cos2 (problem 3.4.1)

Percentage Accurate: 50.0% → 99.8%

Time: 3.6s

Alternatives: 8

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: 50.0% 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

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

FPCore

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

C

double code(double x) {
	return ((sin(x) / x) * tan((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) / x) * tan((0.5d0 * x))) / x
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.4%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift--.f64 N/A

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

   4. lift-cos.f64 N/A

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

   5. flip-- N/A

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

   6. pow2 N/A

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

   7. associate-/l/ N/A

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

   8. lower-/.f64 N/A

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

   9. metadata-eval N/A

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

   10. 1-sub-cos N/A

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

   11. lower-*.f64 N/A

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

   12. lower-sin.f64 N/A

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

   13. lower-sin.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. +-commutative N/A

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

   16. lower-+.f64 N/A

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

   17. lift-cos.f64 N/A

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

   18. pow2 N/A

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

   19. lift-*.f64 77.0

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

4. Applied rewrites 77.0%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-sin.f64 N/A

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

   4. lift-sin.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-+.f64 N/A

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

   7. lift-cos.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. pow2 N/A

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

   10. times-frac N/A

       \[\leadsto \color{blue}{\frac{\sin x}{\cos x + 1} \cdot \frac{\sin x}{{x}^{2}}} \]

   11. *-commutative N/A

       \[\leadsto \color{blue}{\frac{\sin x}{{x}^{2}} \cdot \frac{\sin x}{\cos x + 1}} \]

   12. lower-*.f64 N/A

       \[\leadsto \color{blue}{\frac{\sin x}{{x}^{2}} \cdot \frac{\sin x}{\cos x + 1}} \]

   13. pow2 N/A

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

   14. associate-/r* N/A

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

   15. lower-/.f64 N/A

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

   16. lower-/.f64 N/A

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

   17. lift-sin.f64 N/A

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

   18. +-commutative N/A

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

   19. hang-0p-tan N/A

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

   20. lower-tan.f64 N/A

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

   21. lower-/.f64 99.7

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

6. Applied rewrites 99.7%

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

7. Taylor expanded in x around 0

   \[\leadsto \frac{\frac{\sin x}{x}}{x} \cdot \tan \color{blue}{\left(\frac{1}{2} \cdot x\right)} \]
8. Step-by-step derivation

   1. lower-*.f64 99.7

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

9. Applied rewrites 99.7%

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

    1. lift-*.f64 N/A

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

    2. lift-/.f64 N/A

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

    3. lift-/.f64 N/A

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

    4. lift-sin.f64 N/A

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

    5. associate-*l/ N/A

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

    6. lower-/.f64 N/A

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

    7. lower-*.f64 N/A

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

    8. lift-sin.f64 N/A

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

    9. lift-/.f64 99.8

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

11. Applied rewrites 99.8%

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

Alternative 2: 99.6% accurate, 0.5× speedup

Math

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

FPCore

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

C

double code(double x) {
	return ((sin(x) / x) / x) * tan((0.5 * 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) / x) / x) * tan((0.5d0 * x))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.4%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift--.f64 N/A

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

   4. lift-cos.f64 N/A

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

   5. flip-- N/A

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

   6. pow2 N/A

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

   7. associate-/l/ N/A

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

   8. lower-/.f64 N/A

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

   9. metadata-eval N/A

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

   10. 1-sub-cos N/A

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

   11. lower-*.f64 N/A

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

   12. lower-sin.f64 N/A

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

   13. lower-sin.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. +-commutative N/A

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

   16. lower-+.f64 N/A

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

   17. lift-cos.f64 N/A

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

   18. pow2 N/A

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

   19. lift-*.f64 77.0

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

4. Applied rewrites 77.0%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-sin.f64 N/A

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

   4. lift-sin.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-+.f64 N/A

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

   7. lift-cos.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. pow2 N/A

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

   10. times-frac N/A

       \[\leadsto \color{blue}{\frac{\sin x}{\cos x + 1} \cdot \frac{\sin x}{{x}^{2}}} \]

   11. *-commutative N/A

       \[\leadsto \color{blue}{\frac{\sin x}{{x}^{2}} \cdot \frac{\sin x}{\cos x + 1}} \]

   12. lower-*.f64 N/A

       \[\leadsto \color{blue}{\frac{\sin x}{{x}^{2}} \cdot \frac{\sin x}{\cos x + 1}} \]

   13. pow2 N/A

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

   14. associate-/r* N/A

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

   15. lower-/.f64 N/A

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

   16. lower-/.f64 N/A

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

   17. lift-sin.f64 N/A

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

   18. +-commutative N/A

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

   19. hang-0p-tan N/A

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

   20. lower-tan.f64 N/A

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

   21. lower-/.f64 99.7

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

6. Applied rewrites 99.7%

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

7. Taylor expanded in x around 0

   \[\leadsto \frac{\frac{\sin x}{x}}{x} \cdot \tan \color{blue}{\left(\frac{1}{2} \cdot x\right)} \]
8. Step-by-step derivation

   1. lower-*.f64 99.7

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

9. Applied rewrites 99.7%

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

Alternative 3: 76.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.032:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.001388888888888889, -0.041666666666666664\right), x \cdot x, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.032)
   (fma (fma (* x x) 0.001388888888888889 -0.041666666666666664) (* x x) 0.5)
   (/ (/ (- 1.0 (cos x)) x) x)))

C

double code(double x) {
	double tmp;
	if (x <= 0.032) {
		tmp = fma(fma((x * x), 0.001388888888888889, -0.041666666666666664), (x * x), 0.5);
	} else {
		tmp = ((1.0 - cos(x)) / x) / x;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.032)
		tmp = fma(fma(Float64(x * x), 0.001388888888888889, -0.041666666666666664), Float64(x * x), 0.5);
	else
		tmp = Float64(Float64(Float64(1.0 - cos(x)) / x) / x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 0.032], N[(N[(N[(x * x), $MachinePrecision] * 0.001388888888888889 + -0.041666666666666664), $MachinePrecision] * N[(x * x), $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 < 0.032000000000000001

   1. Initial program 36.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. flip-- N/A

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

      6. pow2 N/A

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

      7. associate-/l/ N/A

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

      8. lower-/.f64 N/A

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

      9. metadata-eval N/A

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

      10. 1-sub-cos N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sin.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. lift-cos.f64 N/A

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

      18. pow2 N/A

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

      19. lift-*.f64 68.5

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

   4. Applied rewrites 68.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. times-frac N/A

          \[\leadsto \color{blue}{\frac{\sin x}{\cos x + 1} \cdot \frac{\sin x}{{x}^{2}}} \]

      11. *-commutative N/A

          \[\leadsto \color{blue}{\frac{\sin x}{{x}^{2}} \cdot \frac{\sin x}{\cos x + 1}} \]

      12. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{\sin x}{{x}^{2}} \cdot \frac{\sin x}{\cos x + 1}} \]

      13. pow2 N/A

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

      14. associate-/r* N/A

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

      15. lower-/.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lift-sin.f64 N/A

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

      18. +-commutative N/A

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

      19. hang-0p-tan N/A

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

      20. lower-tan.f64 N/A

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

      21. lower-/.f64 99.7

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

   6. Applied rewrites 99.7%

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

   7. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{720} \cdot {x}^{2} - \frac{1}{24}\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 65.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.001388888888888889 \cdot \left(x \cdot x\right) - 0.041666666666666664, x \cdot x, 0.5\right)} \]
   10. Step-by-step derivation

       1. lift--.f64 N/A

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

       2. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{720} \cdot \left(x \cdot x\right) - \frac{1}{24}, x \cdot x, \frac{1}{2}\right) \]

       3. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{720} \cdot \left(x \cdot x\right) - \frac{1}{24}, x \cdot x, \frac{1}{2}\right) \]

       4. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{720} \cdot {x}^{2} - \frac{1}{24}, x \cdot x, \frac{1}{2}\right) \]

       5. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{720} \cdot {x}^{2} - \frac{1}{24} \cdot 1, x \cdot x, \frac{1}{2}\right) \]

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

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

       7. *-commutative N/A

          \[\leadsto \mathsf{fma}\left({x}^{2} \cdot \frac{1}{720} + \left(\mathsf{neg}\left(\frac{1}{24}\right)\right) \cdot 1, x \cdot x, \frac{1}{2}\right) \]

       8. metadata-eval N/A

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

       9. metadata-eval N/A

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

       10. lower-fma.f64 N/A

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

       11. pow2 N/A

           \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{1}{720}, \frac{-1}{24}\right), x \cdot x, \frac{1}{2}\right) \]

       12. lift-*.f64 65.8

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

   11. Applied rewrites 65.8%

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

   if 0.032000000000000001 < x

   1. Initial program 98.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift--.f64 99.1

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

   4. Applied rewrites 99.1%

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

Alternative 4: 75.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.032:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.001388888888888889, -0.041666666666666664\right), x \cdot x, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.032)
   (fma (fma (* x x) 0.001388888888888889 -0.041666666666666664) (* x x) 0.5)
   (/ (- 1.0 (cos x)) (* x x))))

C

double code(double x) {
	double tmp;
	if (x <= 0.032) {
		tmp = fma(fma((x * x), 0.001388888888888889, -0.041666666666666664), (x * x), 0.5);
	} else {
		tmp = (1.0 - cos(x)) / (x * x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.032)
		tmp = fma(fma(Float64(x * x), 0.001388888888888889, -0.041666666666666664), Float64(x * x), 0.5);
	else
		tmp = Float64(Float64(1.0 - cos(x)) / Float64(x * x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 0.032], N[(N[(N[(x * x), $MachinePrecision] * 0.001388888888888889 + -0.041666666666666664), $MachinePrecision] * N[(x * x), $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 < 0.032000000000000001

   1. Initial program 36.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. flip-- N/A

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

      6. pow2 N/A

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

      7. associate-/l/ N/A

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

      8. lower-/.f64 N/A

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

      9. metadata-eval N/A

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

      10. 1-sub-cos N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sin.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. lift-cos.f64 N/A

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

      18. pow2 N/A

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

      19. lift-*.f64 68.5

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

   4. Applied rewrites 68.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. times-frac N/A

          \[\leadsto \color{blue}{\frac{\sin x}{\cos x + 1} \cdot \frac{\sin x}{{x}^{2}}} \]

      11. *-commutative N/A

          \[\leadsto \color{blue}{\frac{\sin x}{{x}^{2}} \cdot \frac{\sin x}{\cos x + 1}} \]

      12. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{\sin x}{{x}^{2}} \cdot \frac{\sin x}{\cos x + 1}} \]

      13. pow2 N/A

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

      14. associate-/r* N/A

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

      15. lower-/.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lift-sin.f64 N/A

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

      18. +-commutative N/A

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

      19. hang-0p-tan N/A

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

      20. lower-tan.f64 N/A

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

      21. lower-/.f64 99.7

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

   6. Applied rewrites 99.7%

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

   7. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{720} \cdot {x}^{2} - \frac{1}{24}\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 65.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.001388888888888889 \cdot \left(x \cdot x\right) - 0.041666666666666664, x \cdot x, 0.5\right)} \]
   10. Step-by-step derivation

       1. lift--.f64 N/A

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

       2. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{720} \cdot \left(x \cdot x\right) - \frac{1}{24}, x \cdot x, \frac{1}{2}\right) \]

       3. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{720} \cdot \left(x \cdot x\right) - \frac{1}{24}, x \cdot x, \frac{1}{2}\right) \]

       4. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{720} \cdot {x}^{2} - \frac{1}{24}, x \cdot x, \frac{1}{2}\right) \]

       5. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{720} \cdot {x}^{2} - \frac{1}{24} \cdot 1, x \cdot x, \frac{1}{2}\right) \]

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

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

       7. *-commutative N/A

          \[\leadsto \mathsf{fma}\left({x}^{2} \cdot \frac{1}{720} + \left(\mathsf{neg}\left(\frac{1}{24}\right)\right) \cdot 1, x \cdot x, \frac{1}{2}\right) \]

       8. metadata-eval N/A

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

       9. metadata-eval N/A

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

       10. lower-fma.f64 N/A

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

       11. pow2 N/A

           \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{1}{720}, \frac{-1}{24}\right), x \cdot x, \frac{1}{2}\right) \]

       12. lift-*.f64 65.8

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

   11. Applied rewrites 65.8%

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

   if 0.032000000000000001 < x

   1. Initial program 98.0%

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

Alternative 5: 64.2% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 85000000000000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.001388888888888889, -0.041666666666666664\right), x \cdot x, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{1}{x} - \frac{1}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 85000000000000.0)
   (fma (fma (* x x) 0.001388888888888889 -0.041666666666666664) (* x x) 0.5)
   (- (* (/ 1.0 x) (/ 1.0 x)) (/ 1.0 (* x x)))))

C

double code(double x) {
	double tmp;
	if (x <= 85000000000000.0) {
		tmp = fma(fma((x * x), 0.001388888888888889, -0.041666666666666664), (x * x), 0.5);
	} else {
		tmp = ((1.0 / x) * (1.0 / x)) - (1.0 / (x * x));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 85000000000000.0)
		tmp = fma(fma(Float64(x * x), 0.001388888888888889, -0.041666666666666664), Float64(x * x), 0.5);
	else
		tmp = Float64(Float64(Float64(1.0 / x) * Float64(1.0 / x)) - Float64(1.0 / Float64(x * x)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 85000000000000.0], N[(N[(N[(x * x), $MachinePrecision] * 0.001388888888888889 + -0.041666666666666664), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision], N[(N[(N[(1.0 / x), $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision] - N[(1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 8.5e13

   1. Initial program 38.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. flip-- N/A

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

      6. pow2 N/A

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

      7. associate-/l/ N/A

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

      8. lower-/.f64 N/A

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

      9. metadata-eval N/A

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

      10. 1-sub-cos N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sin.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. lift-cos.f64 N/A

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

      18. pow2 N/A

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

      19. lift-*.f64 69.5

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

   4. Applied rewrites 69.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. times-frac N/A

          \[\leadsto \color{blue}{\frac{\sin x}{\cos x + 1} \cdot \frac{\sin x}{{x}^{2}}} \]

      11. *-commutative N/A

          \[\leadsto \color{blue}{\frac{\sin x}{{x}^{2}} \cdot \frac{\sin x}{\cos x + 1}} \]

      12. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{\sin x}{{x}^{2}} \cdot \frac{\sin x}{\cos x + 1}} \]

      13. pow2 N/A

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

      14. associate-/r* N/A

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

      15. lower-/.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lift-sin.f64 N/A

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

      18. +-commutative N/A

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

      19. hang-0p-tan N/A

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

      20. lower-tan.f64 N/A

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

      21. lower-/.f64 99.7

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

   6. Applied rewrites 99.7%

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

   7. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{720} \cdot {x}^{2} - \frac{1}{24}\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 64.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.001388888888888889 \cdot \left(x \cdot x\right) - 0.041666666666666664, x \cdot x, 0.5\right)} \]
   10. Step-by-step derivation

       1. lift--.f64 N/A

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

       2. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{720} \cdot \left(x \cdot x\right) - \frac{1}{24}, x \cdot x, \frac{1}{2}\right) \]

       3. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{720} \cdot \left(x \cdot x\right) - \frac{1}{24}, x \cdot x, \frac{1}{2}\right) \]

       4. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{720} \cdot {x}^{2} - \frac{1}{24}, x \cdot x, \frac{1}{2}\right) \]

       5. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{720} \cdot {x}^{2} - \frac{1}{24} \cdot 1, x \cdot x, \frac{1}{2}\right) \]

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

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

       7. *-commutative N/A

          \[\leadsto \mathsf{fma}\left({x}^{2} \cdot \frac{1}{720} + \left(\mathsf{neg}\left(\frac{1}{24}\right)\right) \cdot 1, x \cdot x, \frac{1}{2}\right) \]

       8. metadata-eval N/A

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

       9. metadata-eval N/A

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

       10. lower-fma.f64 N/A

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

       11. pow2 N/A

           \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{1}{720}, \frac{-1}{24}\right), x \cdot x, \frac{1}{2}\right) \]

       12. lift-*.f64 64.0

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

   11. Applied rewrites 64.0%

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

   if 8.5e13 < x

   1. Initial program 97.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. pow2 N/A

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

      6. div-sub N/A

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

      7. lower--.f64 N/A

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

      8. pow-flip N/A

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

      9. lower-pow.f64 N/A

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

      10. metadata-eval N/A

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

      11. pow2 N/A

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

      12. associate-/r* N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lift-cos.f64 99.3

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

   4. Applied rewrites 99.3%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. associate-/l/ N/A

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

      5. pow2 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 98.2

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

   6. Applied rewrites 98.2%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 55.8%

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

       1. lift-pow.f64 N/A

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

       2. metadata-eval N/A

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

       3. pow-flip N/A

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

       4. pow2 N/A

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

       5. metadata-eval N/A

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

       6. sqr-neg-rev N/A

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

       7. times-frac N/A

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

       8. lower-*.f64 N/A

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

       9. lower-/.f64 N/A

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

       10. lower-neg.f64 N/A

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

       11. lower-/.f64 N/A

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

       12. lower-neg.f64 56.1

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

   11. Applied rewrites 56.1%

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

4. Final simplification 61.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 85000000000000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.001388888888888889, -0.041666666666666664\right), x \cdot x, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{1}{x} - \frac{1}{x \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 64.2% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5.2 \cdot 10^{+38}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.001388888888888889, -0.041666666666666664\right), x \cdot x, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - 1}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 5.2e+38)
   (fma (fma (* x x) 0.001388888888888889 -0.041666666666666664) (* x x) 0.5)
   (/ (- 1.0 1.0) (* x x))))

C

double code(double x) {
	double tmp;
	if (x <= 5.2e+38) {
		tmp = fma(fma((x * x), 0.001388888888888889, -0.041666666666666664), (x * x), 0.5);
	} else {
		tmp = (1.0 - 1.0) / (x * x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 5.2e+38)
		tmp = fma(fma(Float64(x * x), 0.001388888888888889, -0.041666666666666664), Float64(x * x), 0.5);
	else
		tmp = Float64(Float64(1.0 - 1.0) / Float64(x * x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 5.2e+38], N[(N[(N[(x * x), $MachinePrecision] * 0.001388888888888889 + -0.041666666666666664), $MachinePrecision] * N[(x * x), $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 < 5.1999999999999998e38

   1. Initial program 39.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. flip-- N/A

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

      6. pow2 N/A

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

      7. associate-/l/ N/A

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

      8. lower-/.f64 N/A

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

      9. metadata-eval N/A

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

      10. 1-sub-cos N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sin.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. lift-cos.f64 N/A

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

      18. pow2 N/A

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

      19. lift-*.f64 69.6

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

   4. Applied rewrites 69.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. times-frac N/A

          \[\leadsto \color{blue}{\frac{\sin x}{\cos x + 1} \cdot \frac{\sin x}{{x}^{2}}} \]

      11. *-commutative N/A

          \[\leadsto \color{blue}{\frac{\sin x}{{x}^{2}} \cdot \frac{\sin x}{\cos x + 1}} \]

      12. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{\sin x}{{x}^{2}} \cdot \frac{\sin x}{\cos x + 1}} \]

      13. pow2 N/A

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

      14. associate-/r* N/A

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

      15. lower-/.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lift-sin.f64 N/A

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

      18. +-commutative N/A

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

      19. hang-0p-tan N/A

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

      20. lower-tan.f64 N/A

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

      21. lower-/.f64 99.7

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

   6. Applied rewrites 99.7%

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

   7. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{720} \cdot {x}^{2} - \frac{1}{24}\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 63.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.001388888888888889 \cdot \left(x \cdot x\right) - 0.041666666666666664, x \cdot x, 0.5\right)} \]
   10. Step-by-step derivation

       1. lift--.f64 N/A

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

       2. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{720} \cdot \left(x \cdot x\right) - \frac{1}{24}, x \cdot x, \frac{1}{2}\right) \]

       3. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{720} \cdot \left(x \cdot x\right) - \frac{1}{24}, x \cdot x, \frac{1}{2}\right) \]

       4. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{720} \cdot {x}^{2} - \frac{1}{24}, x \cdot x, \frac{1}{2}\right) \]

       5. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{720} \cdot {x}^{2} - \frac{1}{24} \cdot 1, x \cdot x, \frac{1}{2}\right) \]

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

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

       7. *-commutative N/A

          \[\leadsto \mathsf{fma}\left({x}^{2} \cdot \frac{1}{720} + \left(\mathsf{neg}\left(\frac{1}{24}\right)\right) \cdot 1, x \cdot x, \frac{1}{2}\right) \]

       8. metadata-eval N/A

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

       9. metadata-eval N/A

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

       10. lower-fma.f64 N/A

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

       11. pow2 N/A

           \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{1}{720}, \frac{-1}{24}\right), x \cdot x, \frac{1}{2}\right) \]

       12. lift-*.f64 63.7

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

   11. Applied rewrites 63.7%

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

   if 5.1999999999999998e38 < x

   1. Initial program 97.9%

      \[\frac{1 - \cos x}{x \cdot x} \]
   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 56.1%

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

Alternative 7: 63.8% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.4:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, -0.041666666666666664, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - 1}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 3.4)
   (fma (* x x) -0.041666666666666664 0.5)
   (/ (- 1.0 1.0) (* x x))))

C

double code(double x) {
	double tmp;
	if (x <= 3.4) {
		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 <= 3.4)
		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, 3.4], 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 < 3.39999999999999991

   1. Initial program 36.9%

      \[\frac{1 - \cos x}{x \cdot x} \]
   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. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 65.6

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

   5. Applied rewrites 65.6%

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

   if 3.39999999999999991 < x

   1. Initial program 98.0%

      \[\frac{1 - \cos x}{x \cdot x} \]
   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 51.0%

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

4. Final simplification 61.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.4:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, -0.041666666666666664, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - 1}{x \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 52.5% 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 54.4%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 48.3%

   \[\leadsto \color{blue}{0.5} \]

6. Final simplification 48.3%

   \[\leadsto 0.5 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2025056 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))