cos2 (problem 3.4.1)

Percentage Accurate: 50.3% → 99.6%

Time: 5.4s

Alternatives: 7

Speedup: N/A×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 50.3% 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: 99.6% accurate, N/A× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.102:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(-2.48015873015873 \cdot 10^{-5}, x\_m \cdot x\_m, 0.001388888888888889\right) \cdot x\_m\right) \cdot x\_m - 0.041666666666666664, x\_m \cdot x\_m, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x\_m}{x\_m}}{x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.102)
   (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 <= 0.102) {
		tmp = 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 <= 0.102)
		tmp = fma(Float64(Float64(Float64(fma(-2.48015873015873e-5, Float64(x_m * x_m), 0.001388888888888889) * x_m) * x_m) - 0.041666666666666664), Float64(x_m * x_m), 0.5);
	else
		tmp = Float64(Float64(Float64(1.0 - cos(x_m)) / x_m) / x_m);
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.102], N[(N[(N[(N[(N[(-2.48015873015873e-5 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.001388888888888889), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] - 0.041666666666666664), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 0.5), $MachinePrecision], N[(N[(N[(1.0 - N[Cos[x$95$m], $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision] / x$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.101999999999999993

   1. Initial program 34.6%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. pow2 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 67.5

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

   5. Applied rewrites 67.5%

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

   if 0.101999999999999993 < x

   1. Initial program 96.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. sqr-neg-rev N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-neg.f64 99.3

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

   4. Applied rewrites 99.3%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.102:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(-2.48015873015873 \cdot 10^{-5}, x \cdot x, 0.001388888888888889\right) \cdot x\right) \cdot x - 0.041666666666666664, x \cdot x, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.5% accurate, N/A× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.12:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(-2.48015873015873 \cdot 10^{-5}, x\_m \cdot x\_m, 0.001388888888888889\right) \cdot x\_m\right) \cdot x\_m - 0.041666666666666664, x\_m \cdot x\_m, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{x\_m}^{-1} \cdot -1 - \frac{\cos x\_m}{x\_m} \cdot -1}{-1 \cdot x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.12)
   (fma
    (-
     (*
      (* (fma -2.48015873015873e-5 (* x_m x_m) 0.001388888888888889) x_m)
      x_m)
     0.041666666666666664)
    (* x_m x_m)
    0.5)
   (/ (- (* (pow x_m -1.0) -1.0) (* (/ (cos x_m) x_m) -1.0)) (* -1.0 x_m))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.12) {
		tmp = 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 = ((pow(x_m, -1.0) * -1.0) - ((cos(x_m) / x_m) * -1.0)) / (-1.0 * x_m);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.12)
		tmp = fma(Float64(Float64(Float64(fma(-2.48015873015873e-5, Float64(x_m * x_m), 0.001388888888888889) * x_m) * x_m) - 0.041666666666666664), Float64(x_m * x_m), 0.5);
	else
		tmp = Float64(Float64(Float64((x_m ^ -1.0) * -1.0) - Float64(Float64(cos(x_m) / x_m) * -1.0)) / Float64(-1.0 * x_m));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.12], N[(N[(N[(N[(N[(-2.48015873015873e-5 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.001388888888888889), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] - 0.041666666666666664), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 0.5), $MachinePrecision], N[(N[(N[(N[Power[x$95$m, -1.0], $MachinePrecision] * -1.0), $MachinePrecision] - N[(N[(N[Cos[x$95$m], $MachinePrecision] / x$95$m), $MachinePrecision] * -1.0), $MachinePrecision]), $MachinePrecision] / N[(-1.0 * x$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.12

   1. Initial program 34.6%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. pow2 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 67.5

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

   5. Applied rewrites 67.5%

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

   if 0.12 < x

   1. Initial program 96.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. sqr-neg-rev N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-neg.f64 99.3

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

   4. Applied rewrites 99.3%

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

      1. lift-neg.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. div-sub N/A

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

      6. metadata-eval N/A

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

      7. frac-2neg N/A

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

      8. lower--.f64 N/A

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

      9. frac-2neg N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. mul-1-neg N/A

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

      13. *-commutative N/A

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

      14. times-frac N/A

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

      15. metadata-eval N/A

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

      16. lower-*.f64 N/A

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

      17. inv-pow N/A

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

      18. lower-pow.f64 N/A

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

      19. *-rgt-identity N/A

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

      20. mul-1-neg N/A

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

      21. *-commutative N/A

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

      22. times-frac N/A

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

      23. metadata-eval N/A

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

      24. lower-*.f64 N/A

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

      25. lower-/.f64 N/A

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

      26. lift-cos.f64 99.1

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

   6. Applied rewrites 99.1%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.12:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(-2.48015873015873 \cdot 10^{-5}, x \cdot x, 0.001388888888888889\right) \cdot x\right) \cdot x - 0.041666666666666664, x \cdot x, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{-1} \cdot -1 - \frac{\cos x}{x} \cdot -1}{-1 \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.2% accurate, N/A× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.102:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(-2.48015873015873 \cdot 10^{-5}, x\_m \cdot x\_m, 0.001388888888888889\right) \cdot x\_m\right) \cdot x\_m - 0.041666666666666664, x\_m \cdot x\_m, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 \cdot \left(1 - \cos x\_m\right)}{-1 \cdot \left(x\_m \cdot x\_m\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.102)
   (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 (- 1.0 (cos x_m))) (* -1.0 (* x_m x_m)))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.102) {
		tmp = 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 * (1.0 - cos(x_m))) / (-1.0 * (x_m * x_m));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.102)
		tmp = fma(Float64(Float64(Float64(fma(-2.48015873015873e-5, Float64(x_m * x_m), 0.001388888888888889) * x_m) * x_m) - 0.041666666666666664), Float64(x_m * x_m), 0.5);
	else
		tmp = Float64(Float64(-1.0 * Float64(1.0 - cos(x_m))) / Float64(-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, 0.102], N[(N[(N[(N[(N[(-2.48015873015873e-5 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.001388888888888889), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] - 0.041666666666666664), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 0.5), $MachinePrecision], N[(N[(-1.0 * N[(1.0 - N[Cos[x$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-1.0 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.101999999999999993

   1. Initial program 34.6%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. pow2 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 67.5

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

   5. Applied rewrites 67.5%

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

   if 0.101999999999999993 < x

   1. Initial program 96.7%

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

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.102:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(-2.48015873015873 \cdot 10^{-5}, x \cdot x, 0.001388888888888889\right) \cdot x\right) \cdot x - 0.041666666666666664, x \cdot x, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 \cdot \left(1 - \cos x\right)}{-1 \cdot \left(x \cdot x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.0% accurate, N/A× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.115:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(-2.48015873015873 \cdot 10^{-5}, x\_m \cdot x\_m, 0.001388888888888889\right) \cdot x\_m\right) \cdot x\_m - 0.041666666666666664, x\_m \cdot x\_m, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(x\_m \cdot x\_m\right)}^{-1} - \frac{\frac{\cos x\_m}{x\_m}}{x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.115)
   (fma
    (-
     (*
      (* (fma -2.48015873015873e-5 (* x_m x_m) 0.001388888888888889) x_m)
      x_m)
     0.041666666666666664)
    (* x_m x_m)
    0.5)
   (- (pow (* x_m x_m) -1.0) (/ (/ (cos x_m) x_m) x_m))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.115) {
		tmp = 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 = pow((x_m * x_m), -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 <= 0.115)
		tmp = fma(Float64(Float64(Float64(fma(-2.48015873015873e-5, Float64(x_m * x_m), 0.001388888888888889) * x_m) * x_m) - 0.041666666666666664), Float64(x_m * x_m), 0.5);
	else
		tmp = Float64((Float64(x_m * x_m) ^ -1.0) - Float64(Float64(cos(x_m) / x_m) / x_m));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.115], N[(N[(N[(N[(N[(-2.48015873015873e-5 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.001388888888888889), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] - 0.041666666666666664), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 0.5), $MachinePrecision], N[(N[Power[N[(x$95$m * x$95$m), $MachinePrecision], -1.0], $MachinePrecision] - N[(N[(N[Cos[x$95$m], $MachinePrecision] / x$95$m), $MachinePrecision] / x$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.115000000000000005

   1. Initial program 34.6%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. pow2 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 67.5

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

   5. Applied rewrites 67.5%

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

   if 0.115000000000000005 < x

   1. Initial program 96.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. pow2 N/A

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

      6. div-sub N/A

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

      7. lower--.f64 N/A

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

      8. inv-pow N/A

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

      9. lower-pow.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. associate-/r* N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lift-cos.f64 96.4

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

   4. Applied rewrites 96.4%

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

Alternative 5: 98.9% accurate, N/A× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\cos x\_m, 1, 1\right)\\ \mathbf{if}\;x\_m \leq 0.11:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(-2.48015873015873 \cdot 10^{-5}, x\_m \cdot x\_m, 0.001388888888888889\right) \cdot x\_m\right) \cdot x\_m - 0.041666666666666664, x\_m \cdot x\_m, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{t\_0}^{-1}}{x\_m \cdot x\_m} - \frac{\frac{{\cos x\_m}^{2}}{t\_0}}{x\_m \cdot x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (fma (cos x_m) 1.0 1.0)))
   (if (<= x_m 0.11)
     (fma
      (-
       (*
        (* (fma -2.48015873015873e-5 (* x_m x_m) 0.001388888888888889) x_m)
        x_m)
       0.041666666666666664)
      (* x_m x_m)
      0.5)
     (-
      (/ (pow t_0 -1.0) (* x_m x_m))
      (/ (/ (pow (cos x_m) 2.0) t_0) (* x_m x_m))))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = fma(cos(x_m), 1.0, 1.0);
	double tmp;
	if (x_m <= 0.11) {
		tmp = 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 = (pow(t_0, -1.0) / (x_m * x_m)) - ((pow(cos(x_m), 2.0) / t_0) / (x_m * x_m));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = fma(cos(x_m), 1.0, 1.0)
	tmp = 0.0
	if (x_m <= 0.11)
		tmp = fma(Float64(Float64(Float64(fma(-2.48015873015873e-5, Float64(x_m * x_m), 0.001388888888888889) * x_m) * x_m) - 0.041666666666666664), Float64(x_m * x_m), 0.5);
	else
		tmp = Float64(Float64((t_0 ^ -1.0) / Float64(x_m * x_m)) - Float64(Float64((cos(x_m) ^ 2.0) / t_0) / Float64(x_m * x_m)));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(N[Cos[x$95$m], $MachinePrecision] * 1.0 + 1.0), $MachinePrecision]}, If[LessEqual[x$95$m, 0.11], N[(N[(N[(N[(N[(-2.48015873015873e-5 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.001388888888888889), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] - 0.041666666666666664), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 0.5), $MachinePrecision], N[(N[(N[Power[t$95$0, -1.0], $MachinePrecision] / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] - N[(N[(N[Power[N[Cos[x$95$m], $MachinePrecision], 2.0], $MachinePrecision] / t$95$0), $MachinePrecision] / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 0.110000000000000001

   1. Initial program 34.6%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. pow2 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 67.5

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

   5. Applied rewrites 67.5%

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

   if 0.110000000000000001 < x

   1. Initial program 96.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. sqr-neg-rev N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-neg.f64 99.3

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

   4. Applied rewrites 99.3%

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

      1. lift-neg.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. associate-/l/ N/A

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

      8. sqr-neg-rev N/A

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

      9. pow2 N/A

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

      10. flip-- N/A

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

      11. metadata-eval N/A

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

      12. unpow2 N/A

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

      13. sub-div N/A

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

      14. div-sub N/A

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

   6. Applied rewrites 96.0%

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

Alternative 6: 98.8% accurate, N/A× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\cos x\_m, 1, 1\right)\\ \mathbf{if}\;x\_m \leq 0.11:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(-2.48015873015873 \cdot 10^{-5}, x\_m \cdot x\_m, 0.001388888888888889\right) \cdot x\_m\right) \cdot x\_m - 0.041666666666666664, x\_m \cdot x\_m, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{t\_0}^{-1}}{x\_m \cdot x\_m} - \frac{\frac{\frac{1}{{\left({\cos x\_m}^{2}\right)}^{-1}}}{t\_0}}{{\left({\left(x\_m \cdot x\_m\right)}^{-0.5}\right)}^{-2}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (fma (cos x_m) 1.0 1.0)))
   (if (<= x_m 0.11)
     (fma
      (-
       (*
        (* (fma -2.48015873015873e-5 (* x_m x_m) 0.001388888888888889) x_m)
        x_m)
       0.041666666666666664)
      (* x_m x_m)
      0.5)
     (-
      (/ (pow t_0 -1.0) (* x_m x_m))
      (/
       (/ (/ 1.0 (pow (pow (cos x_m) 2.0) -1.0)) t_0)
       (pow (pow (* x_m x_m) -0.5) -2.0))))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = fma(cos(x_m), 1.0, 1.0);
	double tmp;
	if (x_m <= 0.11) {
		tmp = 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 = (pow(t_0, -1.0) / (x_m * x_m)) - (((1.0 / pow(pow(cos(x_m), 2.0), -1.0)) / t_0) / pow(pow((x_m * x_m), -0.5), -2.0));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = fma(cos(x_m), 1.0, 1.0)
	tmp = 0.0
	if (x_m <= 0.11)
		tmp = fma(Float64(Float64(Float64(fma(-2.48015873015873e-5, Float64(x_m * x_m), 0.001388888888888889) * x_m) * x_m) - 0.041666666666666664), Float64(x_m * x_m), 0.5);
	else
		tmp = Float64(Float64((t_0 ^ -1.0) / Float64(x_m * x_m)) - Float64(Float64(Float64(1.0 / ((cos(x_m) ^ 2.0) ^ -1.0)) / t_0) / ((Float64(x_m * x_m) ^ -0.5) ^ -2.0)));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(N[Cos[x$95$m], $MachinePrecision] * 1.0 + 1.0), $MachinePrecision]}, If[LessEqual[x$95$m, 0.11], N[(N[(N[(N[(N[(-2.48015873015873e-5 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.001388888888888889), $MachinePrecision] * x$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] - 0.041666666666666664), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 0.5), $MachinePrecision], N[(N[(N[Power[t$95$0, -1.0], $MachinePrecision] / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(1.0 / N[Power[N[Power[N[Cos[x$95$m], $MachinePrecision], 2.0], $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[Power[N[Power[N[(x$95$m * x$95$m), $MachinePrecision], -0.5], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 0.110000000000000001

   1. Initial program 34.6%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. pow2 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 67.5

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

   5. Applied rewrites 67.5%

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

   if 0.110000000000000001 < x

   1. Initial program 96.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. sqr-neg-rev N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-neg.f64 99.3

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

   4. Applied rewrites 99.3%

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

      1. lift-neg.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. associate-/l/ N/A

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

      8. sqr-neg-rev N/A

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

      9. pow2 N/A

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

      10. flip-- N/A

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

      11. metadata-eval N/A

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

      12. unpow2 N/A

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

      13. sub-div N/A

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

      14. div-sub N/A

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

   6. Applied rewrites 96.0%

      \[\leadsto \color{blue}{\frac{{\left(\mathsf{fma}\left(\cos x, 1, 1\right)\right)}^{-1}}{x \cdot x} - \frac{\frac{{\cos x}^{2}}{\mathsf{fma}\left(\cos x, 1, 1\right)}}{x \cdot x}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. pow2 N/A

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

      3. metadata-eval N/A

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

      4. pow-pow N/A

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

      5. inv-pow N/A

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

      6. lower-pow.f64 N/A

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

      7. inv-pow N/A

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

      8. metadata-eval N/A

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

      9. pow-pow N/A

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

      10. lower-pow.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 95.9

          \[\leadsto \frac{{\left(\mathsf{fma}\left(\cos x, 1, 1\right)\right)}^{-1}}{x \cdot x} - \frac{\frac{{\cos x}^{2}}{\mathsf{fma}\left(\cos x, 1, 1\right)}}{{\left({\color{blue}{\left(x \cdot x\right)}}^{-0.5}\right)}^{-2}} \]

   8. Applied rewrites 95.9%

      \[\leadsto \frac{{\left(\mathsf{fma}\left(\cos x, 1, 1\right)\right)}^{-1}}{x \cdot x} - \frac{\frac{{\cos x}^{2}}{\mathsf{fma}\left(\cos x, 1, 1\right)}}{\color{blue}{{\left({\left(x \cdot x\right)}^{-0.5}\right)}^{-2}}} \]
   9. Step-by-step derivation

      1. lift-cos.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow1 N/A

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

      4. metadata-eval N/A

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

      5. pow-neg N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-cos.f64 95.7

         \[\leadsto \frac{{\left(\mathsf{fma}\left(\cos x, 1, 1\right)\right)}^{-1}}{x \cdot x} - \frac{\frac{\frac{1}{{\left({\color{blue}{\cos x}}^{2}\right)}^{-1}}}{\mathsf{fma}\left(\cos x, 1, 1\right)}}{{\left({\left(x \cdot x\right)}^{-0.5}\right)}^{-2}} \]

   10. Applied rewrites 95.7%

       \[\leadsto \frac{{\left(\mathsf{fma}\left(\cos x, 1, 1\right)\right)}^{-1}}{x \cdot x} - \frac{\frac{\color{blue}{\frac{1}{{\left({\cos x}^{2}\right)}^{-1}}}}{\mathsf{fma}\left(\cos x, 1, 1\right)}}{{\left({\left(x \cdot x\right)}^{-0.5}\right)}^{-2}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 51.1% accurate, N/A× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \mathsf{fma}\left(\left(\mathsf{fma}\left(-2.48015873015873 \cdot 10^{-5}, x\_m \cdot x\_m, 0.001388888888888889\right) \cdot x\_m\right) \cdot x\_m - 0.041666666666666664, x\_m \cdot x\_m, 0.5\right) \end{array} \]

FPCore

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

C

x_m = fabs(x);
double code(double x_m) {
	return fma((((fma(-2.48015873015873e-5, (x_m * x_m), 0.001388888888888889) * x_m) * x_m) - 0.041666666666666664), (x_m * x_m), 0.5);
}

Julia

x_m = abs(x)
function code(x_m)
	return fma(Float64(Float64(Float64(fma(-2.48015873015873e-5, Float64(x_m * x_m), 0.001388888888888889) * x_m) * x_m) - 0.041666666666666664), Float64(x_m * x_m), 0.5)
end

Wolfram

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

Derivation

1. Initial program 50.3%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. lower--.f64 N/A

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

   5. *-commutative N/A

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

   6. pow2 N/A

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

   7. associate-*r* N/A

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

   8. lower-*.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. +-commutative N/A

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

   11. lower-fma.f64 N/A

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

   12. pow2 N/A

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

   13. lift-*.f64 N/A

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

   14. pow2 N/A

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

   15. lift-*.f64 50.8

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

5. Applied rewrites 50.8%

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

Reproduce

herbie shell --seed 2025065 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))