mixedcos

Percentage Accurate: 66.3% → 97.6%

Time: 8.0s

Alternatives: 13

Speedup: 6.2×

• 30.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))

C

double code(double x, double c, double s) {
	return cos((2.0 * x)) / (pow(c, 2.0) * ((x * pow(s, 2.0)) * 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, c, s)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    code = cos((2.0d0 * x)) / ((c ** 2.0d0) * ((x * (s ** 2.0d0)) * x))
end function

Java

public static double code(double x, double c, double s) {
	return Math.cos((2.0 * x)) / (Math.pow(c, 2.0) * ((x * Math.pow(s, 2.0)) * x));
}

Python

def code(x, c, s):
	return math.cos((2.0 * x)) / (math.pow(c, 2.0) * ((x * math.pow(s, 2.0)) * x))

Julia

function code(x, c, s)
	return Float64(cos(Float64(2.0 * x)) / Float64((c ^ 2.0) * Float64(Float64(x * (s ^ 2.0)) * x)))
end

MATLAB

function tmp = code(x, c, s)
	tmp = cos((2.0 * x)) / ((c ^ 2.0) * ((x * (s ^ 2.0)) * x));
end

Wolfram

code[x_, c_, s_] := N[(N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] / N[(N[Power[c, 2.0], $MachinePrecision] * N[(N[(x * N[Power[s, 2.0], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 66.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))

C

double code(double x, double c, double s) {
	return cos((2.0 * x)) / (pow(c, 2.0) * ((x * pow(s, 2.0)) * 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, c, s)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    code = cos((2.0d0 * x)) / ((c ** 2.0d0) * ((x * (s ** 2.0d0)) * x))
end function

Java

public static double code(double x, double c, double s) {
	return Math.cos((2.0 * x)) / (Math.pow(c, 2.0) * ((x * Math.pow(s, 2.0)) * x));
}

Python

def code(x, c, s):
	return math.cos((2.0 * x)) / (math.pow(c, 2.0) * ((x * math.pow(s, 2.0)) * x))

Julia

function code(x, c, s)
	return Float64(cos(Float64(2.0 * x)) / Float64((c ^ 2.0) * Float64(Float64(x * (s ^ 2.0)) * x)))
end

MATLAB

function tmp = code(x, c, s)
	tmp = cos((2.0 * x)) / ((c ^ 2.0) * ((x * (s ^ 2.0)) * x));
end

Wolfram

code[x_, c_, s_] := N[(N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] / N[(N[Power[c, 2.0], $MachinePrecision] * N[(N[(x * N[Power[s, 2.0], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 97.6% accurate, 2.2× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ x_m = \left|x\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \left(s\_m \cdot x\_m\right) \cdot c\_m\\ \mathbf{if}\;x\_m \leq 5.6 \cdot 10^{-11}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2, x\_m \cdot x\_m, 1\right)}{t\_0 \cdot t\_0}\\ \mathbf{elif}\;x\_m \leq 10^{+114}:\\ \;\;\;\;\frac{\cos \left(x\_m + x\_m\right)}{\left(\left(\left(\left(x\_m \cdot x\_m\right) \cdot c\_m\right) \cdot s\_m\right) \cdot c\_m\right) \cdot s\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\_m\right)}{\left(x\_m \cdot c\_m\right) \cdot \left(\left(s\_m \cdot s\_m\right) \cdot \left(c\_m \cdot x\_m\right)\right)}\\ \end{array} \end{array} \]

FPCore

s_m = (fabs.f64 s)
c_m = (fabs.f64 c)
x_m = (fabs.f64 x)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (let* ((t_0 (* (* s_m x_m) c_m)))
   (if (<= x_m 5.6e-11)
     (/ (fma -2.0 (* x_m x_m) 1.0) (* t_0 t_0))
     (if (<= x_m 1e+114)
       (/ (cos (+ x_m x_m)) (* (* (* (* (* x_m x_m) c_m) s_m) c_m) s_m))
       (/ (cos (* 2.0 x_m)) (* (* x_m c_m) (* (* s_m s_m) (* c_m x_m))))))))

C

s_m = fabs(s);
c_m = fabs(c);
x_m = fabs(x);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double t_0 = (s_m * x_m) * c_m;
	double tmp;
	if (x_m <= 5.6e-11) {
		tmp = fma(-2.0, (x_m * x_m), 1.0) / (t_0 * t_0);
	} else if (x_m <= 1e+114) {
		tmp = cos((x_m + x_m)) / (((((x_m * x_m) * c_m) * s_m) * c_m) * s_m);
	} else {
		tmp = cos((2.0 * x_m)) / ((x_m * c_m) * ((s_m * s_m) * (c_m * x_m)));
	}
	return tmp;
}

Julia

s_m = abs(s)
c_m = abs(c)
x_m = abs(x)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	t_0 = Float64(Float64(s_m * x_m) * c_m)
	tmp = 0.0
	if (x_m <= 5.6e-11)
		tmp = Float64(fma(-2.0, Float64(x_m * x_m), 1.0) / Float64(t_0 * t_0));
	elseif (x_m <= 1e+114)
		tmp = Float64(cos(Float64(x_m + x_m)) / Float64(Float64(Float64(Float64(Float64(x_m * x_m) * c_m) * s_m) * c_m) * s_m));
	else
		tmp = Float64(cos(Float64(2.0 * x_m)) / Float64(Float64(x_m * c_m) * Float64(Float64(s_m * s_m) * Float64(c_m * x_m))));
	end
	return tmp
end

Wolfram

s_m = N[Abs[s], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
x_m = N[Abs[x], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[(N[(s$95$m * x$95$m), $MachinePrecision] * c$95$m), $MachinePrecision]}, If[LessEqual[x$95$m, 5.6e-11], N[(N[(-2.0 * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$95$m, 1e+114], N[(N[Cos[N[(x$95$m + x$95$m), $MachinePrecision]], $MachinePrecision] / N[(N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * c$95$m), $MachinePrecision] * s$95$m), $MachinePrecision] * c$95$m), $MachinePrecision] * s$95$m), $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(2.0 * x$95$m), $MachinePrecision]], $MachinePrecision] / N[(N[(x$95$m * c$95$m), $MachinePrecision] * N[(N[(s$95$m * s$95$m), $MachinePrecision] * N[(c$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < 5.6e-11

   1. Initial program 67.0%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. associate-*r/ N/A

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

      2. div-add-rev N/A

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

      3. +-commutative N/A

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

      4. associate-/l/ N/A

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

      5. associate-*r* N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

      14. *-commutative N/A

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

      15. unpow2 N/A

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

      16. associate-*l* N/A

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

      17. associate-*r* N/A

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

   5. Applied rewrites 59.8%

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

   7. Applied rewrites 76.2%

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

   if 5.6e-11 < x < 1e114

   1. Initial program 65.8%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. pow-prod-down N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-*.f64 88.8

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

   4. Applied rewrites 88.8%

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left({\left(c \cdot s\right)}^{2} \cdot x\right) \cdot x}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(x \cdot \color{blue}{\left(\left(c \cdot s\right) \cdot \left(c \cdot s\right)\right)}\right) \cdot x} \]

      5. sqr-neg-rev N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(x \cdot \color{blue}{\left(\left(\mathsf{neg}\left(c \cdot s\right)\right) \cdot \left(\mathsf{neg}\left(c \cdot s\right)\right)\right)}\right) \cdot x} \]

      6. associate-*r* N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(\left(x \cdot \left(\mathsf{neg}\left(c \cdot s\right)\right)\right) \cdot \left(\mathsf{neg}\left(c \cdot s\right)\right)\right)} \cdot x} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(\left(x \cdot \left(\mathsf{neg}\left(c \cdot s\right)\right)\right) \cdot \left(\mathsf{neg}\left(c \cdot s\right)\right)\right)} \cdot x} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(\color{blue}{\left(x \cdot \left(\mathsf{neg}\left(c \cdot s\right)\right)\right)} \cdot \left(\mathsf{neg}\left(c \cdot s\right)\right)\right) \cdot x} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(\left(x \cdot \left(\mathsf{neg}\left(\color{blue}{c \cdot s}\right)\right)\right) \cdot \left(\mathsf{neg}\left(c \cdot s\right)\right)\right) \cdot x} \]

      10. distribute-lft-neg-in N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(\left(x \cdot \color{blue}{\left(\left(\mathsf{neg}\left(c\right)\right) \cdot s\right)}\right) \cdot \left(\mathsf{neg}\left(c \cdot s\right)\right)\right) \cdot x} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(\left(x \cdot \color{blue}{\left(\left(\mathsf{neg}\left(c\right)\right) \cdot s\right)}\right) \cdot \left(\mathsf{neg}\left(c \cdot s\right)\right)\right) \cdot x} \]

      12. lower-neg.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. distribute-lft-neg-in N/A

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

      15. lower-*.f64 N/A

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

      16. lower-neg.f64 95.9

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

   6. Applied rewrites 95.9%

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(\left(x \cdot \left(\left(-c\right) \cdot s\right)\right) \cdot \left(\left(-c\right) \cdot s\right)\right)} \cdot x} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 95.9

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

   8. Applied rewrites 95.9%

      \[\leadsto \frac{\cos \color{blue}{\left(x + x\right)}}{\left(\left(x \cdot \left(\left(-c\right) \cdot s\right)\right) \cdot \left(\left(-c\right) \cdot s\right)\right) \cdot x} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. swap-sqr N/A

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

      7. lift-*.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. distribute-lft-neg-out N/A

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

      10. lift-*.f64 N/A

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

      11. lift-neg.f64 N/A

          \[\leadsto \frac{\cos \left(x + x\right)}{\left(x \cdot x\right) \cdot \left(\left(\mathsf{neg}\left(c \cdot s\right)\right) \cdot \left(\color{blue}{\left(\mathsf{neg}\left(c\right)\right)} \cdot s\right)\right)} \]

      12. distribute-lft-neg-out N/A

          \[\leadsto \frac{\cos \left(x + x\right)}{\left(x \cdot x\right) \cdot \left(\left(\mathsf{neg}\left(c \cdot s\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(c \cdot s\right)\right)}\right)} \]

      13. sqr-neg-rev N/A

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

      14. associate-*r* N/A

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

      15. associate-*l* N/A

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

      16. associate-*l* N/A

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

      17. lift-*.f64 N/A

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

      18. associate-*r* N/A

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

      19. lower-*.f64 N/A

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

   10. Applied rewrites 88.9%

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

   if 1e114 < x

   1. Initial program 66.4%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 81.4

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 81.4

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

      16. lift-pow.f64 N/A

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

      17. unpow2 N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(x \cdot c\right) \cdot \left(\left(\color{blue}{\left(s \cdot s\right)} \cdot x\right) \cdot c\right)} \]

      18. lower-*.f64 81.4

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(x \cdot c\right) \cdot \left(\left(\color{blue}{\left(s \cdot s\right)} \cdot x\right) \cdot c\right)} \]

   4. Applied rewrites 81.4%

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(x \cdot c\right) \cdot \left(\left(\left(s \cdot s\right) \cdot x\right) \cdot c\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(x \cdot c\right) \cdot \color{blue}{\left(\left(\left(s \cdot s\right) \cdot x\right) \cdot c\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(x \cdot c\right) \cdot \left(\color{blue}{\left(\left(s \cdot s\right) \cdot x\right)} \cdot c\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(x \cdot c\right) \cdot \color{blue}{\left(\left(s \cdot s\right) \cdot \left(x \cdot c\right)\right)}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(x \cdot c\right) \cdot \left(\left(s \cdot s\right) \cdot \color{blue}{\left(x \cdot c\right)}\right)} \]

      5. lower-*.f64 83.7

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(x \cdot c\right) \cdot \color{blue}{\left(\left(s \cdot s\right) \cdot \left(x \cdot c\right)\right)}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(x \cdot c\right) \cdot \left(\left(s \cdot s\right) \cdot \color{blue}{\left(x \cdot c\right)}\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(x \cdot c\right) \cdot \left(\left(s \cdot s\right) \cdot \color{blue}{\left(c \cdot x\right)}\right)} \]

      8. lower-*.f64 83.7

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(x \cdot c\right) \cdot \left(\left(s \cdot s\right) \cdot \color{blue}{\left(c \cdot x\right)}\right)} \]

   6. Applied rewrites 83.7%

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

Alternative 2: 82.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ x_m = \left|x\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{\cos \left(2 \cdot x\_m\right)}{{c\_m}^{2} \cdot \left(\left(x\_m \cdot {s\_m}^{2}\right) \cdot x\_m\right)} \leq -1 \cdot 10^{-99}:\\ \;\;\;\;\frac{\frac{\frac{1}{x\_m}}{x\_m} - 2}{\left(\left(s\_m \cdot c\_m\right) \cdot c\_m\right) \cdot s\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\left(s\_m \cdot x\_m\right) \cdot c\_m\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

s_m = (fabs.f64 s)
c_m = (fabs.f64 c)
x_m = (fabs.f64 x)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (if (<=
      (/ (cos (* 2.0 x_m)) (* (pow c_m 2.0) (* (* x_m (pow s_m 2.0)) x_m)))
      -1e-99)
   (/ (- (/ (/ 1.0 x_m) x_m) 2.0) (* (* (* s_m c_m) c_m) s_m))
   (/ 1.0 (pow (* (* s_m x_m) c_m) 2.0))))

C

s_m = fabs(s);
c_m = fabs(c);
x_m = fabs(x);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double tmp;
	if ((cos((2.0 * x_m)) / (pow(c_m, 2.0) * ((x_m * pow(s_m, 2.0)) * x_m))) <= -1e-99) {
		tmp = (((1.0 / x_m) / x_m) - 2.0) / (((s_m * c_m) * c_m) * s_m);
	} else {
		tmp = 1.0 / pow(((s_m * x_m) * c_m), 2.0);
	}
	return tmp;
}

Fortran

s_m =     private
c_m =     private
x_m =     private
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
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, c_m, s_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: tmp
    if ((cos((2.0d0 * x_m)) / ((c_m ** 2.0d0) * ((x_m * (s_m ** 2.0d0)) * x_m))) <= (-1d-99)) then
        tmp = (((1.0d0 / x_m) / x_m) - 2.0d0) / (((s_m * c_m) * c_m) * s_m)
    else
        tmp = 1.0d0 / (((s_m * x_m) * c_m) ** 2.0d0)
    end if
    code = tmp
end function

Java

s_m = Math.abs(s);
c_m = Math.abs(c);
x_m = Math.abs(x);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	double tmp;
	if ((Math.cos((2.0 * x_m)) / (Math.pow(c_m, 2.0) * ((x_m * Math.pow(s_m, 2.0)) * x_m))) <= -1e-99) {
		tmp = (((1.0 / x_m) / x_m) - 2.0) / (((s_m * c_m) * c_m) * s_m);
	} else {
		tmp = 1.0 / Math.pow(((s_m * x_m) * c_m), 2.0);
	}
	return tmp;
}

Python

s_m = math.fabs(s)
c_m = math.fabs(c)
x_m = math.fabs(x)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	tmp = 0
	if (math.cos((2.0 * x_m)) / (math.pow(c_m, 2.0) * ((x_m * math.pow(s_m, 2.0)) * x_m))) <= -1e-99:
		tmp = (((1.0 / x_m) / x_m) - 2.0) / (((s_m * c_m) * c_m) * s_m)
	else:
		tmp = 1.0 / math.pow(((s_m * x_m) * c_m), 2.0)
	return tmp

Julia

s_m = abs(s)
c_m = abs(c)
x_m = abs(x)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	tmp = 0.0
	if (Float64(cos(Float64(2.0 * x_m)) / Float64((c_m ^ 2.0) * Float64(Float64(x_m * (s_m ^ 2.0)) * x_m))) <= -1e-99)
		tmp = Float64(Float64(Float64(Float64(1.0 / x_m) / x_m) - 2.0) / Float64(Float64(Float64(s_m * c_m) * c_m) * s_m));
	else
		tmp = Float64(1.0 / (Float64(Float64(s_m * x_m) * c_m) ^ 2.0));
	end
	return tmp
end

MATLAB

s_m = abs(s);
c_m = abs(c);
x_m = abs(x);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp_2 = code(x_m, c_m, s_m)
	tmp = 0.0;
	if ((cos((2.0 * x_m)) / ((c_m ^ 2.0) * ((x_m * (s_m ^ 2.0)) * x_m))) <= -1e-99)
		tmp = (((1.0 / x_m) / x_m) - 2.0) / (((s_m * c_m) * c_m) * s_m);
	else
		tmp = 1.0 / (((s_m * x_m) * c_m) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

s_m = N[Abs[s], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
x_m = N[Abs[x], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := If[LessEqual[N[(N[Cos[N[(2.0 * x$95$m), $MachinePrecision]], $MachinePrecision] / N[(N[Power[c$95$m, 2.0], $MachinePrecision] * N[(N[(x$95$m * N[Power[s$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1e-99], N[(N[(N[(N[(1.0 / x$95$m), $MachinePrecision] / x$95$m), $MachinePrecision] - 2.0), $MachinePrecision] / N[(N[(N[(s$95$m * c$95$m), $MachinePrecision] * c$95$m), $MachinePrecision] * s$95$m), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Power[N[(N[(s$95$m * x$95$m), $MachinePrecision] * c$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x))) < -1e-99

   1. Initial program 74.4%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

   4. Applied rewrites 74.3%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 55.7%

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

   if -1e-99 < (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x)))

   1. Initial program 66.3%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

   4. Applied rewrites 86.8%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 80.1%

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

   9. Applied rewrites 85.8%

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

Alternative 3: 79.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ x_m = \left|x\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{\cos \left(2 \cdot x\_m\right)}{{c\_m}^{2} \cdot \left(\left(x\_m \cdot {s\_m}^{2}\right) \cdot x\_m\right)} \leq -1 \cdot 10^{-99}:\\ \;\;\;\;\frac{-2}{\left(s\_m \cdot c\_m\right) \cdot \left(s\_m \cdot c\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x\_m}}{\left(\left(s\_m \cdot c\_m\right) \cdot x\_m\right) \cdot \left(s\_m \cdot c\_m\right)}\\ \end{array} \end{array} \]

FPCore

s_m = (fabs.f64 s)
c_m = (fabs.f64 c)
x_m = (fabs.f64 x)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (if (<=
      (/ (cos (* 2.0 x_m)) (* (pow c_m 2.0) (* (* x_m (pow s_m 2.0)) x_m)))
      -1e-99)
   (/ (- 2.0) (* (* s_m c_m) (* s_m c_m)))
   (/ (/ 1.0 x_m) (* (* (* s_m c_m) x_m) (* s_m c_m)))))

C

s_m = fabs(s);
c_m = fabs(c);
x_m = fabs(x);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double tmp;
	if ((cos((2.0 * x_m)) / (pow(c_m, 2.0) * ((x_m * pow(s_m, 2.0)) * x_m))) <= -1e-99) {
		tmp = -2.0 / ((s_m * c_m) * (s_m * c_m));
	} else {
		tmp = (1.0 / x_m) / (((s_m * c_m) * x_m) * (s_m * c_m));
	}
	return tmp;
}

Fortran

s_m =     private
c_m =     private
x_m =     private
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
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, c_m, s_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: tmp
    if ((cos((2.0d0 * x_m)) / ((c_m ** 2.0d0) * ((x_m * (s_m ** 2.0d0)) * x_m))) <= (-1d-99)) then
        tmp = -2.0d0 / ((s_m * c_m) * (s_m * c_m))
    else
        tmp = (1.0d0 / x_m) / (((s_m * c_m) * x_m) * (s_m * c_m))
    end if
    code = tmp
end function

Java

s_m = Math.abs(s);
c_m = Math.abs(c);
x_m = Math.abs(x);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	double tmp;
	if ((Math.cos((2.0 * x_m)) / (Math.pow(c_m, 2.0) * ((x_m * Math.pow(s_m, 2.0)) * x_m))) <= -1e-99) {
		tmp = -2.0 / ((s_m * c_m) * (s_m * c_m));
	} else {
		tmp = (1.0 / x_m) / (((s_m * c_m) * x_m) * (s_m * c_m));
	}
	return tmp;
}

Python

s_m = math.fabs(s)
c_m = math.fabs(c)
x_m = math.fabs(x)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	tmp = 0
	if (math.cos((2.0 * x_m)) / (math.pow(c_m, 2.0) * ((x_m * math.pow(s_m, 2.0)) * x_m))) <= -1e-99:
		tmp = -2.0 / ((s_m * c_m) * (s_m * c_m))
	else:
		tmp = (1.0 / x_m) / (((s_m * c_m) * x_m) * (s_m * c_m))
	return tmp

Julia

s_m = abs(s)
c_m = abs(c)
x_m = abs(x)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	tmp = 0.0
	if (Float64(cos(Float64(2.0 * x_m)) / Float64((c_m ^ 2.0) * Float64(Float64(x_m * (s_m ^ 2.0)) * x_m))) <= -1e-99)
		tmp = Float64(Float64(-2.0) / Float64(Float64(s_m * c_m) * Float64(s_m * c_m)));
	else
		tmp = Float64(Float64(1.0 / x_m) / Float64(Float64(Float64(s_m * c_m) * x_m) * Float64(s_m * c_m)));
	end
	return tmp
end

MATLAB

s_m = abs(s);
c_m = abs(c);
x_m = abs(x);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp_2 = code(x_m, c_m, s_m)
	tmp = 0.0;
	if ((cos((2.0 * x_m)) / ((c_m ^ 2.0) * ((x_m * (s_m ^ 2.0)) * x_m))) <= -1e-99)
		tmp = -2.0 / ((s_m * c_m) * (s_m * c_m));
	else
		tmp = (1.0 / x_m) / (((s_m * c_m) * x_m) * (s_m * c_m));
	end
	tmp_2 = tmp;
end

Wolfram

s_m = N[Abs[s], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
x_m = N[Abs[x], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := If[LessEqual[N[(N[Cos[N[(2.0 * x$95$m), $MachinePrecision]], $MachinePrecision] / N[(N[Power[c$95$m, 2.0], $MachinePrecision] * N[(N[(x$95$m * N[Power[s$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1e-99], N[((-2.0) / N[(N[(s$95$m * c$95$m), $MachinePrecision] * N[(s$95$m * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / x$95$m), $MachinePrecision] / N[(N[(N[(s$95$m * c$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] * N[(s$95$m * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x))) < -1e-99

   1. Initial program 74.4%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. associate-*r/ N/A

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

      2. div-add-rev N/A

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

      3. +-commutative N/A

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

      4. associate-/l/ N/A

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

      5. associate-*r* N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

      14. *-commutative N/A

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

      15. unpow2 N/A

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

      16. associate-*l* N/A

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

      17. associate-*r* N/A

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

   5. Applied rewrites 41.7%

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

   6. Taylor expanded in x around inf

      \[\leadsto \frac{-2}{\color{blue}{{c}^{2} \cdot {s}^{2}}} \]
   7. Step-by-step derivation

   8. Applied rewrites 55.7%

      \[\leadsto \frac{\frac{\frac{-2}{s \cdot s}}{c}}{\color{blue}{c}} \]
   9. Step-by-step derivation

   10. Applied rewrites 55.6%

       \[\leadsto \frac{2}{\left(\left(-s\right) \cdot s\right) \cdot \color{blue}{\left(c \cdot c\right)}} \]
   11. Step-by-step derivation

   12. Applied rewrites 55.7%

       \[\leadsto \frac{2}{\left(s \cdot \left(-c\right)\right) \cdot \left(s \cdot \color{blue}{c}\right)} \]

   if -1e-99 < (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x)))

   1. Initial program 66.3%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

   4. Applied rewrites 86.8%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 80.1%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. distribute-frac-neg N/A

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

      5. distribute-neg-frac2 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. sqr-neg-rev N/A

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

      11. lift-*.f64 N/A

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

      12. distribute-lft-neg-out N/A

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

      13. lift-neg.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. distribute-lft-neg-out N/A

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

      17. lift-neg.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. associate-*r* N/A

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

      20. *-commutative N/A

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

      21. lift-*.f64 N/A

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

   9. Applied rewrites 85.2%

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

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \leq -1 \cdot 10^{-99}:\\ \;\;\;\;\frac{-2}{\left(s \cdot c\right) \cdot \left(s \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(\left(s \cdot c\right) \cdot x\right) \cdot \left(s \cdot c\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 49.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ x_m = \left|x\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{\cos \left(2 \cdot x\_m\right)}{{c\_m}^{2} \cdot \left(\left(x\_m \cdot {s\_m}^{2}\right) \cdot x\_m\right)} \leq -1 \cdot 10^{-99}:\\ \;\;\;\;\frac{-2}{\left(s\_m \cdot c\_m\right) \cdot \left(s\_m \cdot c\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{2}{s\_m \cdot s\_m}}{c\_m}}{c\_m}\\ \end{array} \end{array} \]

FPCore

s_m = (fabs.f64 s)
c_m = (fabs.f64 c)
x_m = (fabs.f64 x)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (if (<=
      (/ (cos (* 2.0 x_m)) (* (pow c_m 2.0) (* (* x_m (pow s_m 2.0)) x_m)))
      -1e-99)
   (/ (- 2.0) (* (* s_m c_m) (* s_m c_m)))
   (/ (/ (/ 2.0 (* s_m s_m)) c_m) c_m)))

C

s_m = fabs(s);
c_m = fabs(c);
x_m = fabs(x);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double tmp;
	if ((cos((2.0 * x_m)) / (pow(c_m, 2.0) * ((x_m * pow(s_m, 2.0)) * x_m))) <= -1e-99) {
		tmp = -2.0 / ((s_m * c_m) * (s_m * c_m));
	} else {
		tmp = ((2.0 / (s_m * s_m)) / c_m) / c_m;
	}
	return tmp;
}

Fortran

s_m =     private
c_m =     private
x_m =     private
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
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, c_m, s_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: tmp
    if ((cos((2.0d0 * x_m)) / ((c_m ** 2.0d0) * ((x_m * (s_m ** 2.0d0)) * x_m))) <= (-1d-99)) then
        tmp = -2.0d0 / ((s_m * c_m) * (s_m * c_m))
    else
        tmp = ((2.0d0 / (s_m * s_m)) / c_m) / c_m
    end if
    code = tmp
end function

Java

s_m = Math.abs(s);
c_m = Math.abs(c);
x_m = Math.abs(x);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	double tmp;
	if ((Math.cos((2.0 * x_m)) / (Math.pow(c_m, 2.0) * ((x_m * Math.pow(s_m, 2.0)) * x_m))) <= -1e-99) {
		tmp = -2.0 / ((s_m * c_m) * (s_m * c_m));
	} else {
		tmp = ((2.0 / (s_m * s_m)) / c_m) / c_m;
	}
	return tmp;
}

Python

s_m = math.fabs(s)
c_m = math.fabs(c)
x_m = math.fabs(x)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	tmp = 0
	if (math.cos((2.0 * x_m)) / (math.pow(c_m, 2.0) * ((x_m * math.pow(s_m, 2.0)) * x_m))) <= -1e-99:
		tmp = -2.0 / ((s_m * c_m) * (s_m * c_m))
	else:
		tmp = ((2.0 / (s_m * s_m)) / c_m) / c_m
	return tmp

Julia

s_m = abs(s)
c_m = abs(c)
x_m = abs(x)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	tmp = 0.0
	if (Float64(cos(Float64(2.0 * x_m)) / Float64((c_m ^ 2.0) * Float64(Float64(x_m * (s_m ^ 2.0)) * x_m))) <= -1e-99)
		tmp = Float64(Float64(-2.0) / Float64(Float64(s_m * c_m) * Float64(s_m * c_m)));
	else
		tmp = Float64(Float64(Float64(2.0 / Float64(s_m * s_m)) / c_m) / c_m);
	end
	return tmp
end

MATLAB

s_m = abs(s);
c_m = abs(c);
x_m = abs(x);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp_2 = code(x_m, c_m, s_m)
	tmp = 0.0;
	if ((cos((2.0 * x_m)) / ((c_m ^ 2.0) * ((x_m * (s_m ^ 2.0)) * x_m))) <= -1e-99)
		tmp = -2.0 / ((s_m * c_m) * (s_m * c_m));
	else
		tmp = ((2.0 / (s_m * s_m)) / c_m) / c_m;
	end
	tmp_2 = tmp;
end

Wolfram

s_m = N[Abs[s], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
x_m = N[Abs[x], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := If[LessEqual[N[(N[Cos[N[(2.0 * x$95$m), $MachinePrecision]], $MachinePrecision] / N[(N[Power[c$95$m, 2.0], $MachinePrecision] * N[(N[(x$95$m * N[Power[s$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1e-99], N[((-2.0) / N[(N[(s$95$m * c$95$m), $MachinePrecision] * N[(s$95$m * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 / N[(s$95$m * s$95$m), $MachinePrecision]), $MachinePrecision] / c$95$m), $MachinePrecision] / c$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x))) < -1e-99

   1. Initial program 74.4%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. associate-*r/ N/A

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

      2. div-add-rev N/A

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

      3. +-commutative N/A

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

      4. associate-/l/ N/A

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

      5. associate-*r* N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

      14. *-commutative N/A

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

      15. unpow2 N/A

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

      16. associate-*l* N/A

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

      17. associate-*r* N/A

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

   5. Applied rewrites 41.7%

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

   6. Taylor expanded in x around inf

      \[\leadsto \frac{-2}{\color{blue}{{c}^{2} \cdot {s}^{2}}} \]
   7. Step-by-step derivation

   8. Applied rewrites 55.7%

      \[\leadsto \frac{\frac{\frac{-2}{s \cdot s}}{c}}{\color{blue}{c}} \]
   9. Step-by-step derivation

   10. Applied rewrites 55.6%

       \[\leadsto \frac{2}{\left(\left(-s\right) \cdot s\right) \cdot \color{blue}{\left(c \cdot c\right)}} \]
   11. Step-by-step derivation

   12. Applied rewrites 55.7%

       \[\leadsto \frac{2}{\left(s \cdot \left(-c\right)\right) \cdot \left(s \cdot \color{blue}{c}\right)} \]

   if -1e-99 < (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x)))

   1. Initial program 66.3%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. associate-*r/ N/A

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

      2. div-add-rev N/A

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

      3. +-commutative N/A

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

      4. associate-/l/ N/A

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

      5. associate-*r* N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

      14. *-commutative N/A

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

      15. unpow2 N/A

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

      16. associate-*l* N/A

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

      17. associate-*r* N/A

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

   5. Applied rewrites 53.1%

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

   7. Applied rewrites 55.2%

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

   8. Taylor expanded in x around inf

      \[\leadsto \frac{2}{\color{blue}{{c}^{2} \cdot {s}^{2}}} \]
   9. Step-by-step derivation

   10. Applied rewrites 50.6%

       \[\leadsto \frac{\frac{\frac{2}{s \cdot s}}{c}}{\color{blue}{c}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 50.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \leq -1 \cdot 10^{-99}:\\ \;\;\;\;\frac{-2}{\left(s \cdot c\right) \cdot \left(s \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{2}{s \cdot s}}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ x_m = \left|x\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{\cos \left(2 \cdot x\_m\right)}{{c\_m}^{2} \cdot \left(\left(x\_m \cdot {s\_m}^{2}\right) \cdot x\_m\right)} \leq -1 \cdot 10^{-99}:\\ \;\;\;\;\frac{-2}{\left(s\_m \cdot c\_m\right) \cdot \left(s\_m \cdot c\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(x\_m \cdot \left(c\_m \cdot s\_m\right)\right) \cdot \left(c\_m \cdot s\_m\right)\right) \cdot x\_m}\\ \end{array} \end{array} \]

FPCore

s_m = (fabs.f64 s)
c_m = (fabs.f64 c)
x_m = (fabs.f64 x)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (if (<=
      (/ (cos (* 2.0 x_m)) (* (pow c_m 2.0) (* (* x_m (pow s_m 2.0)) x_m)))
      -1e-99)
   (/ (- 2.0) (* (* s_m c_m) (* s_m c_m)))
   (/ 1.0 (* (* (* x_m (* c_m s_m)) (* c_m s_m)) x_m))))

C

s_m = fabs(s);
c_m = fabs(c);
x_m = fabs(x);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double tmp;
	if ((cos((2.0 * x_m)) / (pow(c_m, 2.0) * ((x_m * pow(s_m, 2.0)) * x_m))) <= -1e-99) {
		tmp = -2.0 / ((s_m * c_m) * (s_m * c_m));
	} else {
		tmp = 1.0 / (((x_m * (c_m * s_m)) * (c_m * s_m)) * x_m);
	}
	return tmp;
}

Fortran

s_m =     private
c_m =     private
x_m =     private
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
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, c_m, s_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: tmp
    if ((cos((2.0d0 * x_m)) / ((c_m ** 2.0d0) * ((x_m * (s_m ** 2.0d0)) * x_m))) <= (-1d-99)) then
        tmp = -2.0d0 / ((s_m * c_m) * (s_m * c_m))
    else
        tmp = 1.0d0 / (((x_m * (c_m * s_m)) * (c_m * s_m)) * x_m)
    end if
    code = tmp
end function

Java

s_m = Math.abs(s);
c_m = Math.abs(c);
x_m = Math.abs(x);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	double tmp;
	if ((Math.cos((2.0 * x_m)) / (Math.pow(c_m, 2.0) * ((x_m * Math.pow(s_m, 2.0)) * x_m))) <= -1e-99) {
		tmp = -2.0 / ((s_m * c_m) * (s_m * c_m));
	} else {
		tmp = 1.0 / (((x_m * (c_m * s_m)) * (c_m * s_m)) * x_m);
	}
	return tmp;
}

Python

s_m = math.fabs(s)
c_m = math.fabs(c)
x_m = math.fabs(x)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	tmp = 0
	if (math.cos((2.0 * x_m)) / (math.pow(c_m, 2.0) * ((x_m * math.pow(s_m, 2.0)) * x_m))) <= -1e-99:
		tmp = -2.0 / ((s_m * c_m) * (s_m * c_m))
	else:
		tmp = 1.0 / (((x_m * (c_m * s_m)) * (c_m * s_m)) * x_m)
	return tmp

Julia

s_m = abs(s)
c_m = abs(c)
x_m = abs(x)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	tmp = 0.0
	if (Float64(cos(Float64(2.0 * x_m)) / Float64((c_m ^ 2.0) * Float64(Float64(x_m * (s_m ^ 2.0)) * x_m))) <= -1e-99)
		tmp = Float64(Float64(-2.0) / Float64(Float64(s_m * c_m) * Float64(s_m * c_m)));
	else
		tmp = Float64(1.0 / Float64(Float64(Float64(x_m * Float64(c_m * s_m)) * Float64(c_m * s_m)) * x_m));
	end
	return tmp
end

MATLAB

s_m = abs(s);
c_m = abs(c);
x_m = abs(x);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp_2 = code(x_m, c_m, s_m)
	tmp = 0.0;
	if ((cos((2.0 * x_m)) / ((c_m ^ 2.0) * ((x_m * (s_m ^ 2.0)) * x_m))) <= -1e-99)
		tmp = -2.0 / ((s_m * c_m) * (s_m * c_m));
	else
		tmp = 1.0 / (((x_m * (c_m * s_m)) * (c_m * s_m)) * x_m);
	end
	tmp_2 = tmp;
end

Wolfram

s_m = N[Abs[s], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
x_m = N[Abs[x], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := If[LessEqual[N[(N[Cos[N[(2.0 * x$95$m), $MachinePrecision]], $MachinePrecision] / N[(N[Power[c$95$m, 2.0], $MachinePrecision] * N[(N[(x$95$m * N[Power[s$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1e-99], N[((-2.0) / N[(N[(s$95$m * c$95$m), $MachinePrecision] * N[(s$95$m * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[(x$95$m * N[(c$95$m * s$95$m), $MachinePrecision]), $MachinePrecision] * N[(c$95$m * s$95$m), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x))) < -1e-99

   1. Initial program 74.4%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. associate-*r/ N/A

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

      2. div-add-rev N/A

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

      3. +-commutative N/A

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

      4. associate-/l/ N/A

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

      5. associate-*r* N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

      14. *-commutative N/A

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

      15. unpow2 N/A

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

      16. associate-*l* N/A

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

      17. associate-*r* N/A

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

   5. Applied rewrites 41.7%

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

   6. Taylor expanded in x around inf

      \[\leadsto \frac{-2}{\color{blue}{{c}^{2} \cdot {s}^{2}}} \]
   7. Step-by-step derivation

   8. Applied rewrites 55.7%

      \[\leadsto \frac{\frac{\frac{-2}{s \cdot s}}{c}}{\color{blue}{c}} \]
   9. Step-by-step derivation

   10. Applied rewrites 55.6%

       \[\leadsto \frac{2}{\left(\left(-s\right) \cdot s\right) \cdot \color{blue}{\left(c \cdot c\right)}} \]
   11. Step-by-step derivation

   12. Applied rewrites 55.7%

       \[\leadsto \frac{2}{\left(s \cdot \left(-c\right)\right) \cdot \left(s \cdot \color{blue}{c}\right)} \]

   if -1e-99 < (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x)))

   1. Initial program 66.3%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. pow-prod-down N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-*.f64 86.6

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

   4. Applied rewrites 86.6%

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left({\left(c \cdot s\right)}^{2} \cdot x\right) \cdot x}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(x \cdot \color{blue}{\left(\left(c \cdot s\right) \cdot \left(c \cdot s\right)\right)}\right) \cdot x} \]

      5. sqr-neg-rev N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(x \cdot \color{blue}{\left(\left(\mathsf{neg}\left(c \cdot s\right)\right) \cdot \left(\mathsf{neg}\left(c \cdot s\right)\right)\right)}\right) \cdot x} \]

      6. associate-*r* N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(\left(x \cdot \left(\mathsf{neg}\left(c \cdot s\right)\right)\right) \cdot \left(\mathsf{neg}\left(c \cdot s\right)\right)\right)} \cdot x} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(\left(x \cdot \left(\mathsf{neg}\left(c \cdot s\right)\right)\right) \cdot \left(\mathsf{neg}\left(c \cdot s\right)\right)\right)} \cdot x} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(\color{blue}{\left(x \cdot \left(\mathsf{neg}\left(c \cdot s\right)\right)\right)} \cdot \left(\mathsf{neg}\left(c \cdot s\right)\right)\right) \cdot x} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(\left(x \cdot \left(\mathsf{neg}\left(\color{blue}{c \cdot s}\right)\right)\right) \cdot \left(\mathsf{neg}\left(c \cdot s\right)\right)\right) \cdot x} \]

      10. distribute-lft-neg-in N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(\left(x \cdot \color{blue}{\left(\left(\mathsf{neg}\left(c\right)\right) \cdot s\right)}\right) \cdot \left(\mathsf{neg}\left(c \cdot s\right)\right)\right) \cdot x} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(\left(x \cdot \color{blue}{\left(\left(\mathsf{neg}\left(c\right)\right) \cdot s\right)}\right) \cdot \left(\mathsf{neg}\left(c \cdot s\right)\right)\right) \cdot x} \]

      12. lower-neg.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. distribute-lft-neg-in N/A

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

      15. lower-*.f64 N/A

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

      16. lower-neg.f64 94.6

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

   6. Applied rewrites 94.6%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 85.2%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \leq -1 \cdot 10^{-99}:\\ \;\;\;\;\frac{-2}{\left(s \cdot c\right) \cdot \left(s \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(x \cdot \left(c \cdot s\right)\right) \cdot \left(c \cdot s\right)\right) \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 96.7% accurate, 2.3× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ x_m = \left|x\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \left(s\_m \cdot x\_m\right) \cdot c\_m\\ \mathbf{if}\;x\_m \leq 1.45 \cdot 10^{-8}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2, x\_m \cdot x\_m, 1\right)}{t\_0 \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x\_m + x\_m\right)}{\left(\left(x\_m \cdot \left(s\_m \cdot c\_m\right)\right) \cdot \left(s\_m \cdot c\_m\right)\right) \cdot x\_m}\\ \end{array} \end{array} \]

FPCore

s_m = (fabs.f64 s)
c_m = (fabs.f64 c)
x_m = (fabs.f64 x)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (let* ((t_0 (* (* s_m x_m) c_m)))
   (if (<= x_m 1.45e-8)
     (/ (fma -2.0 (* x_m x_m) 1.0) (* t_0 t_0))
     (/ (cos (+ x_m x_m)) (* (* (* x_m (* s_m c_m)) (* s_m c_m)) x_m)))))

C

s_m = fabs(s);
c_m = fabs(c);
x_m = fabs(x);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double t_0 = (s_m * x_m) * c_m;
	double tmp;
	if (x_m <= 1.45e-8) {
		tmp = fma(-2.0, (x_m * x_m), 1.0) / (t_0 * t_0);
	} else {
		tmp = cos((x_m + x_m)) / (((x_m * (s_m * c_m)) * (s_m * c_m)) * x_m);
	}
	return tmp;
}

Julia

s_m = abs(s)
c_m = abs(c)
x_m = abs(x)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	t_0 = Float64(Float64(s_m * x_m) * c_m)
	tmp = 0.0
	if (x_m <= 1.45e-8)
		tmp = Float64(fma(-2.0, Float64(x_m * x_m), 1.0) / Float64(t_0 * t_0));
	else
		tmp = Float64(cos(Float64(x_m + x_m)) / Float64(Float64(Float64(x_m * Float64(s_m * c_m)) * Float64(s_m * c_m)) * x_m));
	end
	return tmp
end

Wolfram

s_m = N[Abs[s], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
x_m = N[Abs[x], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[(N[(s$95$m * x$95$m), $MachinePrecision] * c$95$m), $MachinePrecision]}, If[LessEqual[x$95$m, 1.45e-8], N[(N[(-2.0 * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(x$95$m + x$95$m), $MachinePrecision]], $MachinePrecision] / N[(N[(N[(x$95$m * N[(s$95$m * c$95$m), $MachinePrecision]), $MachinePrecision] * N[(s$95$m * c$95$m), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.4500000000000001e-8

   1. Initial program 67.0%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. associate-*r/ N/A

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

      2. div-add-rev N/A

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

      3. +-commutative N/A

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

      4. associate-/l/ N/A

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

      5. associate-*r* N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

      14. *-commutative N/A

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

      15. unpow2 N/A

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

      16. associate-*l* N/A

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

      17. associate-*r* N/A

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

   5. Applied rewrites 59.8%

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

   7. Applied rewrites 76.2%

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

   if 1.4500000000000001e-8 < x

   1. Initial program 66.2%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. pow-prod-down N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-*.f64 83.9

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

   4. Applied rewrites 83.9%

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left({\left(c \cdot s\right)}^{2} \cdot x\right) \cdot x}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(x \cdot \color{blue}{\left(\left(c \cdot s\right) \cdot \left(c \cdot s\right)\right)}\right) \cdot x} \]

      5. sqr-neg-rev N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(x \cdot \color{blue}{\left(\left(\mathsf{neg}\left(c \cdot s\right)\right) \cdot \left(\mathsf{neg}\left(c \cdot s\right)\right)\right)}\right) \cdot x} \]

      6. associate-*r* N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(\left(x \cdot \left(\mathsf{neg}\left(c \cdot s\right)\right)\right) \cdot \left(\mathsf{neg}\left(c \cdot s\right)\right)\right)} \cdot x} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(\left(x \cdot \left(\mathsf{neg}\left(c \cdot s\right)\right)\right) \cdot \left(\mathsf{neg}\left(c \cdot s\right)\right)\right)} \cdot x} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(\color{blue}{\left(x \cdot \left(\mathsf{neg}\left(c \cdot s\right)\right)\right)} \cdot \left(\mathsf{neg}\left(c \cdot s\right)\right)\right) \cdot x} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(\left(x \cdot \left(\mathsf{neg}\left(\color{blue}{c \cdot s}\right)\right)\right) \cdot \left(\mathsf{neg}\left(c \cdot s\right)\right)\right) \cdot x} \]

      10. distribute-lft-neg-in N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(\left(x \cdot \color{blue}{\left(\left(\mathsf{neg}\left(c\right)\right) \cdot s\right)}\right) \cdot \left(\mathsf{neg}\left(c \cdot s\right)\right)\right) \cdot x} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(\left(x \cdot \color{blue}{\left(\left(\mathsf{neg}\left(c\right)\right) \cdot s\right)}\right) \cdot \left(\mathsf{neg}\left(c \cdot s\right)\right)\right) \cdot x} \]

      12. lower-neg.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. distribute-lft-neg-in N/A

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

      15. lower-*.f64 N/A

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

      16. lower-neg.f64 93.1

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

   6. Applied rewrites 93.1%

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(\left(x \cdot \left(\left(-c\right) \cdot s\right)\right) \cdot \left(\left(-c\right) \cdot s\right)\right)} \cdot x} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 93.1

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

   8. Applied rewrites 93.1%

      \[\leadsto \frac{\cos \color{blue}{\left(x + x\right)}}{\left(\left(x \cdot \left(\left(-c\right) \cdot s\right)\right) \cdot \left(\left(-c\right) \cdot s\right)\right) \cdot x} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-neg.f64 N/A

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

      6. distribute-lft-neg-out N/A

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

      7. lift-*.f64 N/A

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

      8. lift-neg.f64 N/A

         \[\leadsto \frac{\cos \left(x + x\right)}{\left(x \cdot \left(\left(\mathsf{neg}\left(c \cdot s\right)\right) \cdot \left(\color{blue}{\left(\mathsf{neg}\left(c\right)\right)} \cdot s\right)\right)\right) \cdot x} \]

      9. distribute-lft-neg-out N/A

         \[\leadsto \frac{\cos \left(x + x\right)}{\left(x \cdot \left(\left(\mathsf{neg}\left(c \cdot s\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(c \cdot s\right)\right)}\right)\right) \cdot x} \]

      10. sqr-neg-rev N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 93.1

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

   10. Applied rewrites 93.1%

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

Alternative 7: 94.4% accurate, 2.3× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ x_m = \left|x\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \left(s\_m \cdot x\_m\right) \cdot c\_m\\ \mathbf{if}\;x\_m \leq 5 \cdot 10^{-33}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2, x\_m \cdot x\_m, 1\right)}{t\_0 \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x\_m + x\_m\right)}{\left(\left(s\_m \cdot c\_m\right) \cdot \left(c\_m \cdot \left(s\_m \cdot x\_m\right)\right)\right) \cdot x\_m}\\ \end{array} \end{array} \]

FPCore

s_m = (fabs.f64 s)
c_m = (fabs.f64 c)
x_m = (fabs.f64 x)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (let* ((t_0 (* (* s_m x_m) c_m)))
   (if (<= x_m 5e-33)
     (/ (fma -2.0 (* x_m x_m) 1.0) (* t_0 t_0))
     (/ (cos (+ x_m x_m)) (* (* (* s_m c_m) (* c_m (* s_m x_m))) x_m)))))

C

s_m = fabs(s);
c_m = fabs(c);
x_m = fabs(x);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double t_0 = (s_m * x_m) * c_m;
	double tmp;
	if (x_m <= 5e-33) {
		tmp = fma(-2.0, (x_m * x_m), 1.0) / (t_0 * t_0);
	} else {
		tmp = cos((x_m + x_m)) / (((s_m * c_m) * (c_m * (s_m * x_m))) * x_m);
	}
	return tmp;
}

Julia

s_m = abs(s)
c_m = abs(c)
x_m = abs(x)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	t_0 = Float64(Float64(s_m * x_m) * c_m)
	tmp = 0.0
	if (x_m <= 5e-33)
		tmp = Float64(fma(-2.0, Float64(x_m * x_m), 1.0) / Float64(t_0 * t_0));
	else
		tmp = Float64(cos(Float64(x_m + x_m)) / Float64(Float64(Float64(s_m * c_m) * Float64(c_m * Float64(s_m * x_m))) * x_m));
	end
	return tmp
end

Wolfram

s_m = N[Abs[s], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
x_m = N[Abs[x], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[(N[(s$95$m * x$95$m), $MachinePrecision] * c$95$m), $MachinePrecision]}, If[LessEqual[x$95$m, 5e-33], N[(N[(-2.0 * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(x$95$m + x$95$m), $MachinePrecision]], $MachinePrecision] / N[(N[(N[(s$95$m * c$95$m), $MachinePrecision] * N[(c$95$m * N[(s$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 5.00000000000000028e-33

   1. Initial program 66.6%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. associate-*r/ N/A

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

      2. div-add-rev N/A

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

      3. +-commutative N/A

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

      4. associate-/l/ N/A

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

      5. associate-*r* N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

      14. *-commutative N/A

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

      15. unpow2 N/A

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

      16. associate-*l* N/A

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

      17. associate-*r* N/A

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

   5. Applied rewrites 59.3%

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

   7. Applied rewrites 76.1%

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

   if 5.00000000000000028e-33 < x

   1. Initial program 67.1%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. pow-prod-down N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-*.f64 85.1

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

   4. Applied rewrites 85.1%

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left({\left(c \cdot s\right)}^{2} \cdot x\right) \cdot x}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(x \cdot \color{blue}{\left(\left(c \cdot s\right) \cdot \left(c \cdot s\right)\right)}\right) \cdot x} \]

      5. sqr-neg-rev N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(x \cdot \color{blue}{\left(\left(\mathsf{neg}\left(c \cdot s\right)\right) \cdot \left(\mathsf{neg}\left(c \cdot s\right)\right)\right)}\right) \cdot x} \]

      6. associate-*r* N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(\left(x \cdot \left(\mathsf{neg}\left(c \cdot s\right)\right)\right) \cdot \left(\mathsf{neg}\left(c \cdot s\right)\right)\right)} \cdot x} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(\left(x \cdot \left(\mathsf{neg}\left(c \cdot s\right)\right)\right) \cdot \left(\mathsf{neg}\left(c \cdot s\right)\right)\right)} \cdot x} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(\color{blue}{\left(x \cdot \left(\mathsf{neg}\left(c \cdot s\right)\right)\right)} \cdot \left(\mathsf{neg}\left(c \cdot s\right)\right)\right) \cdot x} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(\left(x \cdot \left(\mathsf{neg}\left(\color{blue}{c \cdot s}\right)\right)\right) \cdot \left(\mathsf{neg}\left(c \cdot s\right)\right)\right) \cdot x} \]

      10. distribute-lft-neg-in N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(\left(x \cdot \color{blue}{\left(\left(\mathsf{neg}\left(c\right)\right) \cdot s\right)}\right) \cdot \left(\mathsf{neg}\left(c \cdot s\right)\right)\right) \cdot x} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(\left(x \cdot \color{blue}{\left(\left(\mathsf{neg}\left(c\right)\right) \cdot s\right)}\right) \cdot \left(\mathsf{neg}\left(c \cdot s\right)\right)\right) \cdot x} \]

      12. lower-neg.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. distribute-lft-neg-in N/A

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

      15. lower-*.f64 N/A

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

      16. lower-neg.f64 93.6

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

   6. Applied rewrites 93.6%

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(\left(x \cdot \left(\left(-c\right) \cdot s\right)\right) \cdot \left(\left(-c\right) \cdot s\right)\right)} \cdot x} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 93.6

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

   8. Applied rewrites 93.6%

      \[\leadsto \frac{\cos \color{blue}{\left(x + x\right)}}{\left(\left(x \cdot \left(\left(-c\right) \cdot s\right)\right) \cdot \left(\left(-c\right) \cdot s\right)\right) \cdot x} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lift-*.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. distribute-lft-neg-out N/A

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

      9. lift-*.f64 N/A

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

      10. lift-neg.f64 N/A

          \[\leadsto \frac{\cos \left(x + x\right)}{\left(\left(\left(\mathsf{neg}\left(c \cdot s\right)\right) \cdot \left(\color{blue}{\left(\mathsf{neg}\left(c\right)\right)} \cdot s\right)\right) \cdot x\right) \cdot x} \]

      11. distribute-lft-neg-out N/A

          \[\leadsto \frac{\cos \left(x + x\right)}{\left(\left(\left(\mathsf{neg}\left(c \cdot s\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(c \cdot s\right)\right)}\right) \cdot x\right) \cdot x} \]

      12. sqr-neg-rev N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*l* N/A

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

      16. lift-*.f64 N/A

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

      17. associate-*l* N/A

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

      18. lift-*.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. associate-*l* N/A

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

      22. lower-*.f64 N/A

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

      23. *-commutative N/A

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

      24. lift-*.f64 N/A

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

      25. lower-*.f64 N/A

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

   10. Applied rewrites 91.0%

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

Alternative 8: 94.0% accurate, 2.3× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ x_m = \left|x\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \left(s\_m \cdot x\_m\right) \cdot c\_m\\ \mathbf{if}\;x\_m \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2, x\_m \cdot x\_m, 1\right)}{t\_0 \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x\_m + x\_m\right)}{\left(c\_m \cdot \left(\left(s\_m \cdot c\_m\right) \cdot \left(s\_m \cdot x\_m\right)\right)\right) \cdot x\_m}\\ \end{array} \end{array} \]

FPCore

s_m = (fabs.f64 s)
c_m = (fabs.f64 c)
x_m = (fabs.f64 x)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (let* ((t_0 (* (* s_m x_m) c_m)))
   (if (<= x_m 2e-8)
     (/ (fma -2.0 (* x_m x_m) 1.0) (* t_0 t_0))
     (/ (cos (+ x_m x_m)) (* (* c_m (* (* s_m c_m) (* s_m x_m))) x_m)))))

C

s_m = fabs(s);
c_m = fabs(c);
x_m = fabs(x);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double t_0 = (s_m * x_m) * c_m;
	double tmp;
	if (x_m <= 2e-8) {
		tmp = fma(-2.0, (x_m * x_m), 1.0) / (t_0 * t_0);
	} else {
		tmp = cos((x_m + x_m)) / ((c_m * ((s_m * c_m) * (s_m * x_m))) * x_m);
	}
	return tmp;
}

Julia

s_m = abs(s)
c_m = abs(c)
x_m = abs(x)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	t_0 = Float64(Float64(s_m * x_m) * c_m)
	tmp = 0.0
	if (x_m <= 2e-8)
		tmp = Float64(fma(-2.0, Float64(x_m * x_m), 1.0) / Float64(t_0 * t_0));
	else
		tmp = Float64(cos(Float64(x_m + x_m)) / Float64(Float64(c_m * Float64(Float64(s_m * c_m) * Float64(s_m * x_m))) * x_m));
	end
	return tmp
end

Wolfram

s_m = N[Abs[s], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
x_m = N[Abs[x], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[(N[(s$95$m * x$95$m), $MachinePrecision] * c$95$m), $MachinePrecision]}, If[LessEqual[x$95$m, 2e-8], N[(N[(-2.0 * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(x$95$m + x$95$m), $MachinePrecision]], $MachinePrecision] / N[(N[(c$95$m * N[(N[(s$95$m * c$95$m), $MachinePrecision] * N[(s$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 2e-8

   1. Initial program 67.0%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. associate-*r/ N/A

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

      2. div-add-rev N/A

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

      3. +-commutative N/A

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

      4. associate-/l/ N/A

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

      5. associate-*r* N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

      14. *-commutative N/A

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

      15. unpow2 N/A

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

      16. associate-*l* N/A

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

      17. associate-*r* N/A

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

   5. Applied rewrites 59.8%

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

   7. Applied rewrites 76.2%

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

   if 2e-8 < x

   1. Initial program 66.2%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. pow-prod-down N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-*.f64 83.9

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

   4. Applied rewrites 83.9%

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left({\left(c \cdot s\right)}^{2} \cdot x\right) \cdot x}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(x \cdot \color{blue}{\left(\left(c \cdot s\right) \cdot \left(c \cdot s\right)\right)}\right) \cdot x} \]

      5. sqr-neg-rev N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(x \cdot \color{blue}{\left(\left(\mathsf{neg}\left(c \cdot s\right)\right) \cdot \left(\mathsf{neg}\left(c \cdot s\right)\right)\right)}\right) \cdot x} \]

      6. associate-*r* N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(\left(x \cdot \left(\mathsf{neg}\left(c \cdot s\right)\right)\right) \cdot \left(\mathsf{neg}\left(c \cdot s\right)\right)\right)} \cdot x} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(\left(x \cdot \left(\mathsf{neg}\left(c \cdot s\right)\right)\right) \cdot \left(\mathsf{neg}\left(c \cdot s\right)\right)\right)} \cdot x} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(\color{blue}{\left(x \cdot \left(\mathsf{neg}\left(c \cdot s\right)\right)\right)} \cdot \left(\mathsf{neg}\left(c \cdot s\right)\right)\right) \cdot x} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(\left(x \cdot \left(\mathsf{neg}\left(\color{blue}{c \cdot s}\right)\right)\right) \cdot \left(\mathsf{neg}\left(c \cdot s\right)\right)\right) \cdot x} \]

      10. distribute-lft-neg-in N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(\left(x \cdot \color{blue}{\left(\left(\mathsf{neg}\left(c\right)\right) \cdot s\right)}\right) \cdot \left(\mathsf{neg}\left(c \cdot s\right)\right)\right) \cdot x} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(\left(x \cdot \color{blue}{\left(\left(\mathsf{neg}\left(c\right)\right) \cdot s\right)}\right) \cdot \left(\mathsf{neg}\left(c \cdot s\right)\right)\right) \cdot x} \]

      12. lower-neg.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. distribute-lft-neg-in N/A

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

      15. lower-*.f64 N/A

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

      16. lower-neg.f64 93.1

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

   6. Applied rewrites 93.1%

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(\left(x \cdot \left(\left(-c\right) \cdot s\right)\right) \cdot \left(\left(-c\right) \cdot s\right)\right)} \cdot x} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 93.1

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

   8. Applied rewrites 93.1%

      \[\leadsto \frac{\cos \color{blue}{\left(x + x\right)}}{\left(\left(x \cdot \left(\left(-c\right) \cdot s\right)\right) \cdot \left(\left(-c\right) \cdot s\right)\right) \cdot x} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lift-*.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. distribute-lft-neg-out N/A

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

      9. lift-*.f64 N/A

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

      10. lift-neg.f64 N/A

          \[\leadsto \frac{\cos \left(x + x\right)}{\left(\left(\left(\mathsf{neg}\left(c \cdot s\right)\right) \cdot \left(\color{blue}{\left(\mathsf{neg}\left(c\right)\right)} \cdot s\right)\right) \cdot x\right) \cdot x} \]

      11. distribute-lft-neg-out N/A

          \[\leadsto \frac{\cos \left(x + x\right)}{\left(\left(\left(\mathsf{neg}\left(c \cdot s\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(c \cdot s\right)\right)}\right) \cdot x\right) \cdot x} \]

      12. sqr-neg-rev N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*l* N/A

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

      16. lift-*.f64 N/A

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

      17. associate-*l* N/A

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

      18. lift-*.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. *-commutative N/A

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

      22. associate-*l* N/A

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

      23. lower-*.f64 N/A

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

      24. lower-*.f64 N/A

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

   10. Applied rewrites 90.8%

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

Alternative 9: 90.4% accurate, 2.3× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ x_m = \left|x\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} \mathbf{if}\;s\_m \leq 4.5 \cdot 10^{+164}:\\ \;\;\;\;\frac{\cos \left(x\_m + x\_m\right)}{\left(\left(\left(\left(c\_m \cdot x\_m\right) \cdot c\_m\right) \cdot s\_m\right) \cdot s\_m\right) \cdot x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\left(s\_m \cdot x\_m\right) \cdot c\_m\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

s_m = (fabs.f64 s)
c_m = (fabs.f64 c)
x_m = (fabs.f64 x)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (if (<= s_m 4.5e+164)
   (/ (cos (+ x_m x_m)) (* (* (* (* (* c_m x_m) c_m) s_m) s_m) x_m))
   (/ 1.0 (pow (* (* s_m x_m) c_m) 2.0))))

C

s_m = fabs(s);
c_m = fabs(c);
x_m = fabs(x);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double tmp;
	if (s_m <= 4.5e+164) {
		tmp = cos((x_m + x_m)) / (((((c_m * x_m) * c_m) * s_m) * s_m) * x_m);
	} else {
		tmp = 1.0 / pow(((s_m * x_m) * c_m), 2.0);
	}
	return tmp;
}

Fortran

s_m =     private
c_m =     private
x_m =     private
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
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, c_m, s_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: tmp
    if (s_m <= 4.5d+164) then
        tmp = cos((x_m + x_m)) / (((((c_m * x_m) * c_m) * s_m) * s_m) * x_m)
    else
        tmp = 1.0d0 / (((s_m * x_m) * c_m) ** 2.0d0)
    end if
    code = tmp
end function

Java

s_m = Math.abs(s);
c_m = Math.abs(c);
x_m = Math.abs(x);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	double tmp;
	if (s_m <= 4.5e+164) {
		tmp = Math.cos((x_m + x_m)) / (((((c_m * x_m) * c_m) * s_m) * s_m) * x_m);
	} else {
		tmp = 1.0 / Math.pow(((s_m * x_m) * c_m), 2.0);
	}
	return tmp;
}

Python

s_m = math.fabs(s)
c_m = math.fabs(c)
x_m = math.fabs(x)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	tmp = 0
	if s_m <= 4.5e+164:
		tmp = math.cos((x_m + x_m)) / (((((c_m * x_m) * c_m) * s_m) * s_m) * x_m)
	else:
		tmp = 1.0 / math.pow(((s_m * x_m) * c_m), 2.0)
	return tmp

Julia

s_m = abs(s)
c_m = abs(c)
x_m = abs(x)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	tmp = 0.0
	if (s_m <= 4.5e+164)
		tmp = Float64(cos(Float64(x_m + x_m)) / Float64(Float64(Float64(Float64(Float64(c_m * x_m) * c_m) * s_m) * s_m) * x_m));
	else
		tmp = Float64(1.0 / (Float64(Float64(s_m * x_m) * c_m) ^ 2.0));
	end
	return tmp
end

MATLAB

s_m = abs(s);
c_m = abs(c);
x_m = abs(x);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp_2 = code(x_m, c_m, s_m)
	tmp = 0.0;
	if (s_m <= 4.5e+164)
		tmp = cos((x_m + x_m)) / (((((c_m * x_m) * c_m) * s_m) * s_m) * x_m);
	else
		tmp = 1.0 / (((s_m * x_m) * c_m) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

s_m = N[Abs[s], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
x_m = N[Abs[x], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := If[LessEqual[s$95$m, 4.5e+164], N[(N[Cos[N[(x$95$m + x$95$m), $MachinePrecision]], $MachinePrecision] / N[(N[(N[(N[(N[(c$95$m * x$95$m), $MachinePrecision] * c$95$m), $MachinePrecision] * s$95$m), $MachinePrecision] * s$95$m), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Power[N[(N[(s$95$m * x$95$m), $MachinePrecision] * c$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if s < 4.49999999999999975e164

   1. Initial program 67.5%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. pow-prod-down N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-*.f64 85.6

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

   4. Applied rewrites 85.6%

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left({\left(c \cdot s\right)}^{2} \cdot x\right) \cdot x}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(x \cdot \color{blue}{\left(\left(c \cdot s\right) \cdot \left(c \cdot s\right)\right)}\right) \cdot x} \]

      5. sqr-neg-rev N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(x \cdot \color{blue}{\left(\left(\mathsf{neg}\left(c \cdot s\right)\right) \cdot \left(\mathsf{neg}\left(c \cdot s\right)\right)\right)}\right) \cdot x} \]

      6. associate-*r* N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(\left(x \cdot \left(\mathsf{neg}\left(c \cdot s\right)\right)\right) \cdot \left(\mathsf{neg}\left(c \cdot s\right)\right)\right)} \cdot x} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(\left(x \cdot \left(\mathsf{neg}\left(c \cdot s\right)\right)\right) \cdot \left(\mathsf{neg}\left(c \cdot s\right)\right)\right)} \cdot x} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(\color{blue}{\left(x \cdot \left(\mathsf{neg}\left(c \cdot s\right)\right)\right)} \cdot \left(\mathsf{neg}\left(c \cdot s\right)\right)\right) \cdot x} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(\left(x \cdot \left(\mathsf{neg}\left(\color{blue}{c \cdot s}\right)\right)\right) \cdot \left(\mathsf{neg}\left(c \cdot s\right)\right)\right) \cdot x} \]

      10. distribute-lft-neg-in N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(\left(x \cdot \color{blue}{\left(\left(\mathsf{neg}\left(c\right)\right) \cdot s\right)}\right) \cdot \left(\mathsf{neg}\left(c \cdot s\right)\right)\right) \cdot x} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(\left(x \cdot \color{blue}{\left(\left(\mathsf{neg}\left(c\right)\right) \cdot s\right)}\right) \cdot \left(\mathsf{neg}\left(c \cdot s\right)\right)\right) \cdot x} \]

      12. lower-neg.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. distribute-lft-neg-in N/A

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

      15. lower-*.f64 N/A

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

      16. lower-neg.f64 93.9

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

   6. Applied rewrites 93.9%

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(\left(x \cdot \left(\left(-c\right) \cdot s\right)\right) \cdot \left(\left(-c\right) \cdot s\right)\right)} \cdot x} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 93.9

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

   8. Applied rewrites 93.9%

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

   9. Taylor expanded in x around 0

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. unpow2 N/A

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

       4. associate-*r* N/A

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

       5. lower-*.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. *-commutative N/A

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

       8. unpow2 N/A

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

       9. associate-*r* N/A

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

       10. *-commutative N/A

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

       11. lower-*.f64 N/A

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

       12. lower-*.f64 85.6

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

   11. Applied rewrites 85.6%

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

   if 4.49999999999999975e164 < s

   1. Initial program 61.7%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

   4. Applied rewrites 87.7%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 81.9%

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

   9. Applied rewrites 93.0%

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

Alternative 10: 79.5% accurate, 6.2× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ x_m = \left|x\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \begin{array}{l} t_0 := \left(s\_m \cdot x\_m\right) \cdot c\_m\\ \mathbf{if}\;x\_m \leq 3.3 \cdot 10^{+42}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2, x\_m \cdot x\_m, 1\right)}{t\_0 \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x\_m}}{\left(\left(s\_m \cdot c\_m\right) \cdot x\_m\right) \cdot \left(s\_m \cdot c\_m\right)}\\ \end{array} \end{array} \]

FPCore

s_m = (fabs.f64 s)
c_m = (fabs.f64 c)
x_m = (fabs.f64 x)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (let* ((t_0 (* (* s_m x_m) c_m)))
   (if (<= x_m 3.3e+42)
     (/ (fma -2.0 (* x_m x_m) 1.0) (* t_0 t_0))
     (/ (/ 1.0 x_m) (* (* (* s_m c_m) x_m) (* s_m c_m))))))

C

s_m = fabs(s);
c_m = fabs(c);
x_m = fabs(x);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	double t_0 = (s_m * x_m) * c_m;
	double tmp;
	if (x_m <= 3.3e+42) {
		tmp = fma(-2.0, (x_m * x_m), 1.0) / (t_0 * t_0);
	} else {
		tmp = (1.0 / x_m) / (((s_m * c_m) * x_m) * (s_m * c_m));
	}
	return tmp;
}

Julia

s_m = abs(s)
c_m = abs(c)
x_m = abs(x)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	t_0 = Float64(Float64(s_m * x_m) * c_m)
	tmp = 0.0
	if (x_m <= 3.3e+42)
		tmp = Float64(fma(-2.0, Float64(x_m * x_m), 1.0) / Float64(t_0 * t_0));
	else
		tmp = Float64(Float64(1.0 / x_m) / Float64(Float64(Float64(s_m * c_m) * x_m) * Float64(s_m * c_m)));
	end
	return tmp
end

Wolfram

s_m = N[Abs[s], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
x_m = N[Abs[x], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[(N[(s$95$m * x$95$m), $MachinePrecision] * c$95$m), $MachinePrecision]}, If[LessEqual[x$95$m, 3.3e+42], N[(N[(-2.0 * N[(x$95$m * x$95$m), $MachinePrecision] + 1.0), $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / x$95$m), $MachinePrecision] / N[(N[(N[(s$95$m * c$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] * N[(s$95$m * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 3.2999999999999999e42

   1. Initial program 66.9%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. associate-*r/ N/A

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

      2. div-add-rev N/A

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

      3. +-commutative N/A

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

      4. associate-/l/ N/A

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

      5. associate-*r* N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

      14. *-commutative N/A

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

      15. unpow2 N/A

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

      16. associate-*l* N/A

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

      17. associate-*r* N/A

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

   5. Applied rewrites 60.3%

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

   7. Applied rewrites 75.7%

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

   if 3.2999999999999999e42 < x

   1. Initial program 66.0%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

   4. Applied rewrites 81.3%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 58.4%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. distribute-frac-neg N/A

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

      5. distribute-neg-frac2 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. sqr-neg-rev N/A

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

      11. lift-*.f64 N/A

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

      12. distribute-lft-neg-out N/A

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

      13. lift-neg.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. distribute-lft-neg-out N/A

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

      17. lift-neg.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. associate-*r* N/A

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

      20. *-commutative N/A

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

      21. lift-*.f64 N/A

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

   9. Applied rewrites 59.4%

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

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.3 \cdot 10^{+42}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2, x \cdot x, 1\right)}{\left(\left(s \cdot x\right) \cdot c\right) \cdot \left(\left(s \cdot x\right) \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(\left(s \cdot c\right) \cdot x\right) \cdot \left(s \cdot c\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 32.2% accurate, 11.5× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ x_m = \left|x\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \frac{2}{\left(\left(\left(-s\_m\right) \cdot s\_m\right) \cdot c\_m\right) \cdot c\_m} \end{array} \]

FPCore

s_m = (fabs.f64 s)
c_m = (fabs.f64 c)
x_m = (fabs.f64 x)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (/ 2.0 (* (* (* (- s_m) s_m) c_m) c_m)))

C

s_m = fabs(s);
c_m = fabs(c);
x_m = fabs(x);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	return 2.0 / (((-s_m * s_m) * c_m) * c_m);
}

Fortran

s_m =     private
c_m =     private
x_m =     private
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
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, c_m, s_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    code = 2.0d0 / (((-s_m * s_m) * c_m) * c_m)
end function

Java

s_m = Math.abs(s);
c_m = Math.abs(c);
x_m = Math.abs(x);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	return 2.0 / (((-s_m * s_m) * c_m) * c_m);
}

Python

s_m = math.fabs(s)
c_m = math.fabs(c)
x_m = math.fabs(x)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	return 2.0 / (((-s_m * s_m) * c_m) * c_m)

Julia

s_m = abs(s)
c_m = abs(c)
x_m = abs(x)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	return Float64(2.0 / Float64(Float64(Float64(Float64(-s_m) * s_m) * c_m) * c_m))
end

MATLAB

s_m = abs(s);
c_m = abs(c);
x_m = abs(x);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp = code(x_m, c_m, s_m)
	tmp = 2.0 / (((-s_m * s_m) * c_m) * c_m);
end

Wolfram

s_m = N[Abs[s], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
x_m = N[Abs[x], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := N[(2.0 / N[(N[(N[((-s$95$m) * s$95$m), $MachinePrecision] * c$95$m), $MachinePrecision] * c$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 66.8%

   \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

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

   1. associate-*r/ N/A

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

   2. div-add-rev N/A

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

   3. +-commutative N/A

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

   4. associate-/l/ N/A

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

   5. associate-*r* N/A

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

   6. lower-/.f64 N/A

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

   7. +-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. *-commutative N/A

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

   12. *-commutative N/A

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

   13. associate-*l* N/A

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

   14. *-commutative N/A

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

   15. unpow2 N/A

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

   16. associate-*l* N/A

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

   17. associate-*r* N/A

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

5. Applied rewrites 52.4%

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

6. Taylor expanded in x around inf

   \[\leadsto \frac{-2}{\color{blue}{{c}^{2} \cdot {s}^{2}}} \]
7. Step-by-step derivation

8. Applied rewrites 27.9%

   \[\leadsto \frac{\frac{\frac{-2}{s \cdot s}}{c}}{\color{blue}{c}} \]
9. Step-by-step derivation

10. Applied rewrites 27.5%

    \[\leadsto \frac{2}{\left(\left(-s\right) \cdot s\right) \cdot \color{blue}{\left(c \cdot c\right)}} \]
11. Step-by-step derivation

12. Applied rewrites 28.1%

    \[\leadsto \frac{2}{\left(\left(\left(-s\right) \cdot s\right) \cdot c\right) \cdot c} \]
13. Add Preprocessing

Alternative 12: 28.9% accurate, 11.5× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ x_m = \left|x\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \frac{-2}{\left(s\_m \cdot s\_m\right) \cdot \left(c\_m \cdot c\_m\right)} \end{array} \]

FPCore

s_m = (fabs.f64 s)
c_m = (fabs.f64 c)
x_m = (fabs.f64 x)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (/ (- 2.0) (* (* s_m s_m) (* c_m c_m))))

C

s_m = fabs(s);
c_m = fabs(c);
x_m = fabs(x);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	return -2.0 / ((s_m * s_m) * (c_m * c_m));
}

Fortran

s_m =     private
c_m =     private
x_m =     private
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
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, c_m, s_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    code = -2.0d0 / ((s_m * s_m) * (c_m * c_m))
end function

Java

s_m = Math.abs(s);
c_m = Math.abs(c);
x_m = Math.abs(x);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	return -2.0 / ((s_m * s_m) * (c_m * c_m));
}

Python

s_m = math.fabs(s)
c_m = math.fabs(c)
x_m = math.fabs(x)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	return -2.0 / ((s_m * s_m) * (c_m * c_m))

Julia

s_m = abs(s)
c_m = abs(c)
x_m = abs(x)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	return Float64(Float64(-2.0) / Float64(Float64(s_m * s_m) * Float64(c_m * c_m)))
end

MATLAB

s_m = abs(s);
c_m = abs(c);
x_m = abs(x);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp = code(x_m, c_m, s_m)
	tmp = -2.0 / ((s_m * s_m) * (c_m * c_m));
end

Wolfram

s_m = N[Abs[s], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
x_m = N[Abs[x], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := N[((-2.0) / N[(N[(s$95$m * s$95$m), $MachinePrecision] * N[(c$95$m * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 66.8%

   \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

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

   1. associate-*r/ N/A

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

   2. div-add-rev N/A

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

   3. +-commutative N/A

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

   4. associate-/l/ N/A

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

   5. associate-*r* N/A

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

   6. lower-/.f64 N/A

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

   7. +-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. *-commutative N/A

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

   12. *-commutative N/A

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

   13. associate-*l* N/A

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

   14. *-commutative N/A

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

   15. unpow2 N/A

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

   16. associate-*l* N/A

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

   17. associate-*r* N/A

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

5. Applied rewrites 52.4%

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

6. Taylor expanded in x around inf

   \[\leadsto \frac{-2}{\color{blue}{{c}^{2} \cdot {s}^{2}}} \]
7. Step-by-step derivation

8. Applied rewrites 27.9%

   \[\leadsto \frac{\frac{\frac{-2}{s \cdot s}}{c}}{\color{blue}{c}} \]
9. Step-by-step derivation

10. Applied rewrites 27.5%

    \[\leadsto \frac{2}{\left(\left(-s\right) \cdot s\right) \cdot \color{blue}{\left(c \cdot c\right)}} \]

11. Final simplification 27.5%

    \[\leadsto \frac{-2}{\left(s \cdot s\right) \cdot \left(c \cdot c\right)} \]
12. Add Preprocessing

Alternative 13: 26.7% accurate, 11.5× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ c_m = \left|c\right| \\ x_m = \left|x\right| \\ [x_m, c_m, s_m] = \mathsf{sort}([x_m, c_m, s_m])\\ \\ \frac{-2}{\left(s\_m \cdot c\_m\right) \cdot \left(s\_m \cdot c\_m\right)} \end{array} \]

FPCore

s_m = (fabs.f64 s)
c_m = (fabs.f64 c)
x_m = (fabs.f64 x)
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x_m c_m s_m)
 :precision binary64
 (/ (- 2.0) (* (* s_m c_m) (* s_m c_m))))

C

s_m = fabs(s);
c_m = fabs(c);
x_m = fabs(x);
assert(x_m < c_m && c_m < s_m);
double code(double x_m, double c_m, double s_m) {
	return -2.0 / ((s_m * c_m) * (s_m * c_m));
}

Fortran

s_m =     private
c_m =     private
x_m =     private
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
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, c_m, s_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    code = -2.0d0 / ((s_m * c_m) * (s_m * c_m))
end function

Java

s_m = Math.abs(s);
c_m = Math.abs(c);
x_m = Math.abs(x);
assert x_m < c_m && c_m < s_m;
public static double code(double x_m, double c_m, double s_m) {
	return -2.0 / ((s_m * c_m) * (s_m * c_m));
}

Python

s_m = math.fabs(s)
c_m = math.fabs(c)
x_m = math.fabs(x)
[x_m, c_m, s_m] = sort([x_m, c_m, s_m])
def code(x_m, c_m, s_m):
	return -2.0 / ((s_m * c_m) * (s_m * c_m))

Julia

s_m = abs(s)
c_m = abs(c)
x_m = abs(x)
x_m, c_m, s_m = sort([x_m, c_m, s_m])
function code(x_m, c_m, s_m)
	return Float64(Float64(-2.0) / Float64(Float64(s_m * c_m) * Float64(s_m * c_m)))
end

MATLAB

s_m = abs(s);
c_m = abs(c);
x_m = abs(x);
x_m, c_m, s_m = num2cell(sort([x_m, c_m, s_m])){:}
function tmp = code(x_m, c_m, s_m)
	tmp = -2.0 / ((s_m * c_m) * (s_m * c_m));
end

Wolfram

s_m = N[Abs[s], $MachinePrecision]
c_m = N[Abs[c], $MachinePrecision]
x_m = N[Abs[x], $MachinePrecision]
NOTE: x_m, c_m, and s_m should be sorted in increasing order before calling this function.
code[x$95$m_, c$95$m_, s$95$m_] := N[((-2.0) / N[(N[(s$95$m * c$95$m), $MachinePrecision] * N[(s$95$m * c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 66.8%

   \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

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

   1. associate-*r/ N/A

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

   2. div-add-rev N/A

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

   3. +-commutative N/A

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

   4. associate-/l/ N/A

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

   5. associate-*r* N/A

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

   6. lower-/.f64 N/A

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

   7. +-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. *-commutative N/A

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

   12. *-commutative N/A

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

   13. associate-*l* N/A

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

   14. *-commutative N/A

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

   15. unpow2 N/A

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

   16. associate-*l* N/A

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

   17. associate-*r* N/A

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

5. Applied rewrites 52.4%

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

6. Taylor expanded in x around inf

   \[\leadsto \frac{-2}{\color{blue}{{c}^{2} \cdot {s}^{2}}} \]
7. Step-by-step derivation

8. Applied rewrites 27.9%

   \[\leadsto \frac{\frac{\frac{-2}{s \cdot s}}{c}}{\color{blue}{c}} \]
9. Step-by-step derivation

10. Applied rewrites 27.5%

    \[\leadsto \frac{2}{\left(\left(-s\right) \cdot s\right) \cdot \color{blue}{\left(c \cdot c\right)}} \]
11. Step-by-step derivation

12. Applied rewrites 25.6%

    \[\leadsto \frac{2}{\left(s \cdot \left(-c\right)\right) \cdot \left(s \cdot \color{blue}{c}\right)} \]

13. Final simplification 25.6%

    \[\leadsto \frac{-2}{\left(s \cdot c\right) \cdot \left(s \cdot c\right)} \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2025003 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))