cos2 (problem 3.4.1)

Percentage Accurate: 51.2% → 99.6%

Time: 3.8s

Alternatives: 7

Speedup: 120.0×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 51.2% 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, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.1)
		tmp = fma(fma(fma(-2.48015873015873e-5, Float64(x_m * x_m), 0.001388888888888889), Float64(x_m * x_m), -0.041666666666666664), Float64(x_m * x_m), 0.5);
	else
		tmp = Float64(Float64(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, 0.1], N[(N[(N[(-2.48015873015873e-5 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.001388888888888889), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + -0.041666666666666664), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 0.5), $MachinePrecision], N[(N[(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 < 0.10000000000000001

   1. Initial program 36.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. flip-- N/A

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

      6. pow2 N/A

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

      7. associate-/l/ N/A

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

      8. lower-/.f64 N/A

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

      9. metadata-eval N/A

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

      10. 1-sub-cos N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sin.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. lift-cos.f64 N/A

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

      18. pow2 N/A

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

      19. lift-*.f64 70.1

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

   4. Applied rewrites 70.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. *-commutative N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-sin.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 N/A

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

      17. +-commutative N/A

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

      18. hang-0p-tan N/A

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

      19. lower-tan.f64 N/A

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

      20. lower-/.f64 70.4

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

   6. Applied rewrites 70.4%

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

   7. Taylor expanded in x around 0

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

      1. lift-sin.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. pow2 N/A

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

      4. associate-*l/ N/A

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

      5. pow2 N/A

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

      6. hang-0p-tan-rev N/A

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

      7. associate-*r/ N/A

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

      8. 1-sub-cos N/A

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

      9. metadata-eval N/A

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

      10. flip-- N/A

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

      11. associate-/l/ N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

   9. Applied rewrites 66.1%

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

   10. Taylor expanded in x around 0

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

       1. div-sub N/A

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

       2. sub-div N/A

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

       3. associate-/r* N/A

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

       4. +-commutative N/A

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

       5. *-commutative N/A

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

       6. lower-fma.f64 N/A

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

   12. Applied rewrites 66.0%

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

   if 0.10000000000000001 < x

   1. Initial program 99.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift--.f64 99.5

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

   4. Applied rewrites 99.5%

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

4. Final simplification 73.7%

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

Alternative 2: 99.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 0.10000000000000001

   1. Initial program 36.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. flip-- N/A

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

      6. pow2 N/A

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

      7. associate-/l/ N/A

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

      8. lower-/.f64 N/A

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

      9. metadata-eval N/A

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

      10. 1-sub-cos N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sin.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. lift-cos.f64 N/A

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

      18. pow2 N/A

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

      19. lift-*.f64 70.1

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

   4. Applied rewrites 70.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. *-commutative N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-sin.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 N/A

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

      17. +-commutative N/A

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

      18. hang-0p-tan N/A

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

      19. lower-tan.f64 N/A

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

      20. lower-/.f64 70.4

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

   6. Applied rewrites 70.4%

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

   7. Taylor expanded in x around 0

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

      1. lift-sin.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. pow2 N/A

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

      4. associate-*l/ N/A

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

      5. pow2 N/A

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

      6. hang-0p-tan-rev N/A

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

      7. associate-*r/ N/A

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

      8. 1-sub-cos N/A

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

      9. metadata-eval N/A

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

      10. flip-- N/A

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

      11. associate-/l/ N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

   9. Applied rewrites 66.1%

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

   10. Taylor expanded in x around 0

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

       1. div-sub N/A

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

       2. sub-div N/A

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

       3. associate-/r* N/A

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

       4. +-commutative N/A

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

       5. *-commutative N/A

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

       6. lower-fma.f64 N/A

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

   12. Applied rewrites 66.0%

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

   if 0.10000000000000001 < x

   1. Initial program 99.5%

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

4. Final simplification 73.7%

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

Alternative 3: 76.4% accurate, 2.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 2.45 \cdot 10^{+33}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x\_m \cdot x\_m, 0.001388888888888889, -0.041666666666666664\right), x\_m \cdot x\_m, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x\_m} \cdot \frac{-1}{x\_m} - \frac{1}{x\_m \cdot x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 2.45e+33)
   (fma
    (fma (* x_m x_m) 0.001388888888888889 -0.041666666666666664)
    (* x_m x_m)
    0.5)
   (- (* (/ -1.0 x_m) (/ -1.0 x_m)) (/ 1.0 (* x_m x_m)))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 2.45e+33) {
		tmp = fma(fma((x_m * x_m), 0.001388888888888889, -0.041666666666666664), (x_m * x_m), 0.5);
	} else {
		tmp = ((-1.0 / x_m) * (-1.0 / 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 <= 2.45e+33)
		tmp = fma(fma(Float64(x_m * x_m), 0.001388888888888889, -0.041666666666666664), Float64(x_m * x_m), 0.5);
	else
		tmp = Float64(Float64(Float64(-1.0 / x_m) * Float64(-1.0 / 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, 2.45e+33], N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.001388888888888889 + -0.041666666666666664), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 0.5), $MachinePrecision], N[(N[(N[(-1.0 / x$95$m), $MachinePrecision] * N[(-1.0 / x$95$m), $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 < 2.45000000000000007e33

   1. Initial program 37.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. flip-- N/A

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

      6. pow2 N/A

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

      7. associate-/l/ N/A

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

      8. lower-/.f64 N/A

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

      9. metadata-eval N/A

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

      10. 1-sub-cos N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sin.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. lift-cos.f64 N/A

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

      18. pow2 N/A

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

      19. lift-*.f64 70.7

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

   4. Applied rewrites 70.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. *-commutative N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-sin.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 N/A

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

      17. +-commutative N/A

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

      18. hang-0p-tan N/A

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

      19. lower-tan.f64 N/A

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

      20. lower-/.f64 71.0

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

   6. Applied rewrites 71.0%

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

   7. Taylor expanded in x around 0

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

      1. lift-sin.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. pow2 N/A

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

      4. associate-*l/ N/A

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

      5. pow2 N/A

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

      6. hang-0p-tan-rev N/A

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

      7. associate-*r/ N/A

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

      8. 1-sub-cos N/A

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

      9. metadata-eval N/A

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

      10. flip-- N/A

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

      11. associate-/l/ N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

   9. Applied rewrites 64.9%

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

       1. lift--.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. pow2 N/A

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

       5. metadata-eval N/A

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

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

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

       7. *-commutative N/A

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

       8. metadata-eval N/A

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

       9. metadata-eval N/A

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

       10. lower-fma.f64 N/A

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

       11. pow2 N/A

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

       12. lift-*.f64 64.9

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

   11. Applied rewrites 64.9%

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

   if 2.45000000000000007e33 < x

   1. Initial program 99.5%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 55.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift--.f64 N/A

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

      5. div-sub N/A

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

      6. lower--.f64 N/A

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

      7. pow-flip N/A

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

      8. lower-pow.f64 N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 55.8

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

   7. Applied rewrites 55.8%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-flip N/A

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

      4. metadata-eval N/A

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

      5. pow2 N/A

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

      6. sqr-neg-rev N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. metadata-eval N/A

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

      10. frac-2neg-rev N/A

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

      11. lower-/.f64 N/A

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

      12. metadata-eval N/A

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

      13. frac-2neg-rev N/A

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

      14. lower-/.f64 55.7

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

   9. Applied rewrites 55.7%

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

Alternative 4: 76.4% accurate, 2.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 2.45 \cdot 10^{+33}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x\_m \cdot x\_m, 0.001388888888888889, -0.041666666666666664\right), x\_m \cdot x\_m, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;-\left(\frac{\frac{-1}{x\_m}}{x\_m} + \frac{1}{x\_m \cdot x\_m}\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 2.45e+33)
   (fma
    (fma (* x_m x_m) 0.001388888888888889 -0.041666666666666664)
    (* x_m x_m)
    0.5)
   (- (+ (/ (/ -1.0 x_m) x_m) (/ 1.0 (* x_m x_m))))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 2.45e+33) {
		tmp = fma(fma((x_m * x_m), 0.001388888888888889, -0.041666666666666664), (x_m * x_m), 0.5);
	} else {
		tmp = -(((-1.0 / x_m) / 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 <= 2.45e+33)
		tmp = fma(fma(Float64(x_m * x_m), 0.001388888888888889, -0.041666666666666664), Float64(x_m * x_m), 0.5);
	else
		tmp = Float64(-Float64(Float64(Float64(-1.0 / x_m) / 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, 2.45e+33], N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.001388888888888889 + -0.041666666666666664), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 0.5), $MachinePrecision], (-N[(N[(N[(-1.0 / x$95$m), $MachinePrecision] / x$95$m), $MachinePrecision] + N[(1.0 / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if x < 2.45000000000000007e33

   1. Initial program 37.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. flip-- N/A

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

      6. pow2 N/A

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

      7. associate-/l/ N/A

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

      8. lower-/.f64 N/A

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

      9. metadata-eval N/A

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

      10. 1-sub-cos N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sin.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. lift-cos.f64 N/A

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

      18. pow2 N/A

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

      19. lift-*.f64 70.7

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

   4. Applied rewrites 70.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. *-commutative N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-sin.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 N/A

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

      17. +-commutative N/A

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

      18. hang-0p-tan N/A

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

      19. lower-tan.f64 N/A

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

      20. lower-/.f64 71.0

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

   6. Applied rewrites 71.0%

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

   7. Taylor expanded in x around 0

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

      1. lift-sin.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. pow2 N/A

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

      4. associate-*l/ N/A

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

      5. pow2 N/A

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

      6. hang-0p-tan-rev N/A

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

      7. associate-*r/ N/A

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

      8. 1-sub-cos N/A

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

      9. metadata-eval N/A

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

      10. flip-- N/A

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

      11. associate-/l/ N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

   9. Applied rewrites 64.9%

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

       1. lift--.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. pow2 N/A

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

       5. metadata-eval N/A

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

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

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

       7. *-commutative N/A

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

       8. metadata-eval N/A

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

       9. metadata-eval N/A

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

       10. lower-fma.f64 N/A

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

       11. pow2 N/A

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

       12. lift-*.f64 64.9

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

   11. Applied rewrites 64.9%

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

   if 2.45000000000000007e33 < x

   1. Initial program 99.5%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 55.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift--.f64 N/A

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

      5. div-sub N/A

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

      6. lower--.f64 N/A

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

      7. pow-flip N/A

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

      8. lower-pow.f64 N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 55.8

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

   7. Applied rewrites 55.8%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-flip N/A

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

      4. pow2 N/A

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

      5. associate-/r* N/A

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

      6. frac-2neg N/A

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

      7. distribute-frac-neg2 N/A

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

      8. lower-/.f64 N/A

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

      9. metadata-eval N/A

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

      10. frac-2neg-rev N/A

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

      11. lower-/.f64 N/A

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

      12. lift-neg.f64 55.6

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

   9. Applied rewrites 55.6%

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

4. Final simplification 62.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.45 \cdot 10^{+33}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.001388888888888889, -0.041666666666666664\right), x \cdot x, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;-\left(\frac{\frac{-1}{x}}{x} + \frac{1}{x \cdot x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.3% accurate, 4.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 6.6 \cdot 10^{+38}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x\_m \cdot x\_m, 0.001388888888888889, -0.041666666666666664\right), x\_m \cdot x\_m, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - 1}{x\_m \cdot x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 6.6e+38)
   (fma
    (fma (* x_m x_m) 0.001388888888888889 -0.041666666666666664)
    (* x_m x_m)
    0.5)
   (/ (- 1.0 1.0) (* x_m x_m))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 6.6e+38) {
		tmp = fma(fma((x_m * x_m), 0.001388888888888889, -0.041666666666666664), (x_m * x_m), 0.5);
	} else {
		tmp = (1.0 - 1.0) / (x_m * x_m);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 6.6e+38)
		tmp = fma(fma(Float64(x_m * x_m), 0.001388888888888889, -0.041666666666666664), Float64(x_m * x_m), 0.5);
	else
		tmp = Float64(Float64(1.0 - 1.0) / Float64(x_m * x_m));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 6.6e+38], N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.001388888888888889 + -0.041666666666666664), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 0.5), $MachinePrecision], N[(N[(1.0 - 1.0), $MachinePrecision] / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 6.5999999999999998e38

   1. Initial program 37.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. flip-- N/A

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

      6. pow2 N/A

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

      7. associate-/l/ N/A

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

      8. lower-/.f64 N/A

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

      9. metadata-eval N/A

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

      10. 1-sub-cos N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sin.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. lift-cos.f64 N/A

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

      18. pow2 N/A

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

      19. lift-*.f64 70.7

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

   4. Applied rewrites 70.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. *-commutative N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-sin.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 N/A

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

      17. +-commutative N/A

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

      18. hang-0p-tan N/A

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

      19. lower-tan.f64 N/A

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

      20. lower-/.f64 71.0

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

   6. Applied rewrites 71.0%

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

   7. Taylor expanded in x around 0

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

      1. lift-sin.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. pow2 N/A

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

      4. associate-*l/ N/A

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

      5. pow2 N/A

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

      6. hang-0p-tan-rev N/A

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

      7. associate-*r/ N/A

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

      8. 1-sub-cos N/A

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

      9. metadata-eval N/A

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

      10. flip-- N/A

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

      11. associate-/l/ N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

   9. Applied rewrites 64.9%

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

       1. lift--.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. pow2 N/A

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

       5. metadata-eval N/A

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

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

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

       7. *-commutative N/A

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

       8. metadata-eval N/A

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

       9. metadata-eval N/A

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

       10. lower-fma.f64 N/A

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

       11. pow2 N/A

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

       12. lift-*.f64 64.9

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

   11. Applied rewrites 64.9%

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

   if 6.5999999999999998e38 < x

   1. Initial program 99.5%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 55.5%

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

Alternative 6: 76.0% accurate, 4.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 3.5:\\ \;\;\;\;\mathsf{fma}\left(x\_m \cdot x\_m, -0.041666666666666664, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - 1}{x\_m \cdot x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 3.5)
   (fma (* x_m x_m) -0.041666666666666664 0.5)
   (/ (- 1.0 1.0) (* x_m x_m))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 3.5) {
		tmp = fma((x_m * x_m), -0.041666666666666664, 0.5);
	} else {
		tmp = (1.0 - 1.0) / (x_m * x_m);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 3.5)
		tmp = fma(Float64(x_m * x_m), -0.041666666666666664, 0.5);
	else
		tmp = Float64(Float64(1.0 - 1.0) / Float64(x_m * x_m));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 3.5], N[(N[(x$95$m * x$95$m), $MachinePrecision] * -0.041666666666666664 + 0.5), $MachinePrecision], N[(N[(1.0 - 1.0), $MachinePrecision] / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.5

   1. Initial program 36.1%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 65.8

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

   5. Applied rewrites 65.8%

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

   if 3.5 < x

   1. Initial program 99.5%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 52.0%

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

4. Final simplification 62.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.5:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, -0.041666666666666664, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - 1}{x \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 51.2% accurate, 120.0× speedup

Math

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

FPCore

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

C

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

Fortran

x_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.7%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 51.8%

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

6. Final simplification 51.8%

   \[\leadsto 0.5 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2025064 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))