mixedcos

Percentage Accurate: 66.9% → 98.9%

Time: 19.2s

Alternatives: 13

Speedup: 17.1×

• 31.5% 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: 66.9% 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: 98.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ [x, c, s_m] = \mathsf{sort}([x, c, s_m])\\ \\ \begin{array}{l} t_0 := \left(s\_m \cdot x\right) \cdot c\\ t_1 := \cos \left(2 \cdot x\right)\\ t_2 := \left(c \cdot s\_m\right) \cdot x\\ \mathbf{if}\;\frac{t\_1}{{c}^{2} \cdot \left(\left(x \cdot {s\_m}^{2}\right) \cdot x\right)} \leq \infty:\\ \;\;\;\;\frac{t\_1}{t\_0 \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x + x\right)}{t\_2 \cdot t\_2}\\ \end{array} \end{array} \]

FPCore

s_m = (fabs.f64 s)
NOTE: x, c, and s_m should be sorted in increasing order before calling this function.
(FPCore (x c s_m)
 :precision binary64
 (let* ((t_0 (* (* s_m x) c)) (t_1 (cos (* 2.0 x))) (t_2 (* (* c s_m) x)))
   (if (<= (/ t_1 (* (pow c 2.0) (* (* x (pow s_m 2.0)) x))) INFINITY)
     (/ t_1 (* t_0 t_0))
     (/ (cos (+ x x)) (* t_2 t_2)))))

C

s_m = fabs(s);
assert(x < c && c < s_m);
double code(double x, double c, double s_m) {
	double t_0 = (s_m * x) * c;
	double t_1 = cos((2.0 * x));
	double t_2 = (c * s_m) * x;
	double tmp;
	if ((t_1 / (pow(c, 2.0) * ((x * pow(s_m, 2.0)) * x))) <= ((double) INFINITY)) {
		tmp = t_1 / (t_0 * t_0);
	} else {
		tmp = cos((x + x)) / (t_2 * t_2);
	}
	return tmp;
}

Java

s_m = Math.abs(s);
assert x < c && c < s_m;
public static double code(double x, double c, double s_m) {
	double t_0 = (s_m * x) * c;
	double t_1 = Math.cos((2.0 * x));
	double t_2 = (c * s_m) * x;
	double tmp;
	if ((t_1 / (Math.pow(c, 2.0) * ((x * Math.pow(s_m, 2.0)) * x))) <= Double.POSITIVE_INFINITY) {
		tmp = t_1 / (t_0 * t_0);
	} else {
		tmp = Math.cos((x + x)) / (t_2 * t_2);
	}
	return tmp;
}

Python

s_m = math.fabs(s)
[x, c, s_m] = sort([x, c, s_m])
def code(x, c, s_m):
	t_0 = (s_m * x) * c
	t_1 = math.cos((2.0 * x))
	t_2 = (c * s_m) * x
	tmp = 0
	if (t_1 / (math.pow(c, 2.0) * ((x * math.pow(s_m, 2.0)) * x))) <= math.inf:
		tmp = t_1 / (t_0 * t_0)
	else:
		tmp = math.cos((x + x)) / (t_2 * t_2)
	return tmp

Julia

s_m = abs(s)
x, c, s_m = sort([x, c, s_m])
function code(x, c, s_m)
	t_0 = Float64(Float64(s_m * x) * c)
	t_1 = cos(Float64(2.0 * x))
	t_2 = Float64(Float64(c * s_m) * x)
	tmp = 0.0
	if (Float64(t_1 / Float64((c ^ 2.0) * Float64(Float64(x * (s_m ^ 2.0)) * x))) <= Inf)
		tmp = Float64(t_1 / Float64(t_0 * t_0));
	else
		tmp = Float64(cos(Float64(x + x)) / Float64(t_2 * t_2));
	end
	return tmp
end

MATLAB

s_m = abs(s);
x, c, s_m = num2cell(sort([x, c, s_m])){:}
function tmp_2 = code(x, c, s_m)
	t_0 = (s_m * x) * c;
	t_1 = cos((2.0 * x));
	t_2 = (c * s_m) * x;
	tmp = 0.0;
	if ((t_1 / ((c ^ 2.0) * ((x * (s_m ^ 2.0)) * x))) <= Inf)
		tmp = t_1 / (t_0 * t_0);
	else
		tmp = cos((x + x)) / (t_2 * t_2);
	end
	tmp_2 = tmp;
end

Wolfram

s_m = N[Abs[s], $MachinePrecision]
NOTE: x, c, and s_m should be sorted in increasing order before calling this function.
code[x_, c_, s$95$m_] := Block[{t$95$0 = N[(N[(s$95$m * x), $MachinePrecision] * c), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(c * s$95$m), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[N[(t$95$1 / N[(N[Power[c, 2.0], $MachinePrecision] * N[(N[(x * N[Power[s$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(t$95$1 / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] / N[(t$95$2 * t$95$2), $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))) < +inf.0

   1. Initial program 84.7%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.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. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. unpow2 N/A

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

      9. associate-*r* N/A

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

      10. pow-prod-down N/A

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

      11. pow-prod-down N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 97.3

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

   4. Applied rewrites 97.3%

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

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. unpow-prod-down N/A

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

      6. lift-*.f64 N/A

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

      7. unpow-prod-down N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. unpow-prod-down N/A

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

      12. pow2 N/A

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

      13. sqr-neg-rev N/A

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

      14. pow2 N/A

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

      15. unswap-sqr N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 N/A

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

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

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

      19. lower-*.f64 N/A

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

      20. lower-neg.f64 N/A

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

      21. lower-*.f64 N/A

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

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

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

      23. lower-*.f64 N/A

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

      24. lower-neg.f64 99.3

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

   6. Applied rewrites 99.3%

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

   if +inf.0 < (/.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 0.0%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.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. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. unpow2 N/A

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

      9. associate-*r* N/A

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

      10. pow-prod-down N/A

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

      11. pow-prod-down N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 96.5

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

   4. Applied rewrites 96.5%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 96.4

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 96.4

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 96.4

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

   6. Applied rewrites 96.4%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 96.4

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

   8. Applied rewrites 96.4%

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

4. Final simplification 98.5%

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

Alternative 2: 82.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ [x, c, s_m] = \mathsf{sort}([x, c, s_m])\\ \\ \begin{array}{l} t_0 := \left(c \cdot s\_m\right) \cdot x\\ t_1 := \frac{\frac{-1}{c}}{s\_m \cdot x}\\ \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s\_m}^{2}\right) \cdot x\right)} \leq -4 \cdot 10^{-119}:\\ \;\;\;\;\frac{-2 \cdot \left(x \cdot x\right) - -1}{t\_0 \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot t\_1\\ \end{array} \end{array} \]

FPCore

s_m = (fabs.f64 s)
NOTE: x, c, and s_m should be sorted in increasing order before calling this function.
(FPCore (x c s_m)
 :precision binary64
 (let* ((t_0 (* (* c s_m) x)) (t_1 (/ (/ -1.0 c) (* s_m x))))
   (if (<=
        (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s_m 2.0)) x)))
        -4e-119)
     (/ (- (* -2.0 (* x x)) -1.0) (* t_0 t_0))
     (* t_1 t_1))))

C

s_m = fabs(s);
assert(x < c && c < s_m);
double code(double x, double c, double s_m) {
	double t_0 = (c * s_m) * x;
	double t_1 = (-1.0 / c) / (s_m * x);
	double tmp;
	if ((cos((2.0 * x)) / (pow(c, 2.0) * ((x * pow(s_m, 2.0)) * x))) <= -4e-119) {
		tmp = ((-2.0 * (x * x)) - -1.0) / (t_0 * t_0);
	} else {
		tmp = t_1 * t_1;
	}
	return tmp;
}

Fortran

s_m =     private
NOTE: x, c, 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, c, s_m)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (c * s_m) * x
    t_1 = ((-1.0d0) / c) / (s_m * x)
    if ((cos((2.0d0 * x)) / ((c ** 2.0d0) * ((x * (s_m ** 2.0d0)) * x))) <= (-4d-119)) then
        tmp = (((-2.0d0) * (x * x)) - (-1.0d0)) / (t_0 * t_0)
    else
        tmp = t_1 * t_1
    end if
    code = tmp
end function

Java

s_m = Math.abs(s);
assert x < c && c < s_m;
public static double code(double x, double c, double s_m) {
	double t_0 = (c * s_m) * x;
	double t_1 = (-1.0 / c) / (s_m * x);
	double tmp;
	if ((Math.cos((2.0 * x)) / (Math.pow(c, 2.0) * ((x * Math.pow(s_m, 2.0)) * x))) <= -4e-119) {
		tmp = ((-2.0 * (x * x)) - -1.0) / (t_0 * t_0);
	} else {
		tmp = t_1 * t_1;
	}
	return tmp;
}

Python

s_m = math.fabs(s)
[x, c, s_m] = sort([x, c, s_m])
def code(x, c, s_m):
	t_0 = (c * s_m) * x
	t_1 = (-1.0 / c) / (s_m * x)
	tmp = 0
	if (math.cos((2.0 * x)) / (math.pow(c, 2.0) * ((x * math.pow(s_m, 2.0)) * x))) <= -4e-119:
		tmp = ((-2.0 * (x * x)) - -1.0) / (t_0 * t_0)
	else:
		tmp = t_1 * t_1
	return tmp

Julia

s_m = abs(s)
x, c, s_m = sort([x, c, s_m])
function code(x, c, s_m)
	t_0 = Float64(Float64(c * s_m) * x)
	t_1 = Float64(Float64(-1.0 / c) / Float64(s_m * x))
	tmp = 0.0
	if (Float64(cos(Float64(2.0 * x)) / Float64((c ^ 2.0) * Float64(Float64(x * (s_m ^ 2.0)) * x))) <= -4e-119)
		tmp = Float64(Float64(Float64(-2.0 * Float64(x * x)) - -1.0) / Float64(t_0 * t_0));
	else
		tmp = Float64(t_1 * t_1);
	end
	return tmp
end

MATLAB

s_m = abs(s);
x, c, s_m = num2cell(sort([x, c, s_m])){:}
function tmp_2 = code(x, c, s_m)
	t_0 = (c * s_m) * x;
	t_1 = (-1.0 / c) / (s_m * x);
	tmp = 0.0;
	if ((cos((2.0 * x)) / ((c ^ 2.0) * ((x * (s_m ^ 2.0)) * x))) <= -4e-119)
		tmp = ((-2.0 * (x * x)) - -1.0) / (t_0 * t_0);
	else
		tmp = t_1 * t_1;
	end
	tmp_2 = tmp;
end

Wolfram

s_m = N[Abs[s], $MachinePrecision]
NOTE: x, c, and s_m should be sorted in increasing order before calling this function.
code[x_, c_, s$95$m_] := Block[{t$95$0 = N[(N[(c * s$95$m), $MachinePrecision] * x), $MachinePrecision]}, Block[{t$95$1 = N[(N[(-1.0 / c), $MachinePrecision] / N[(s$95$m * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] / N[(N[Power[c, 2.0], $MachinePrecision] * N[(N[(x * N[Power[s$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -4e-119], N[(N[(N[(-2.0 * N[(x * x), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * t$95$1), $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))) < -4.00000000000000005e-119

   1. Initial program 76.4%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   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-pow.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. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. unpow2 N/A

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

      9. associate-*r* N/A

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

      10. pow-prod-down N/A

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

      11. pow-prod-down N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 99.2

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

   4. Applied rewrites 99.2%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 99.2

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 99.2

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 99.2

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

   6. Applied rewrites 99.2%

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

   7. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 52.5

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

   9. Applied rewrites 52.5%

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

   if -4.00000000000000005e-119 < (/.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 61.5%

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

   3. Taylor expanded in x around 0

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

      1. associate-/r* N/A

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

      2. metadata-eval N/A

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

      3. unpow2 N/A

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

      4. times-frac N/A

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

      5. pow-prod-down N/A

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

      6. unpow2 N/A

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

      7. lower-ratio-of-squares.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 59.2

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

   5. Applied rewrites 59.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-ratio-of-squares.f64 N/A

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

      4. sqr-neg-rev N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. distribute-neg-frac N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. distribute-neg-frac N/A

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

      14. metadata-eval N/A

          \[\leadsto \frac{\frac{-1}{c}}{s \cdot x} \cdot \frac{\frac{-1}{c}}{s \cdot x} \]

      15. lower-/.f64 N/A

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

      16. lift-*.f64 81.1

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

   7. Applied rewrites 81.1%

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

Alternative 3: 81.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ [x, c, s_m] = \mathsf{sort}([x, c, s_m])\\ \\ \begin{array}{l} t_0 := \left(c \cdot s\_m\right) \cdot x\\ \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s\_m}^{2}\right) \cdot x\right)} \leq -4 \cdot 10^{-119}:\\ \;\;\;\;\frac{-2 \cdot \left(x \cdot x\right) - -1}{t\_0 \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{t\_0}}{t\_0}\\ \end{array} \end{array} \]

FPCore

s_m = (fabs.f64 s)
NOTE: x, c, and s_m should be sorted in increasing order before calling this function.
(FPCore (x c s_m)
 :precision binary64
 (let* ((t_0 (* (* c s_m) x)))
   (if (<=
        (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s_m 2.0)) x)))
        -4e-119)
     (/ (- (* -2.0 (* x x)) -1.0) (* t_0 t_0))
     (/ (/ 1.0 t_0) t_0))))

C

s_m = fabs(s);
assert(x < c && c < s_m);
double code(double x, double c, double s_m) {
	double t_0 = (c * s_m) * x;
	double tmp;
	if ((cos((2.0 * x)) / (pow(c, 2.0) * ((x * pow(s_m, 2.0)) * x))) <= -4e-119) {
		tmp = ((-2.0 * (x * x)) - -1.0) / (t_0 * t_0);
	} else {
		tmp = (1.0 / t_0) / t_0;
	}
	return tmp;
}

Fortran

s_m =     private
NOTE: x, c, 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, c, s_m)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (c * s_m) * x
    if ((cos((2.0d0 * x)) / ((c ** 2.0d0) * ((x * (s_m ** 2.0d0)) * x))) <= (-4d-119)) then
        tmp = (((-2.0d0) * (x * x)) - (-1.0d0)) / (t_0 * t_0)
    else
        tmp = (1.0d0 / t_0) / t_0
    end if
    code = tmp
end function

Java

s_m = Math.abs(s);
assert x < c && c < s_m;
public static double code(double x, double c, double s_m) {
	double t_0 = (c * s_m) * x;
	double tmp;
	if ((Math.cos((2.0 * x)) / (Math.pow(c, 2.0) * ((x * Math.pow(s_m, 2.0)) * x))) <= -4e-119) {
		tmp = ((-2.0 * (x * x)) - -1.0) / (t_0 * t_0);
	} else {
		tmp = (1.0 / t_0) / t_0;
	}
	return tmp;
}

Python

s_m = math.fabs(s)
[x, c, s_m] = sort([x, c, s_m])
def code(x, c, s_m):
	t_0 = (c * s_m) * x
	tmp = 0
	if (math.cos((2.0 * x)) / (math.pow(c, 2.0) * ((x * math.pow(s_m, 2.0)) * x))) <= -4e-119:
		tmp = ((-2.0 * (x * x)) - -1.0) / (t_0 * t_0)
	else:
		tmp = (1.0 / t_0) / t_0
	return tmp

Julia

s_m = abs(s)
x, c, s_m = sort([x, c, s_m])
function code(x, c, s_m)
	t_0 = Float64(Float64(c * s_m) * x)
	tmp = 0.0
	if (Float64(cos(Float64(2.0 * x)) / Float64((c ^ 2.0) * Float64(Float64(x * (s_m ^ 2.0)) * x))) <= -4e-119)
		tmp = Float64(Float64(Float64(-2.0 * Float64(x * x)) - -1.0) / Float64(t_0 * t_0));
	else
		tmp = Float64(Float64(1.0 / t_0) / t_0);
	end
	return tmp
end

MATLAB

s_m = abs(s);
x, c, s_m = num2cell(sort([x, c, s_m])){:}
function tmp_2 = code(x, c, s_m)
	t_0 = (c * s_m) * x;
	tmp = 0.0;
	if ((cos((2.0 * x)) / ((c ^ 2.0) * ((x * (s_m ^ 2.0)) * x))) <= -4e-119)
		tmp = ((-2.0 * (x * x)) - -1.0) / (t_0 * t_0);
	else
		tmp = (1.0 / t_0) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

s_m = N[Abs[s], $MachinePrecision]
NOTE: x, c, and s_m should be sorted in increasing order before calling this function.
code[x_, c_, s$95$m_] := Block[{t$95$0 = N[(N[(c * s$95$m), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[N[(N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] / N[(N[Power[c, 2.0], $MachinePrecision] * N[(N[(x * N[Power[s$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -4e-119], N[(N[(N[(-2.0 * N[(x * x), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / t$95$0), $MachinePrecision] / t$95$0), $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))) < -4.00000000000000005e-119

   1. Initial program 76.4%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   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-pow.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. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. unpow2 N/A

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

      9. associate-*r* N/A

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

      10. pow-prod-down N/A

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

      11. pow-prod-down N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 99.2

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

   4. Applied rewrites 99.2%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 99.2

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 99.2

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 99.2

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

   6. Applied rewrites 99.2%

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

   7. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 52.5

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

   9. Applied rewrites 52.5%

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

   if -4.00000000000000005e-119 < (/.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 61.5%

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

   3. Taylor expanded in x around 0

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

      1. associate-/r* N/A

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

      2. metadata-eval N/A

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

      3. unpow2 N/A

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

      4. times-frac N/A

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

      5. pow-prod-down N/A

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

      6. unpow2 N/A

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

      7. lower-ratio-of-squares.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 59.2

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

   5. Applied rewrites 59.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-ratio-of-squares.f64 N/A

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

      4. frac-times N/A

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

      5. metadata-eval N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. unpow-prod-down N/A

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

      9. associate-/r* N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. unpow-prod-down N/A

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

      13. lift-*.f64 N/A

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

      14. unpow-prod-down N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. unpow2 N/A

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

      19. associate-/r* N/A

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

   7. Applied rewrites 80.3%

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

Alternative 4: 97.3% accurate, 2.3× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ [x, c, s_m] = \mathsf{sort}([x, c, s_m])\\ \\ \begin{array}{l} t_0 := \left(c \cdot s\_m\right) \cdot x\\ \frac{\frac{\cos \left(-2 \cdot x\right)}{t\_0}}{t\_0} \end{array} \end{array} \]

FPCore

s_m = (fabs.f64 s)
NOTE: x, c, and s_m should be sorted in increasing order before calling this function.
(FPCore (x c s_m)
 :precision binary64
 (let* ((t_0 (* (* c s_m) x))) (/ (/ (cos (* -2.0 x)) t_0) t_0)))

C

s_m = fabs(s);
assert(x < c && c < s_m);
double code(double x, double c, double s_m) {
	double t_0 = (c * s_m) * x;
	return (cos((-2.0 * x)) / t_0) / t_0;
}

Fortran

s_m =     private
NOTE: x, c, 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, c, s_m)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s_m
    real(8) :: t_0
    t_0 = (c * s_m) * x
    code = (cos(((-2.0d0) * x)) / t_0) / t_0
end function

Java

s_m = Math.abs(s);
assert x < c && c < s_m;
public static double code(double x, double c, double s_m) {
	double t_0 = (c * s_m) * x;
	return (Math.cos((-2.0 * x)) / t_0) / t_0;
}

Python

s_m = math.fabs(s)
[x, c, s_m] = sort([x, c, s_m])
def code(x, c, s_m):
	t_0 = (c * s_m) * x
	return (math.cos((-2.0 * x)) / t_0) / t_0

Julia

s_m = abs(s)
x, c, s_m = sort([x, c, s_m])
function code(x, c, s_m)
	t_0 = Float64(Float64(c * s_m) * x)
	return Float64(Float64(cos(Float64(-2.0 * x)) / t_0) / t_0)
end

MATLAB

s_m = abs(s);
x, c, s_m = num2cell(sort([x, c, s_m])){:}
function tmp = code(x, c, s_m)
	t_0 = (c * s_m) * x;
	tmp = (cos((-2.0 * x)) / t_0) / t_0;
end

Wolfram

s_m = N[Abs[s], $MachinePrecision]
NOTE: x, c, and s_m should be sorted in increasing order before calling this function.
code[x_, c_, s$95$m_] := Block[{t$95$0 = N[(N[(c * s$95$m), $MachinePrecision] * x), $MachinePrecision]}, N[(N[(N[Cos[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]]

Derivation

1. Initial program 62.5%

   \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
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-pow.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. lift-*.f64 N/A

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

   5. lift-pow.f64 N/A

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

   6. *-commutative N/A

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

   7. associate-*r* N/A

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

   8. unpow2 N/A

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

   9. associate-*r* N/A

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

   10. pow-prod-down N/A

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

   11. pow-prod-down N/A

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

   12. lower-pow.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 97.1

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

4. Applied rewrites 97.1%

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

   1. lift-*.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. metadata-eval N/A

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

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

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

   5. cos-neg-rev N/A

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

   6. lift-pow.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. unpow-prod-down N/A

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

   11. lift-*.f64 N/A

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

   12. unpow-prod-down N/A

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

   13. *-commutative N/A

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

   14. associate-*r* N/A

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

   15. lower-/.f64 N/A

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

   16. *-commutative N/A

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

   17. *-commutative N/A

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

   18. associate-*r* N/A

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

6. Applied rewrites 97.4%

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

Alternative 5: 97.0% accurate, 2.4× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ [x, c, s_m] = \mathsf{sort}([x, c, s_m])\\ \\ \begin{array}{l} t_0 := \left(c \cdot s\_m\right) \cdot x\\ \frac{\cos \left(x + x\right)}{t\_0 \cdot t\_0} \end{array} \end{array} \]

FPCore

s_m = (fabs.f64 s)
NOTE: x, c, and s_m should be sorted in increasing order before calling this function.
(FPCore (x c s_m)
 :precision binary64
 (let* ((t_0 (* (* c s_m) x))) (/ (cos (+ x x)) (* t_0 t_0))))

C

s_m = fabs(s);
assert(x < c && c < s_m);
double code(double x, double c, double s_m) {
	double t_0 = (c * s_m) * x;
	return cos((x + x)) / (t_0 * t_0);
}

Fortran

s_m =     private
NOTE: x, c, 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, c, s_m)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s_m
    real(8) :: t_0
    t_0 = (c * s_m) * x
    code = cos((x + x)) / (t_0 * t_0)
end function

Java

s_m = Math.abs(s);
assert x < c && c < s_m;
public static double code(double x, double c, double s_m) {
	double t_0 = (c * s_m) * x;
	return Math.cos((x + x)) / (t_0 * t_0);
}

Python

s_m = math.fabs(s)
[x, c, s_m] = sort([x, c, s_m])
def code(x, c, s_m):
	t_0 = (c * s_m) * x
	return math.cos((x + x)) / (t_0 * t_0)

Julia

s_m = abs(s)
x, c, s_m = sort([x, c, s_m])
function code(x, c, s_m)
	t_0 = Float64(Float64(c * s_m) * x)
	return Float64(cos(Float64(x + x)) / Float64(t_0 * t_0))
end

MATLAB

s_m = abs(s);
x, c, s_m = num2cell(sort([x, c, s_m])){:}
function tmp = code(x, c, s_m)
	t_0 = (c * s_m) * x;
	tmp = cos((x + x)) / (t_0 * t_0);
end

Wolfram

s_m = N[Abs[s], $MachinePrecision]
NOTE: x, c, and s_m should be sorted in increasing order before calling this function.
code[x_, c_, s$95$m_] := Block[{t$95$0 = N[(N[(c * s$95$m), $MachinePrecision] * x), $MachinePrecision]}, N[(N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 62.5%

   \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
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-pow.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. lift-*.f64 N/A

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

   5. lift-pow.f64 N/A

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

   6. *-commutative N/A

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

   7. associate-*r* N/A

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

   8. unpow2 N/A

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

   9. associate-*r* N/A

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

   10. pow-prod-down N/A

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

   11. pow-prod-down N/A

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

   12. lower-pow.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 97.1

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

4. Applied rewrites 97.1%

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

   1. lift-pow.f64 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f64 97.1

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 97.1

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 97.1

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

6. Applied rewrites 97.1%

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

   1. lift-*.f64 N/A

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

   2. count-2-rev N/A

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

   3. lower-+.f64 97.1

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

8. Applied rewrites 97.1%

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

Alternative 6: 77.2% accurate, 2.4× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ [x, c, s_m] = \mathsf{sort}([x, c, s_m])\\ \\ \begin{array}{l} \mathbf{if}\;{s\_m}^{2} \leq 10^{-122}:\\ \;\;\;\;\frac{\mathsf{ratio\_of\_squares}\left(\left(\frac{-1}{c}\right), s\_m\right)}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{ratio\_of\_squares}\left(\left(\frac{1}{s\_m \cdot x}\right), c\right)\\ \end{array} \end{array} \]

FPCore

s_m = (fabs.f64 s)
NOTE: x, c, and s_m should be sorted in increasing order before calling this function.
(FPCore (x c s_m)
 :precision binary64
 (if (<= (pow s_m 2.0) 1e-122)
   (/ (ratio-of-squares (/ -1.0 c) s_m) (* x x))
   (ratio-of-squares (/ 1.0 (* s_m x)) c)))

Derivation

1. Split input into 2 regimes

2. if (pow.f64 s #s(literal 2 binary64)) < 1.00000000000000006e-122

   1. Initial program 62.9%

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

   3. Taylor expanded in x around 0

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

      1. associate-/r* N/A

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

      2. metadata-eval N/A

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

      3. unpow2 N/A

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

      4. times-frac N/A

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

      5. pow-prod-down N/A

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

      6. unpow2 N/A

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

      7. lower-ratio-of-squares.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 42.2

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

   5. Applied rewrites 42.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-ratio-of-squares.f64 N/A

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

      4. frac-times N/A

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

      5. metadata-eval N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. unpow-prod-down N/A

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

      9. associate-/r* N/A

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

      10. associate-/r* N/A

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

      11. metadata-eval N/A

          \[\leadsto \frac{\frac{1 \cdot 1}{{c}^{2} \cdot {s}^{2}}}{{x}^{2}} \]

      12. *-commutative N/A

          \[\leadsto \frac{\frac{1 \cdot 1}{{s}^{2} \cdot {c}^{2}}}{{x}^{2}} \]

      13. unpow-prod-down N/A

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

      14. lift-*.f64 N/A

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

      15. unpow2 N/A

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

      16. times-frac N/A

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

      17. pow2 N/A

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

      18. lower-ratio-of-squares.f64 N/A

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

      19. lower-/.f64 51.8

          \[\leadsto \mathsf{ratio\_of\_squares}\left(\left(\frac{1}{s \cdot c}\right), x\right) \]

      20. lift-*.f64 N/A

          \[\leadsto \mathsf{ratio\_of\_squares}\left(\left(\frac{1}{s \cdot c}\right), x\right) \]

      21. *-commutative N/A

          \[\leadsto \mathsf{ratio\_of\_squares}\left(\left(\frac{1}{c \cdot s}\right), x\right) \]

      22. lower-*.f64 51.8

          \[\leadsto \mathsf{ratio\_of\_squares}\left(\left(\frac{1}{c \cdot s}\right), x\right) \]

   7. Applied rewrites 51.8%

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

      1. lift-ratio-of-squares.f64 N/A

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

      2. lift-*.f64 N/A

         \[\leadsto \frac{\frac{1}{c \cdot s} \cdot \frac{1}{c \cdot s}}{x \cdot x} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{\frac{1}{c \cdot s} \cdot \frac{1}{c \cdot s}}{x \cdot x} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\frac{1}{c \cdot s} \cdot \frac{1}{c \cdot s}}{x \cdot x} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{\frac{1}{c \cdot s} \cdot \frac{1}{c \cdot s}}{x \cdot x} \]

      6. frac-times N/A

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

      7. metadata-eval N/A

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

      8. pow2 N/A

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

      9. pow-prod-down N/A

         \[\leadsto \frac{\frac{1}{{c}^{2} \cdot {s}^{2}}}{x \cdot x} \]

      10. pow2 N/A

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

      11. lower-/.f64 N/A

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

   9. Applied rewrites 62.8%

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

   if 1.00000000000000006e-122 < (pow.f64 s #s(literal 2 binary64))

   1. Initial program 62.3%

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

   3. Taylor expanded in x around 0

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

      1. associate-/r* N/A

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

      2. metadata-eval N/A

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

      3. unpow2 N/A

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

      4. times-frac N/A

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

      5. pow-prod-down N/A

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

      6. unpow2 N/A

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

      7. lower-ratio-of-squares.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 64.0

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

   5. Applied rewrites 64.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-ratio-of-squares.f64 N/A

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

      4. frac-times N/A

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

      5. metadata-eval N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. unpow-prod-down N/A

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

      9. associate-/r* N/A

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

      10. *-commutative N/A

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

      11. associate-/r* N/A

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

      12. metadata-eval N/A

          \[\leadsto \frac{\frac{1 \cdot 1}{{s}^{2} \cdot {x}^{2}}}{{c}^{2}} \]

      13. unpow-prod-down N/A

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

      14. pow2 N/A

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

      15. times-frac N/A

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

      16. pow2 N/A

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

      17. lower-ratio-of-squares.f64 N/A

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

      18. lower-/.f64 N/A

          \[\leadsto \mathsf{ratio\_of\_squares}\left(\left(\frac{1}{s \cdot x}\right), c\right) \]

      19. lift-*.f64 79.2

          \[\leadsto \mathsf{ratio\_of\_squares}\left(\left(\frac{1}{s \cdot x}\right), c\right) \]

   7. Applied rewrites 79.2%

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

Alternative 7: 71.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ [x, c, s_m] = \mathsf{sort}([x, c, s_m])\\ \\ \begin{array}{l} \mathbf{if}\;{s\_m}^{2} \leq 2 \cdot 10^{-144}:\\ \;\;\;\;\mathsf{ratio\_of\_squares}\left(\left(\frac{1}{c}\right), \left(s\_m \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{ratio\_of\_squares}\left(\left(\frac{1}{c \cdot s\_m}\right), x\right)\\ \end{array} \end{array} \]

FPCore

s_m = (fabs.f64 s)
NOTE: x, c, and s_m should be sorted in increasing order before calling this function.
(FPCore (x c s_m)
 :precision binary64
 (if (<= (pow s_m 2.0) 2e-144)
   (ratio-of-squares (/ 1.0 c) (* s_m x))
   (ratio-of-squares (/ 1.0 (* c s_m)) x)))

Derivation

1. Split input into 2 regimes

2. if (pow.f64 s #s(literal 2 binary64)) < 1.9999999999999999e-144

   1. Initial program 61.4%

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

   3. Taylor expanded in x around 0

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

      1. associate-/r* N/A

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

      2. metadata-eval N/A

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

      3. unpow2 N/A

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

      4. times-frac N/A

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

      5. pow-prod-down N/A

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

      6. unpow2 N/A

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

      7. lower-ratio-of-squares.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 39.9

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

   5. Applied rewrites 39.9%

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

   if 1.9999999999999999e-144 < (pow.f64 s #s(literal 2 binary64))

   1. Initial program 63.3%

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

   3. Taylor expanded in x around 0

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

      1. associate-/r* N/A

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

      2. metadata-eval N/A

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

      3. unpow2 N/A

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

      4. times-frac N/A

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

      5. pow-prod-down N/A

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

      6. unpow2 N/A

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

      7. lower-ratio-of-squares.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 64.9

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

   5. Applied rewrites 64.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-ratio-of-squares.f64 N/A

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

      4. frac-times N/A

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

      5. metadata-eval N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. unpow-prod-down N/A

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

      9. associate-/r* N/A

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

      10. associate-/r* N/A

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

      11. metadata-eval N/A

          \[\leadsto \frac{\frac{1 \cdot 1}{{c}^{2} \cdot {s}^{2}}}{{x}^{2}} \]

      12. *-commutative N/A

          \[\leadsto \frac{\frac{1 \cdot 1}{{s}^{2} \cdot {c}^{2}}}{{x}^{2}} \]

      13. unpow-prod-down N/A

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

      14. lift-*.f64 N/A

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

      15. unpow2 N/A

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

      16. times-frac N/A

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

      17. pow2 N/A

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

      18. lower-ratio-of-squares.f64 N/A

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

      19. lower-/.f64 77.1

          \[\leadsto \mathsf{ratio\_of\_squares}\left(\left(\frac{1}{s \cdot c}\right), x\right) \]

      20. lift-*.f64 N/A

          \[\leadsto \mathsf{ratio\_of\_squares}\left(\left(\frac{1}{s \cdot c}\right), x\right) \]

      21. *-commutative N/A

          \[\leadsto \mathsf{ratio\_of\_squares}\left(\left(\frac{1}{c \cdot s}\right), x\right) \]

      22. lower-*.f64 77.1

          \[\leadsto \mathsf{ratio\_of\_squares}\left(\left(\frac{1}{c \cdot s}\right), x\right) \]

   7. Applied rewrites 77.1%

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

Alternative 8: 78.4% accurate, 7.8× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ [x, c, s_m] = \mathsf{sort}([x, c, s_m])\\ \\ \begin{array}{l} t_0 := \left(c \cdot s\_m\right) \cdot x\\ \frac{\frac{1}{t\_0}}{t\_0} \end{array} \end{array} \]

FPCore

s_m = (fabs.f64 s)
NOTE: x, c, and s_m should be sorted in increasing order before calling this function.
(FPCore (x c s_m)
 :precision binary64
 (let* ((t_0 (* (* c s_m) x))) (/ (/ 1.0 t_0) t_0)))

C

s_m = fabs(s);
assert(x < c && c < s_m);
double code(double x, double c, double s_m) {
	double t_0 = (c * s_m) * x;
	return (1.0 / t_0) / t_0;
}

Fortran

s_m =     private
NOTE: x, c, 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, c, s_m)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s_m
    real(8) :: t_0
    t_0 = (c * s_m) * x
    code = (1.0d0 / t_0) / t_0
end function

Java

s_m = Math.abs(s);
assert x < c && c < s_m;
public static double code(double x, double c, double s_m) {
	double t_0 = (c * s_m) * x;
	return (1.0 / t_0) / t_0;
}

Python

s_m = math.fabs(s)
[x, c, s_m] = sort([x, c, s_m])
def code(x, c, s_m):
	t_0 = (c * s_m) * x
	return (1.0 / t_0) / t_0

Julia

s_m = abs(s)
x, c, s_m = sort([x, c, s_m])
function code(x, c, s_m)
	t_0 = Float64(Float64(c * s_m) * x)
	return Float64(Float64(1.0 / t_0) / t_0)
end

MATLAB

s_m = abs(s);
x, c, s_m = num2cell(sort([x, c, s_m])){:}
function tmp = code(x, c, s_m)
	t_0 = (c * s_m) * x;
	tmp = (1.0 / t_0) / t_0;
end

Wolfram

s_m = N[Abs[s], $MachinePrecision]
NOTE: x, c, and s_m should be sorted in increasing order before calling this function.
code[x_, c_, s$95$m_] := Block[{t$95$0 = N[(N[(c * s$95$m), $MachinePrecision] * x), $MachinePrecision]}, N[(N[(1.0 / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]]

Derivation

1. Initial program 62.5%

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

3. Taylor expanded in x around 0

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

   1. associate-/r* N/A

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

   2. metadata-eval N/A

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

   3. unpow2 N/A

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

   4. times-frac N/A

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

   5. pow-prod-down N/A

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

   6. unpow2 N/A

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

   7. lower-ratio-of-squares.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-*.f64 55.1

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

5. Applied rewrites 55.1%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-ratio-of-squares.f64 N/A

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

   4. frac-times N/A

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

   5. metadata-eval N/A

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

   6. pow2 N/A

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

   7. pow2 N/A

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

   8. unpow-prod-down N/A

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

   9. associate-/r* N/A

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

   10. associate-*r* N/A

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

   11. *-commutative N/A

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

   12. unpow-prod-down N/A

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

   13. lift-*.f64 N/A

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

   14. unpow-prod-down N/A

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

   15. lift-*.f64 N/A

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

   16. lift-*.f64 N/A

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

   17. lift-*.f64 N/A

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

   18. unpow2 N/A

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

   19. associate-/r* N/A

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

7. Applied rewrites 74.7%

   \[\leadsto \frac{\frac{1}{\left(c \cdot s\right) \cdot x}}{\color{blue}{\left(c \cdot s\right) \cdot x}} \]
8. Add Preprocessing

Alternative 9: 78.5% accurate, 7.8× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ [x, c, s_m] = \mathsf{sort}([x, c, s_m])\\ \\ \begin{array}{l} \mathbf{if}\;s\_m \leq 1.65 \cdot 10^{+62}:\\ \;\;\;\;\frac{\frac{\mathsf{ratio\_of\_squares}\left(\left(\frac{-1}{c}\right), x\right)}{s\_m}}{s\_m}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{ratio\_of\_squares}\left(\left(\frac{1}{s\_m \cdot x}\right), c\right)\\ \end{array} \end{array} \]

FPCore

s_m = (fabs.f64 s)
NOTE: x, c, and s_m should be sorted in increasing order before calling this function.
(FPCore (x c s_m)
 :precision binary64
 (if (<= s_m 1.65e+62)
   (/ (/ (ratio-of-squares (/ -1.0 c) x) s_m) s_m)
   (ratio-of-squares (/ 1.0 (* s_m x)) c)))

Derivation

1. Split input into 2 regimes

2. if s < 1.65e62

   1. Initial program 61.5%

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

   3. Taylor expanded in x around 0

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

      1. associate-/r* N/A

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

      2. metadata-eval N/A

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

      3. unpow2 N/A

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

      4. times-frac N/A

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

      5. pow-prod-down N/A

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

      6. unpow2 N/A

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

      7. lower-ratio-of-squares.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 52.5

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

   5. Applied rewrites 52.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-ratio-of-squares.f64 N/A

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

      4. frac-times N/A

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

      5. metadata-eval N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. unpow-prod-down N/A

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

      9. associate-/r* N/A

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

      10. associate-/r* N/A

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

      11. metadata-eval N/A

          \[\leadsto \frac{\frac{1 \cdot 1}{{c}^{2} \cdot {s}^{2}}}{{x}^{2}} \]

      12. *-commutative N/A

          \[\leadsto \frac{\frac{1 \cdot 1}{{s}^{2} \cdot {c}^{2}}}{{x}^{2}} \]

      13. unpow-prod-down N/A

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

      14. lift-*.f64 N/A

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

      15. unpow2 N/A

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

      16. times-frac N/A

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

      17. pow2 N/A

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

      18. lower-ratio-of-squares.f64 N/A

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

      19. lower-/.f64 63.1

          \[\leadsto \mathsf{ratio\_of\_squares}\left(\left(\frac{1}{s \cdot c}\right), x\right) \]

      20. lift-*.f64 N/A

          \[\leadsto \mathsf{ratio\_of\_squares}\left(\left(\frac{1}{s \cdot c}\right), x\right) \]

      21. *-commutative N/A

          \[\leadsto \mathsf{ratio\_of\_squares}\left(\left(\frac{1}{c \cdot s}\right), x\right) \]

      22. lower-*.f64 63.1

          \[\leadsto \mathsf{ratio\_of\_squares}\left(\left(\frac{1}{c \cdot s}\right), x\right) \]

   7. Applied rewrites 63.1%

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

      1. lift-ratio-of-squares.f64 N/A

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

      2. lift-*.f64 N/A

         \[\leadsto \frac{\frac{1}{c \cdot s} \cdot \frac{1}{c \cdot s}}{x \cdot x} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{\frac{1}{c \cdot s} \cdot \frac{1}{c \cdot s}}{x \cdot x} \]

      4. associate-/r* N/A

         \[\leadsto \frac{\frac{\frac{1}{c}}{s} \cdot \frac{1}{c \cdot s}}{x \cdot x} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\frac{1}{c}}{s} \cdot \frac{1}{c \cdot s}}{x \cdot x} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{\frac{\frac{1}{c}}{s} \cdot \frac{1}{c \cdot s}}{x \cdot x} \]

      7. associate-/r* N/A

         \[\leadsto \frac{\frac{\frac{1}{c}}{s} \cdot \frac{\frac{1}{c}}{s}}{x \cdot x} \]

      8. frac-times N/A

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

      9. times-frac N/A

         \[\leadsto \frac{\frac{\frac{1 \cdot 1}{c \cdot c}}{s \cdot s}}{x \cdot x} \]

      10. metadata-eval N/A

          \[\leadsto \frac{\frac{\frac{1}{c \cdot c}}{s \cdot s}}{x \cdot x} \]

      11. pow2 N/A

          \[\leadsto \frac{\frac{\frac{1}{{c}^{2}}}{s \cdot s}}{x \cdot x} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\frac{1}{{c}^{2}}}{s \cdot s}}{x \cdot x} \]

      13. pow2 N/A

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

      14. associate-/r* N/A

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

      15. *-commutative N/A

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

      16. associate-/r* N/A

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

   9. Applied rewrites 66.1%

      \[\leadsto \frac{\frac{\mathsf{ratio\_of\_squares}\left(\left(\frac{-1}{c}\right), x\right)}{s}}{\color{blue}{s}} \]

   if 1.65e62 < s

   1. Initial program 68.1%

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

   3. Taylor expanded in x around 0

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

      1. associate-/r* N/A

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

      2. metadata-eval N/A

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

      3. unpow2 N/A

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

      4. times-frac N/A

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

      5. pow-prod-down N/A

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

      6. unpow2 N/A

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

      7. lower-ratio-of-squares.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 69.2

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

   5. Applied rewrites 69.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-ratio-of-squares.f64 N/A

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

      4. frac-times N/A

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

      5. metadata-eval N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. unpow-prod-down N/A

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

      9. associate-/r* N/A

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

      10. *-commutative N/A

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

      11. associate-/r* N/A

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

      12. metadata-eval N/A

          \[\leadsto \frac{\frac{1 \cdot 1}{{s}^{2} \cdot {x}^{2}}}{{c}^{2}} \]

      13. unpow-prod-down N/A

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

      14. pow2 N/A

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

      15. times-frac N/A

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

      16. pow2 N/A

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

      17. lower-ratio-of-squares.f64 N/A

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

      18. lower-/.f64 N/A

          \[\leadsto \mathsf{ratio\_of\_squares}\left(\left(\frac{1}{s \cdot x}\right), c\right) \]

      19. lift-*.f64 89.1

          \[\leadsto \mathsf{ratio\_of\_squares}\left(\left(\frac{1}{s \cdot x}\right), c\right) \]

   7. Applied rewrites 89.1%

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

Alternative 10: 78.3% accurate, 9.0× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ [x, c, s_m] = \mathsf{sort}([x, c, s_m])\\ \\ \begin{array}{l} t_0 := \left(c \cdot s\_m\right) \cdot x\\ \frac{1}{t\_0 \cdot t\_0} \end{array} \end{array} \]

FPCore

s_m = (fabs.f64 s)
NOTE: x, c, and s_m should be sorted in increasing order before calling this function.
(FPCore (x c s_m)
 :precision binary64
 (let* ((t_0 (* (* c s_m) x))) (/ 1.0 (* t_0 t_0))))

C

s_m = fabs(s);
assert(x < c && c < s_m);
double code(double x, double c, double s_m) {
	double t_0 = (c * s_m) * x;
	return 1.0 / (t_0 * t_0);
}

Fortran

s_m =     private
NOTE: x, c, 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, c, s_m)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s_m
    real(8) :: t_0
    t_0 = (c * s_m) * x
    code = 1.0d0 / (t_0 * t_0)
end function

Java

s_m = Math.abs(s);
assert x < c && c < s_m;
public static double code(double x, double c, double s_m) {
	double t_0 = (c * s_m) * x;
	return 1.0 / (t_0 * t_0);
}

Python

s_m = math.fabs(s)
[x, c, s_m] = sort([x, c, s_m])
def code(x, c, s_m):
	t_0 = (c * s_m) * x
	return 1.0 / (t_0 * t_0)

Julia

s_m = abs(s)
x, c, s_m = sort([x, c, s_m])
function code(x, c, s_m)
	t_0 = Float64(Float64(c * s_m) * x)
	return Float64(1.0 / Float64(t_0 * t_0))
end

MATLAB

s_m = abs(s);
x, c, s_m = num2cell(sort([x, c, s_m])){:}
function tmp = code(x, c, s_m)
	t_0 = (c * s_m) * x;
	tmp = 1.0 / (t_0 * t_0);
end

Wolfram

s_m = N[Abs[s], $MachinePrecision]
NOTE: x, c, and s_m should be sorted in increasing order before calling this function.
code[x_, c_, s$95$m_] := Block[{t$95$0 = N[(N[(c * s$95$m), $MachinePrecision] * x), $MachinePrecision]}, N[(1.0 / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 62.5%

   \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
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-pow.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. lift-*.f64 N/A

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

   5. lift-pow.f64 N/A

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

   6. *-commutative N/A

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

   7. associate-*r* N/A

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

   8. unpow2 N/A

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

   9. associate-*r* N/A

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

   10. pow-prod-down N/A

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

   11. pow-prod-down N/A

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

   12. lower-pow.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 97.1

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

4. Applied rewrites 97.1%

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

   1. lift-pow.f64 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f64 97.1

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 97.1

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 97.1

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

6. Applied rewrites 97.1%

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

7. Taylor expanded in x around 0

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

9. Applied rewrites 74.7%

   \[\leadsto \frac{\color{blue}{1}}{\left(\left(c \cdot s\right) \cdot x\right) \cdot \left(\left(c \cdot s\right) \cdot x\right)} \]
10. Add Preprocessing

Alternative 11: 77.5% accurate, 9.1× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ [x, c, s_m] = \mathsf{sort}([x, c, s_m])\\ \\ \begin{array}{l} \mathbf{if}\;s\_m \leq 5 \cdot 10^{+21}:\\ \;\;\;\;\frac{\mathsf{ratio\_of\_squares}\left(\left(\frac{-1}{c}\right), x\right)}{s\_m \cdot s\_m}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{ratio\_of\_squares}\left(\left(\frac{1}{s\_m \cdot x}\right), c\right)\\ \end{array} \end{array} \]

FPCore

s_m = (fabs.f64 s)
NOTE: x, c, and s_m should be sorted in increasing order before calling this function.
(FPCore (x c s_m)
 :precision binary64
 (if (<= s_m 5e+21)
   (/ (ratio-of-squares (/ -1.0 c) x) (* s_m s_m))
   (ratio-of-squares (/ 1.0 (* s_m x)) c)))

Derivation

1. Split input into 2 regimes

2. if s < 5e21

   1. Initial program 60.9%

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

   3. Taylor expanded in x around 0

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

      1. associate-/r* N/A

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

      2. metadata-eval N/A

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

      3. unpow2 N/A

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

      4. times-frac N/A

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

      5. pow-prod-down N/A

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

      6. unpow2 N/A

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

      7. lower-ratio-of-squares.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 51.6

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

   5. Applied rewrites 51.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-ratio-of-squares.f64 N/A

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

      4. frac-times N/A

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

      5. metadata-eval N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. unpow-prod-down N/A

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

      9. *-commutative N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

      12. metadata-eval N/A

          \[\leadsto \frac{\frac{\frac{1 \cdot 1}{{c}^{2}}}{{x}^{2}}}{{s}^{2}} \]

      13. pow2 N/A

          \[\leadsto \frac{\frac{\frac{1 \cdot 1}{c \cdot c}}{{x}^{2}}}{{s}^{2}} \]

      14. frac-times N/A

          \[\leadsto \frac{\frac{\frac{1}{c} \cdot \frac{1}{c}}{{x}^{2}}}{{s}^{2}} \]

      15. sqr-neg-rev N/A

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

      16. pow2 N/A

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

      17. lower-ratio-of-squares.f64 N/A

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

      18. distribute-neg-frac N/A

          \[\leadsto \frac{\mathsf{ratio\_of\_squares}\left(\left(\frac{\mathsf{neg}\left(1\right)}{c}\right), x\right)}{{s}^{2}} \]

      19. metadata-eval N/A

          \[\leadsto \frac{\mathsf{ratio\_of\_squares}\left(\left(\frac{-1}{c}\right), x\right)}{{s}^{2}} \]

      20. lower-/.f64 N/A

          \[\leadsto \frac{\mathsf{ratio\_of\_squares}\left(\left(\frac{-1}{c}\right), x\right)}{{s}^{2}} \]

      21. pow2 N/A

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

      22. lift-*.f64 59.8

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

   7. Applied rewrites 59.8%

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

   if 5e21 < s

   1. Initial program 70.1%

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

   3. Taylor expanded in x around 0

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

      1. associate-/r* N/A

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

      2. metadata-eval N/A

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

      3. unpow2 N/A

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

      4. times-frac N/A

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

      5. pow-prod-down N/A

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

      6. unpow2 N/A

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

      7. lower-ratio-of-squares.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 71.3

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

   5. Applied rewrites 71.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-ratio-of-squares.f64 N/A

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

      4. frac-times N/A

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

      5. metadata-eval N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. unpow-prod-down N/A

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

      9. associate-/r* N/A

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

      10. *-commutative N/A

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

      11. associate-/r* N/A

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

      12. metadata-eval N/A

          \[\leadsto \frac{\frac{1 \cdot 1}{{s}^{2} \cdot {x}^{2}}}{{c}^{2}} \]

      13. unpow-prod-down N/A

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

      14. pow2 N/A

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

      15. times-frac N/A

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

      16. pow2 N/A

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

      17. lower-ratio-of-squares.f64 N/A

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

      18. lower-/.f64 N/A

          \[\leadsto \mathsf{ratio\_of\_squares}\left(\left(\frac{1}{s \cdot x}\right), c\right) \]

      19. lift-*.f64 88.5

          \[\leadsto \mathsf{ratio\_of\_squares}\left(\left(\frac{1}{s \cdot x}\right), c\right) \]

   7. Applied rewrites 88.5%

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

Alternative 12: 71.6% accurate, 17.1× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ [x, c, s_m] = \mathsf{sort}([x, c, s_m])\\ \\ \mathsf{ratio\_of\_squares}\left(\left(\frac{1}{s\_m \cdot x}\right), c\right) \end{array} \]

FPCore

s_m = (fabs.f64 s)
NOTE: x, c, and s_m should be sorted in increasing order before calling this function.
(FPCore (x c s_m) :precision binary64 (ratio-of-squares (/ 1.0 (* s_m x)) c))

Derivation

1. Initial program 62.5%

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

3. Taylor expanded in x around 0

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

   1. associate-/r* N/A

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

   2. metadata-eval N/A

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

   3. unpow2 N/A

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

   4. times-frac N/A

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

   5. pow-prod-down N/A

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

   6. unpow2 N/A

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

   7. lower-ratio-of-squares.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-*.f64 55.1

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

5. Applied rewrites 55.1%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-ratio-of-squares.f64 N/A

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

   4. frac-times N/A

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

   5. metadata-eval N/A

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

   6. pow2 N/A

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

   7. pow2 N/A

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

   8. unpow-prod-down N/A

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

   9. associate-/r* N/A

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

   10. *-commutative N/A

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

   11. associate-/r* N/A

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

   12. metadata-eval N/A

       \[\leadsto \frac{\frac{1 \cdot 1}{{s}^{2} \cdot {x}^{2}}}{{c}^{2}} \]

   13. unpow-prod-down N/A

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

   14. pow2 N/A

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

   15. times-frac N/A

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

   16. pow2 N/A

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

   17. lower-ratio-of-squares.f64 N/A

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

   18. lower-/.f64 N/A

       \[\leadsto \mathsf{ratio\_of\_squares}\left(\left(\frac{1}{s \cdot x}\right), c\right) \]

   19. lift-*.f64 63.2

       \[\leadsto \mathsf{ratio\_of\_squares}\left(\left(\frac{1}{s \cdot x}\right), c\right) \]

7. Applied rewrites 63.2%

   \[\leadsto \mathsf{ratio\_of\_squares}\left(\left(\frac{1}{s \cdot x}\right), \color{blue}{c}\right) \]
8. Add Preprocessing

Alternative 13: 57.4% accurate, 17.1× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ [x, c, s_m] = \mathsf{sort}([x, c, s_m])\\ \\ \mathsf{ratio\_of\_squares}\left(\left(\frac{1}{c}\right), \left(s\_m \cdot x\right)\right) \end{array} \]

FPCore

s_m = (fabs.f64 s)
NOTE: x, c, and s_m should be sorted in increasing order before calling this function.
(FPCore (x c s_m) :precision binary64 (ratio-of-squares (/ 1.0 c) (* s_m x)))

Derivation

1. Initial program 62.5%

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

3. Taylor expanded in x around 0

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

   1. associate-/r* N/A

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

   2. metadata-eval N/A

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

   3. unpow2 N/A

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

   4. times-frac N/A

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

   5. pow-prod-down N/A

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

   6. unpow2 N/A

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

   7. lower-ratio-of-squares.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-*.f64 55.1

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

5. Applied rewrites 55.1%

   \[\leadsto \color{blue}{\mathsf{ratio\_of\_squares}\left(\left(\frac{1}{c}\right), \left(s \cdot x\right)\right)} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2025058 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))