cos2 (problem 3.4.1)

Percentage Accurate: 50.6% → 99.6%

Time: 9.9s

Alternatives: 3

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: 50.6% 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

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.09:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(-2.48015873015873 \cdot 10^{-5}, x\_m \cdot x\_m, 0.001388888888888889\right) \cdot x\_m\right) \cdot x\_m - 0.041666666666666664, x\_m \cdot x\_m, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x\_m}{-x\_m}}{-x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.09)
   (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 <= 0.09) {
		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 <= 0.09)
		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)) / Float64(-x_m)) / Float64(-x_m));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.09], 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 < 0.089999999999999997

   1. Initial program 2.9%

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

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{40320} \cdot {x}^{2}\right) - \frac{1}{24}\right)} \]
   3. 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 100.0

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

   4. Applied rewrites 100.0%

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

   if 0.089999999999999997 < x

   1. Initial program 98.2%

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

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

   3. Applied rewrites 99.3%

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

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

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.122:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(-2.48015873015873 \cdot 10^{-5}, x\_m \cdot x\_m, 0.001388888888888889\right) \cdot x\_m\right) \cdot x\_m - 0.041666666666666664, x\_m \cdot x\_m, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{x\_m}^{-1}}{x\_m} - \frac{\frac{\cos x\_m}{x\_m}}{x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.122)
   (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) x_m) (/ (/ (cos x_m) x_m) x_m))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.122) {
		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) / x_m) - ((cos(x_m) / x_m) / x_m);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.122)
		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 ^ -1.0) / x_m) - 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, 0.122], 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[x$95$m, -1.0], $MachinePrecision] / x$95$m), $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 < 0.122

   1. Initial program 2.9%

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

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{40320} \cdot {x}^{2}\right) - \frac{1}{24}\right)} \]
   3. 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 100.0

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

   4. Applied rewrites 100.0%

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

   if 0.122 < x

   1. Initial program 98.2%

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

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

   3. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\frac{\frac{1 - \cos x}{-x}}{-x}} \]
   4. 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. distribute-frac-neg2 N/A

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

      8. frac-2neg-rev N/A

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

      9. div-sub N/A

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

      10. div-sub N/A

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

      11. lower--.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. inv-pow N/A

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

      14. lower-pow.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lift-cos.f64 99.1

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

   5. Applied rewrites 99.1%

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

Alternative 3: 50.8% 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 50.6%

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

2. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{1}{2} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{40320} \cdot {x}^{2}\right) - \frac{1}{24}\right)} \]
3. 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

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

4. Applied rewrites 50.8%

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

Reproduce

herbie shell --seed 2025093 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))