888888888889 * x), x, -0.041666666666666664), 0.5);
	} else {
		tmp = ((1.0 - cos(x)) / x) / x;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.031)
		tmp = fma(Float64(x * x), fma(Float64(0.001388888888888889 * x), x, -0.041666666666666664), 0.5);
	else
		tmp = Float64(Float64(Float64(1.0 - cos(x)) / x) / x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 0.031], N[(N[(x * x), $MachinePrecision] * N[(N[(0.001388888888888889 * x), $MachinePrecision] * x + -0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision], N[(N[(N[(1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.031

   1. Initial program 26.8%

      \[\frac{1 - \cos x}{x \cdot x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{720} \cdot {x}^{2} - \frac{1}{24}\right)} \]

   5. Applied rewrites 76.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.001388888888888889 \cdot x, x, -0.041666666666666664\right), 0.5\right)} \]

   if 0.031 < x

   1. Initial program 98.9%

      \[\frac{1 - \cos x}{x \cdot x} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1 - \cos x}{x \cdot x}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{1 - \cos x}{\color{blue}{x \cdot x}} \]

      3. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{1 - \cos x}{x}}{x}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{1 - \cos x}{x}}{x}} \]

      5. lower-/.f64 98.9

         \[\leadsto \frac{\color{blue}{\frac{1 - \cos x}{x}}}{x} \]

   4. Applied rewrites 98.9%

      \[\leadsto \color{blue}{\frac{\frac{1 - \cos x}{x}}{x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 75.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.031:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.001388888888888889 \cdot x, x, -0.041666666666666664\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.031)
   (fma (* x x) (fma (* 0.001388888888888889 x) x -0.041666666666666664) 0.5)
   (/ (- 1.0 (cos x)) (* x x))))

C

double code(double x) {
	double tmp;
	if (x <= 0.031) {
		tmp = fma((x * x), fma((0.001388888888888889 * x), x, -0.041666666666666664), 0.5);
	} else {
		tmp = (1.0 - cos(x)) / (x * x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.031)
		tmp = fma(Float64(x * x), fma(Float64(0.001388888888888889 * x), x, -0.041666666666666664), 0.5);
	else
		tmp = Float64(Float64(1.0 - cos(x)) / Float64(x * x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 0.031], N[(N[(x * x), $MachinePrecision] * N[(N[(0.001388888888888889 * x), $MachinePrecision] * x + -0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision], N[(N[(1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.031

   1. Initial program 26.8%

      \[\frac{1 - \cos x}{x \cdot x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{720} \cdot {x}^{2} - \frac{1}{24}\right)} \]

   5. Applied rewrites 76.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.001388888888888889 \cdot x, x, -0.041666666666666664\right), 0.5\right)} \]

   if 0.031 < x

   1. Initial program 98.9%

      \[\frac{1 - \cos x}{x \cdot x} \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 64.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4 \cdot 10^{+24}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.001388888888888889 \cdot x, x, -0.041666666666666664\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x \cdot x} - \frac{\frac{-1}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 4e+24)
   (fma (* x x) (fma (* 0.001388888888888889 x) x -0.041666666666666664) 0.5)
   (- (/ -1.0 (* x x)) (/ (/ -1.0 x) x))))

C

double code(double x) {
	double tmp;
	if (x <= 4e+24) {
		tmp = fma((x * x), fma((0.001388888888888889 * x), x, -0.041666666666666664), 0.5);
	} else {
		tmp = (-1.0 / (x * x)) - ((-1.0 / x) / x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 4e+24)
		tmp = fma(Float64(x * x), fma(Float64(0.001388888888888889 * x), x, -0.041666666666666664), 0.5);
	else
		tmp = Float64(Float64(-1.0 / Float64(x * x)) - Float64(Float64(-1.0 / x) / x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 4e+24], N[(N[(x * x), $MachinePrecision] * N[(N[(0.001388888888888889 * x), $MachinePrecision] * x + -0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision], N[(N[(-1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision] - N[(N[(-1.0 / x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.9999999999999999e24

   1. Initial program 30.0%

      \[\frac{1 - \cos x}{x \cdot x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{720} \cdot {x}^{2} - \frac{1}{24}\right)} \]

   5. Applied rewrites 73.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.001388888888888889 \cdot x, x, -0.041666666666666664\right), 0.5\right)} \]

   if 3.9999999999999999e24 < x

   1. Initial program 99.1%

      \[\frac{1 - \cos x}{x \cdot x} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1 - \cos x}{x \cdot x}} \]

      2. lift--.f64 N/A

         \[\leadsto \frac{\color{blue}{1 - \cos x}}{x \cdot x} \]

      3. flip-- N/A

         \[\leadsto \frac{\color{blue}{\frac{1 \cdot 1 - \cos x \cdot \cos x}{1 + \cos x}}}{x \cdot x} \]

      4. associate-/l/ N/A

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

      5. metadata-eval N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. 1-sub-cos N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sin.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*r* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 99.2

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

   4. Applied rewrites 99.2%

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

   6. Applied rewrites 98.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 98.7

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

   8. Applied rewrites 98.7%

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

   9. Taylor expanded in x around 0

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

       1. lower-/.f64 N/A

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

       2. unpow2 N/A

          \[\leadsto \frac{-1}{\color{blue}{x \cdot x}} - \frac{\frac{-1}{x}}{x} \]

       3. lower-*.f64 54.1

          \[\leadsto \frac{-1}{\color{blue}{x \cdot x}} - \frac{\frac{-1}{x}}{x} \]

   11. Applied rewrites 54.1%

       \[\leadsto \color{blue}{\frac{-1}{x \cdot x}} - \frac{\frac{-1}{x}}{x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 64.5% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.1 \cdot 10^{+38}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.001388888888888889 \cdot x, x, -0.041666666666666664\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - 1}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 4.1e+38)
   (fma (* x x) (fma (* 0.001388888888888889 x) x -0.041666666666666664) 0.5)
   (/ (- 1.0 1.0) (* x x))))

C

double code(double x) {
	double tmp;
	if (x <= 4.1e+38) {
		tmp = fma((x * x), fma((0.001388888888888889 * x), x, -0.041666666666666664), 0.5);
	} else {
		tmp = (1.0 - 1.0) / (x * x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 4.1e+38)
		tmp = fma(Float64(x * x), fma(Float64(0.001388888888888889 * x), x, -0.041666666666666664), 0.5);
	else
		tmp = Float64(Float64(1.0 - 1.0) / Float64(x * x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 4.1e+38], N[(N[(x * x), $MachinePrecision] * N[(N[(0.001388888888888889 * x), $MachinePrecision] * x + -0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision], N[(N[(1.0 - 1.0), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.1000000000000003e38

   1. Initial program 31.3%

      \[\frac{1 - \cos x}{x \cdot x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{720} \cdot {x}^{2} - \frac{1}{24}\right)} \]

   5. Applied rewrites 71.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.001388888888888889 \cdot x, x, -0.041666666666666664\right), 0.5\right)} \]

   if 4.1000000000000003e38 < x

   1. Initial program 99.2%

      \[\frac{1 - \cos x}{x \cdot x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 58.1%

      \[\leadsto \frac{1 - \color{blue}{1}}{x \cdot x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 64.1% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.45:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, -0.041666666666666664, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - 1}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 3.45)
   (fma (* x x) -0.041666666666666664 0.5)
   (/ (- 1.0 1.0) (* x x))))

C

double code(double x) {
	double tmp;
	if (x <= 3.45) {
		tmp = fma((x * x), -0.041666666666666664, 0.5);
	} else {
		tmp = (1.0 - 1.0) / (x * x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 3.45)
		tmp = fma(Float64(x * x), -0.041666666666666664, 0.5);
	else
		tmp = Float64(Float64(1.0 - 1.0) / Float64(x * x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 3.45], N[(N[(x * x), $MachinePrecision] * -0.041666666666666664 + 0.5), $MachinePrecision], N[(N[(1.0 - 1.0), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.4500000000000002

   1. Initial program 26.8%

      \[\frac{1 - \cos x}{x \cdot x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} + \frac{-1}{24} \cdot {x}^{2}} \]
   4. Step-by-step derivation

      1. fp-cancel-sign-sub-inv N/A

         \[\leadsto \color{blue}{\frac{1}{2} - \left(\mathsf{neg}\left(\frac{-1}{24}\right)\right) \cdot {x}^{2}} \]

      2. metadata-eval N/A

         \[\leadsto \frac{1}{2} - \color{blue}{\frac{1}{24}} \cdot {x}^{2} \]

      3. *-commutative N/A

         \[\leadsto \frac{1}{2} - \color{blue}{{x}^{2} \cdot \frac{1}{24}} \]

      4. fp-cancel-sub-sign-inv N/A

         \[\leadsto \color{blue}{\frac{1}{2} + \left(\mathsf{neg}\left({x}^{2}\right)\right) \cdot \frac{1}{24}} \]

      5. +-commutative N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left({x}^{2}\right)\right) \cdot \frac{1}{24} + \frac{1}{2}} \]

      6. distribute-lft-neg-out N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left({x}^{2} \cdot \frac{1}{24}\right)\right)} + \frac{1}{2} \]

      7. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{{x}^{2} \cdot \left(\mathsf{neg}\left(\frac{1}{24}\right)\right)} + \frac{1}{2} \]

      8. metadata-eval N/A

         \[\leadsto {x}^{2} \cdot \color{blue}{\frac{-1}{24}} + \frac{1}{2} \]

      9. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{-1}{24}, \frac{1}{2}\right)} \]

      10. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{24}, \frac{1}{2}\right) \]

      11. lower-*.f64 75.5

          \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, -0.041666666666666664, 0.5\right) \]

   5. Applied rewrites 75.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, -0.041666666666666664, 0.5\right)} \]

   if 3.4500000000000002 < x

   1. Initial program 98.9%

      \[\frac{1 - \cos x}{x \cdot x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 46.3%

      \[\leadsto \frac{1 - \color{blue}{1}}{x \cdot x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 52.1% accurate, 120.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.5)

C

double code(double x) {
	return 0.5;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    code = 0.5d0
end function

Java

public static double code(double x) {
	return 0.5;
}

Python

def code(x):
	return 0.5

Julia

function code(x)
	return 0.5
end

MATLAB

function tmp = code(x)
	tmp = 0.5;
end

Wolfram

code[x_] := 0.5

Derivation

1. Initial program 43.7%

   \[\frac{1 - \cos x}{x \cdot x} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

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

5. Applied rewrites 58.8%

   \[\leadsto \color{blue}{0.5} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024360 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))