cos2 (problem 3.4.1)

Percentage Accurate: 51.1% → 99.4%

Time: 6.9s

Alternatives: 10

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: 51.1% 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: 99.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.02:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-2.48015873015873 \cdot 10^{-5}, x\_m \cdot x\_m, 0.001388888888888889\right), x\_m \cdot x\_m, -0.041666666666666664\right), x\_m \cdot x\_m, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin x\_m}{x\_m \cdot x\_m} \cdot \tan \left(\frac{x\_m}{2}\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.02)
   (fma
    (fma
     (fma -2.48015873015873e-5 (* x_m x_m) 0.001388888888888889)
     (* x_m x_m)
     -0.041666666666666664)
    (* x_m x_m)
    0.5)
   (* (/ (sin x_m) (* x_m x_m)) (tan (/ x_m 2.0)))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.02) {
		tmp = fma(fma(fma(-2.48015873015873e-5, (x_m * x_m), 0.001388888888888889), (x_m * x_m), -0.041666666666666664), (x_m * x_m), 0.5);
	} else {
		tmp = (sin(x_m) / (x_m * x_m)) * tan((x_m / 2.0));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.02)
		tmp = fma(fma(fma(-2.48015873015873e-5, Float64(x_m * x_m), 0.001388888888888889), Float64(x_m * x_m), -0.041666666666666664), Float64(x_m * x_m), 0.5);
	else
		tmp = Float64(Float64(sin(x_m) / Float64(x_m * x_m)) * tan(Float64(x_m / 2.0)));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.02], N[(N[(N[(-2.48015873015873e-5 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.001388888888888889), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + -0.041666666666666664), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 0.5), $MachinePrecision], N[(N[(N[Sin[x$95$m], $MachinePrecision] / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] * N[Tan[N[(x$95$m / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0200000000000000004

   1. Initial program 30.6%

      \[\frac{1 - \cos x}{x \cdot x} \]
   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. lower-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. lift-cos.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. 1-sub-cos N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sin.f64 N/A

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

      12. lower-sin.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 64.5

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

   4. Applied rewrites 64.5%

      \[\leadsto \color{blue}{\frac{\sin x \cdot \sin x}{\left(\cos x + 1\right) \cdot \left(x \cdot x\right)}} \]
   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-*.f64 N/A

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

      4. *-commutative N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-cos.f64 N/A

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

      15. hang-0p-tan N/A

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

      16. lower-tan.f64 N/A

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

      17. 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

      \[\leadsto \color{blue}{\frac{1}{2} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{40320} \cdot {x}^{2}\right) - \frac{1}{24}\right)} \]
   8. Step-by-step derivation

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

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

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 71.4

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

   9. Applied rewrites 71.4%

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

   if 0.0200000000000000004 < x

   1. Initial program 99.5%

      \[\frac{1 - \cos x}{x \cdot x} \]
   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. lower-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. lift-cos.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. 1-sub-cos N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sin.f64 N/A

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

      12. lower-sin.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 99.0

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

   4. Applied rewrites 99.0%

      \[\leadsto \color{blue}{\frac{\sin x \cdot \sin x}{\left(\cos x + 1\right) \cdot \left(x \cdot x\right)}} \]
   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-*.f64 N/A

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

      4. *-commutative N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-cos.f64 N/A

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

      15. hang-0p-tan N/A

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

      16. lower-tan.f64 N/A

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

      17. 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. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. lower-/.f64 99.8

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

   8. Applied rewrites 99.8%

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

Alternative 2: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \frac{\frac{\sin x\_m}{x\_m}}{x\_m} \cdot \tan \left(\frac{x\_m}{2}\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (* (/ (/ (sin x_m) x_m) x_m) (tan (/ x_m 2.0))))

C

x_m = fabs(x);
double code(double x_m) {
	return ((sin(x_m) / x_m) / x_m) * tan((x_m / 2.0));
}

Fortran

x_m =     private
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_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_m
    code = ((sin(x_m) / x_m) / x_m) * tan((x_m / 2.0d0))
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return ((Math.sin(x_m) / x_m) / x_m) * Math.tan((x_m / 2.0));
}

Python

x_m = math.fabs(x)
def code(x_m):
	return ((math.sin(x_m) / x_m) / x_m) * math.tan((x_m / 2.0))

Julia

x_m = abs(x)
function code(x_m)
	return Float64(Float64(Float64(sin(x_m) / x_m) / x_m) * tan(Float64(x_m / 2.0)))
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = ((sin(x_m) / x_m) / x_m) * tan((x_m / 2.0));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(N[(N[(N[Sin[x$95$m], $MachinePrecision] / x$95$m), $MachinePrecision] / x$95$m), $MachinePrecision] * N[Tan[N[(x$95$m / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 47.6%

   \[\frac{1 - \cos x}{x \cdot x} \]
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. lower-/.f64 N/A

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

   6. metadata-eval N/A

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

   7. lift-cos.f64 N/A

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

   8. lift-cos.f64 N/A

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

   9. 1-sub-cos N/A

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

   10. lower-*.f64 N/A

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

   11. lower-sin.f64 N/A

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

   12. lower-sin.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. +-commutative N/A

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

   15. lower-+.f64 73.0

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

4. Applied rewrites 73.0%

   \[\leadsto \color{blue}{\frac{\sin x \cdot \sin x}{\left(\cos x + 1\right) \cdot \left(x \cdot x\right)}} \]
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-*.f64 N/A

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

   4. *-commutative N/A

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

   5. times-frac N/A

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

   6. lower-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. associate-/r* N/A

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

   9. lower-/.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lift-sin.f64 N/A

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

   12. lift-+.f64 N/A

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

   13. +-commutative N/A

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

   14. lift-cos.f64 N/A

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

   15. hang-0p-tan N/A

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

   16. lower-tan.f64 N/A

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

   17. 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. Add Preprocessing

Alternative 3: 99.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.02:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-2.48015873015873 \cdot 10^{-5}, x\_m \cdot x\_m, 0.001388888888888889\right), x\_m \cdot x\_m, -0.041666666666666664\right), x\_m \cdot x\_m, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan \left(0.5 \cdot x\_m\right)}{x\_m \cdot x\_m} \cdot \sin x\_m\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.02)
   (fma
    (fma
     (fma -2.48015873015873e-5 (* x_m x_m) 0.001388888888888889)
     (* x_m x_m)
     -0.041666666666666664)
    (* x_m x_m)
    0.5)
   (* (/ (tan (* 0.5 x_m)) (* x_m x_m)) (sin x_m))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.02) {
		tmp = fma(fma(fma(-2.48015873015873e-5, (x_m * x_m), 0.001388888888888889), (x_m * x_m), -0.041666666666666664), (x_m * x_m), 0.5);
	} else {
		tmp = (tan((0.5 * x_m)) / (x_m * x_m)) * sin(x_m);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.02)
		tmp = fma(fma(fma(-2.48015873015873e-5, Float64(x_m * x_m), 0.001388888888888889), Float64(x_m * x_m), -0.041666666666666664), Float64(x_m * x_m), 0.5);
	else
		tmp = Float64(Float64(tan(Float64(0.5 * x_m)) / Float64(x_m * x_m)) * sin(x_m));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.02], N[(N[(N[(-2.48015873015873e-5 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.001388888888888889), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + -0.041666666666666664), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 0.5), $MachinePrecision], N[(N[(N[Tan[N[(0.5 * x$95$m), $MachinePrecision]], $MachinePrecision] / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] * N[Sin[x$95$m], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0200000000000000004

   1. Initial program 30.6%

      \[\frac{1 - \cos x}{x \cdot x} \]
   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. lower-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. lift-cos.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. 1-sub-cos N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sin.f64 N/A

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

      12. lower-sin.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 64.5

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

   4. Applied rewrites 64.5%

      \[\leadsto \color{blue}{\frac{\sin x \cdot \sin x}{\left(\cos x + 1\right) \cdot \left(x \cdot x\right)}} \]
   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-*.f64 N/A

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

      4. *-commutative N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-cos.f64 N/A

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

      15. hang-0p-tan N/A

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

      16. lower-tan.f64 N/A

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

      17. 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

      \[\leadsto \color{blue}{\frac{1}{2} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{40320} \cdot {x}^{2}\right) - \frac{1}{24}\right)} \]
   8. Step-by-step derivation

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

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

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 71.4

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

   9. Applied rewrites 71.4%

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

   if 0.0200000000000000004 < x

   1. Initial program 99.5%

      \[\frac{1 - \cos x}{x \cdot x} \]
   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. lower-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. lift-cos.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. 1-sub-cos N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sin.f64 N/A

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

      12. lower-sin.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 99.0

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. hang-0p-tan N/A

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

      8. *-rgt-identity N/A

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

      9. associate-/l* N/A

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

      10. metadata-eval N/A

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

      11. *-commutative N/A

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

      12. lower-tan.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-sin.f64 99.6

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

   7. Applied rewrites 99.6%

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.02:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-2.48015873015873 \cdot 10^{-5}, x \cdot x, 0.001388888888888889\right), x \cdot x, -0.041666666666666664\right), x \cdot x, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan \left(0.5 \cdot x\right)}{x \cdot x} \cdot \sin x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.106:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-2.48015873015873 \cdot 10^{-5}, x\_m \cdot x\_m, 0.001388888888888889\right), x\_m \cdot x\_m, -0.041666666666666664\right), x\_m \cdot x\_m, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x\_m - \cos x\_m \cdot x\_m}{x\_m}}{x\_m \cdot x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.106)
   (fma
    (fma
     (fma -2.48015873015873e-5 (* x_m x_m) 0.001388888888888889)
     (* x_m x_m)
     -0.041666666666666664)
    (* x_m x_m)
    0.5)
   (/ (/ (- x_m (* (cos x_m) x_m)) x_m) (* x_m x_m))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.106) {
		tmp = fma(fma(fma(-2.48015873015873e-5, (x_m * x_m), 0.001388888888888889), (x_m * x_m), -0.041666666666666664), (x_m * x_m), 0.5);
	} else {
		tmp = ((x_m - (cos(x_m) * x_m)) / x_m) / (x_m * x_m);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.106)
		tmp = fma(fma(fma(-2.48015873015873e-5, Float64(x_m * x_m), 0.001388888888888889), Float64(x_m * x_m), -0.041666666666666664), Float64(x_m * x_m), 0.5);
	else
		tmp = Float64(Float64(Float64(x_m - Float64(cos(x_m) * x_m)) / x_m) / Float64(x_m * x_m));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.106], N[(N[(N[(-2.48015873015873e-5 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.001388888888888889), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + -0.041666666666666664), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 0.5), $MachinePrecision], N[(N[(N[(x$95$m - N[(N[Cos[x$95$m], $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision] / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.105999999999999997

   1. Initial program 30.6%

      \[\frac{1 - \cos x}{x \cdot x} \]
   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. lower-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. lift-cos.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. 1-sub-cos N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sin.f64 N/A

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

      12. lower-sin.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 64.5

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

   4. Applied rewrites 64.5%

      \[\leadsto \color{blue}{\frac{\sin x \cdot \sin x}{\left(\cos x + 1\right) \cdot \left(x \cdot x\right)}} \]
   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-*.f64 N/A

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

      4. *-commutative N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-cos.f64 N/A

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

      15. hang-0p-tan N/A

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

      16. lower-tan.f64 N/A

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

      17. 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

      \[\leadsto \color{blue}{\frac{1}{2} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{40320} \cdot {x}^{2}\right) - \frac{1}{24}\right)} \]
   8. Step-by-step derivation

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

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

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 71.4

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

   9. Applied rewrites 71.4%

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

   if 0.105999999999999997 < x

   1. Initial program 99.5%

      \[\frac{1 - \cos x}{x \cdot x} \]
   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 99.5

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

   4. Applied rewrites 99.5%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. frac-sub N/A

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

      5. *-lft-identity N/A

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

      6. lift-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 99.5

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

   6. Applied rewrites 99.5%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. associate-/l/ N/A

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

      6. lift-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 99.5

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

   8. Applied rewrites 99.5%

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

Alternative 5: 99.2% accurate, 1.0× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.1)
   (fma
    (fma
     (fma -2.48015873015873e-5 (* x_m x_m) 0.001388888888888889)
     (* x_m x_m)
     -0.041666666666666664)
    (* x_m x_m)
    0.5)
   (/ (- 1.0 (cos x_m)) (* x_m x_m))))

C

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

Julia

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.1], N[(N[(N[(-2.48015873015873e-5 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.001388888888888889), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + -0.041666666666666664), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 0.5), $MachinePrecision], N[(N[(1.0 - N[Cos[x$95$m], $MachinePrecision]), $MachinePrecision] / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.10000000000000001

   1. Initial program 30.6%

      \[\frac{1 - \cos x}{x \cdot x} \]
   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. lower-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. lift-cos.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. 1-sub-cos N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sin.f64 N/A

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

      12. lower-sin.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 64.5

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

   4. Applied rewrites 64.5%

      \[\leadsto \color{blue}{\frac{\sin x \cdot \sin x}{\left(\cos x + 1\right) \cdot \left(x \cdot x\right)}} \]
   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-*.f64 N/A

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

      4. *-commutative N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-cos.f64 N/A

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

      15. hang-0p-tan N/A

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

      16. lower-tan.f64 N/A

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

      17. 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

      \[\leadsto \color{blue}{\frac{1}{2} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{40320} \cdot {x}^{2}\right) - \frac{1}{24}\right)} \]
   8. Step-by-step derivation

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

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

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 71.4

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

   9. Applied rewrites 71.4%

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

   if 0.10000000000000001 < x

   1. Initial program 99.5%

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

Alternative 6: 76.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 7.5 \cdot 10^{+38}:\\ \;\;\;\;\mathsf{fma}\left(x\_m \cdot x\_m, \mathsf{fma}\left(0.001388888888888889 \cdot x\_m, x\_m, -0.041666666666666664\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{\left(\frac{1}{x\_m} \cdot x\_m\right) \cdot x\_m}{x\_m}}{x\_m \cdot x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 7.5e+38)
   (fma
    (* x_m x_m)
    (fma (* 0.001388888888888889 x_m) x_m -0.041666666666666664)
    0.5)
   (/ (- 1.0 (/ (* (* (/ 1.0 x_m) x_m) x_m) x_m)) (* x_m x_m))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 7.5e+38) {
		tmp = fma((x_m * x_m), fma((0.001388888888888889 * x_m), x_m, -0.041666666666666664), 0.5);
	} else {
		tmp = (1.0 - ((((1.0 / x_m) * x_m) * x_m) / x_m)) / (x_m * x_m);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 7.5e+38)
		tmp = fma(Float64(x_m * x_m), fma(Float64(0.001388888888888889 * x_m), x_m, -0.041666666666666664), 0.5);
	else
		tmp = Float64(Float64(1.0 - Float64(Float64(Float64(Float64(1.0 / x_m) * x_m) * x_m) / x_m)) / Float64(x_m * x_m));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 7.5e+38], N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(N[(0.001388888888888889 * x$95$m), $MachinePrecision] * x$95$m + -0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision], N[(N[(1.0 - N[(N[(N[(N[(1.0 / x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] / x$95$m), $MachinePrecision]), $MachinePrecision] / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 7.4999999999999999e38

   1. Initial program 34.3%

      \[\frac{1 - \cos x}{x \cdot x} \]
   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 68.3%

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

   if 7.4999999999999999e38 < x

   1. Initial program 99.6%

      \[\frac{1 - \cos x}{x \cdot x} \]
   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 69.2%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. *-rgt-identity N/A

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

      5. associate-/r* N/A

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

      6. frac-sub N/A

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

      7. *-rgt-identity N/A

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

      8. lower-/.f64 N/A

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

   7. Applied rewrites 2.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 2.1

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

   9. Applied rewrites 2.1%

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

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

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

       3. lift-*.f64 N/A

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

       6. *-lft-identity N/A

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

       7. *-commutative N/A

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

       8. *-inverses N/A

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

       9. associate-/l* N/A

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

       10. lift-*.f64 N/A

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

       11. associate-/l* N/A

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

       12. lift-*.f64 N/A

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

       13. lift-/.f64 N/A

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

       14. *-commutative N/A

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

       15. lower-*.f64 2.1

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

   11. Applied rewrites 69.6%

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

Alternative 7: 76.3% accurate, 2.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 7.5 \cdot 10^{+38}:\\ \;\;\;\;\mathsf{fma}\left(x\_m \cdot x\_m, \mathsf{fma}\left(0.001388888888888889 \cdot x\_m, x\_m, -0.041666666666666664\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{1}{x\_m} \cdot x\_m}{x\_m \cdot x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 7.5e+38)
   (fma
    (* x_m x_m)
    (fma (* 0.001388888888888889 x_m) x_m -0.041666666666666664)
    0.5)
   (/ (- 1.0 (* (/ 1.0 x_m) x_m)) (* x_m x_m))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 7.5e+38) {
		tmp = fma((x_m * x_m), fma((0.001388888888888889 * x_m), x_m, -0.041666666666666664), 0.5);
	} else {
		tmp = (1.0 - ((1.0 / x_m) * x_m)) / (x_m * x_m);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 7.5e+38)
		tmp = fma(Float64(x_m * x_m), fma(Float64(0.001388888888888889 * x_m), x_m, -0.041666666666666664), 0.5);
	else
		tmp = Float64(Float64(1.0 - Float64(Float64(1.0 / x_m) * x_m)) / Float64(x_m * x_m));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 7.5e+38], N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(N[(0.001388888888888889 * x$95$m), $MachinePrecision] * x$95$m + -0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision], N[(N[(1.0 - N[(N[(1.0 / x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision] / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 7.4999999999999999e38

   1. Initial program 34.3%

      \[\frac{1 - \cos x}{x \cdot x} \]
   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 68.3%

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

   if 7.4999999999999999e38 < x

   1. Initial program 99.6%

      \[\frac{1 - \cos x}{x \cdot x} \]
   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 69.2%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. *-rgt-identity N/A

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

      5. associate-/r* N/A

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

      6. frac-sub N/A

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

      7. *-rgt-identity N/A

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

      8. lower-/.f64 N/A

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

   7. Applied rewrites 2.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 2.1

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

   9. Applied rewrites 2.1%

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

       1. lift-/.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-/l* N/A

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

       4. lift-*.f64 N/A

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

       5. associate-/l* N/A

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

       6. *-inverses N/A

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

       7. *-commutative N/A

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

       8. *-lft-identity N/A

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

       9. lower-*.f64 69.6

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

   11. Applied rewrites 69.6%

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

Alternative 8: 76.1% accurate, 4.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 7.5 \cdot 10^{+38}:\\ \;\;\;\;\mathsf{fma}\left(x\_m \cdot x\_m, \mathsf{fma}\left(0.001388888888888889 \cdot x\_m, x\_m, -0.041666666666666664\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - 1}{x\_m \cdot x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 7.5e+38)
   (fma
    (* x_m x_m)
    (fma (* 0.001388888888888889 x_m) x_m -0.041666666666666664)
    0.5)
   (/ (- 1.0 1.0) (* x_m x_m))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 7.5e+38) {
		tmp = fma((x_m * x_m), fma((0.001388888888888889 * x_m), x_m, -0.041666666666666664), 0.5);
	} else {
		tmp = (1.0 - 1.0) / (x_m * x_m);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 7.5e+38)
		tmp = fma(Float64(x_m * x_m), fma(Float64(0.001388888888888889 * x_m), x_m, -0.041666666666666664), 0.5);
	else
		tmp = Float64(Float64(1.0 - 1.0) / Float64(x_m * x_m));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 7.5e+38], N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(N[(0.001388888888888889 * x$95$m), $MachinePrecision] * x$95$m + -0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision], N[(N[(1.0 - 1.0), $MachinePrecision] / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 7.4999999999999999e38

   1. Initial program 34.3%

      \[\frac{1 - \cos x}{x \cdot x} \]
   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 68.3%

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

   if 7.4999999999999999e38 < x

   1. Initial program 99.6%

      \[\frac{1 - \cos x}{x \cdot x} \]
   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 69.2%

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

Alternative 9: 75.9% accurate, 4.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 5.2 \cdot 10^{+76}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - 1}{x\_m \cdot x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 5.2e+76) 0.5 (/ (- 1.0 1.0) (* x_m x_m))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 5.2e+76) {
		tmp = 0.5;
	} else {
		tmp = (1.0 - 1.0) / (x_m * x_m);
	}
	return tmp;
}

Fortran

x_m =     private
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_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 5.2d+76) then
        tmp = 0.5d0
    else
        tmp = (1.0d0 - 1.0d0) / (x_m * x_m)
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 5.2e+76) {
		tmp = 0.5;
	} else {
		tmp = (1.0 - 1.0) / (x_m * x_m);
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 5.2e+76:
		tmp = 0.5
	else:
		tmp = (1.0 - 1.0) / (x_m * x_m)
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 5.2e+76)
		tmp = 0.5;
	else
		tmp = Float64(Float64(1.0 - 1.0) / Float64(x_m * x_m));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 5.2e+76)
		tmp = 0.5;
	else
		tmp = (1.0 - 1.0) / (x_m * x_m);
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 5.2e+76], 0.5, N[(N[(1.0 - 1.0), $MachinePrecision] / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.1999999999999999e76

   1. Initial program 36.1%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 66.5%

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

   if 5.1999999999999999e76 < x

   1. Initial program 99.9%

      \[\frac{1 - \cos x}{x \cdot x} \]
   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 77.7%

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

Alternative 10: 51.4% accurate, 120.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ 0.5 \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 0.5)

C

x_m = fabs(x);
double code(double x_m) {
	return 0.5;
}

Fortran

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

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return 0.5;
}

Python

x_m = math.fabs(x)
def code(x_m):
	return 0.5

Julia

x_m = abs(x)
function code(x_m)
	return 0.5
end

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := 0.5

Derivation

1. Initial program 47.6%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 55.1%

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

Reproduce

herbie shell --seed 2025008 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))