mixedcos

Percentage Accurate: 65.6% → 97.0%

Time: 4.0s

Alternatives: 12

Speedup: 4.2×

• 31.6% 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: 65.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.0% accurate, 1.5× speedup

Math

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

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 (* (* s_m x) c))) (/ (/ (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 = (s_m * x) * c;
	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 = (s_m * x) * c
    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 = (s_m * x) * c;
	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 = (s_m * x) * c
	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(s_m * x) * c)
	return Float64(Float64(cos(Float64(x + 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 = (s_m * x) * c;
	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[(s$95$m * x), $MachinePrecision] * c), $MachinePrecision]}, N[(N[(N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]]

Derivation

1. Initial program 65.6%

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

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

   7. *-commutative N/A

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

   8. associate-*l* N/A

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

   9. unpow2 N/A

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

   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. pow-prod-down N/A

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

   12. pow-prod-down N/A

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

   13. lower-pow.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. lower-*.f64 96.6

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

3. Applied rewrites 96.6%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. lift-pow.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. unpow2 N/A

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

   8. associate-/r* N/A

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

   9. lower-/.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lift-cos.f64 N/A

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

   12. count-2-rev N/A

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

   13. lift-+.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. lift-*.f64 N/A

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

   16. lift-*.f64 N/A

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

   17. lift-*.f64 97.0

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

5. Applied rewrites 97.0%

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

Alternative 2: 96.6% accurate, 1.5× speedup

Math

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

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 (* (* s_m x) c))) (/ (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 = (s_m * x) * c;
	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 = (s_m * x) * c
    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 = (s_m * x) * c;
	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 = (s_m * x) * c
	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(s_m * x) * c)
	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 = (s_m * x) * c;
	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[(s$95$m * x), $MachinePrecision] * c), $MachinePrecision]}, N[(N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 65.6%

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

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

   7. *-commutative N/A

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

   8. associate-*l* N/A

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

   9. unpow2 N/A

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

   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. pow-prod-down N/A

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

   12. pow-prod-down N/A

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

   13. lower-pow.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. lower-*.f64 96.6

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

3. Applied rewrites 96.6%

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

   1. lift-*.f64 N/A

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

   2. count-2-rev N/A

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

   3. lift-+.f64 96.6

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

   4. lift-pow.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. unpow2 N/A

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

   8. *-commutative N/A

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

   9. associate-*r* N/A

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

   10. lower-*.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. lift-*.f64 92.8

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

5. Applied rewrites 92.8%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*l* N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. lift-*.f64 96.6

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

7. Applied rewrites 96.6%

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

Alternative 3: 86.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ [x, c, s_m] = \mathsf{sort}([x, c, s_m])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 2.8 \cdot 10^{-38}:\\ \;\;\;\;\frac{1}{{\left(\left(s\_m \cdot x\right) \cdot c\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x + x\right)}{s\_m \cdot \left(\left(c \cdot s\_m\right) \cdot \left(\left(c \cdot x\right) \cdot x\right)\right)}\\ \end{array} \end{array} \]

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 (<= x 2.8e-38)
   (/ 1.0 (pow (* (* s_m x) c) 2.0))
   (/ (cos (+ x x)) (* s_m (* (* c s_m) (* (* c x) x))))))

C

s_m = fabs(s);
assert(x < c && c < s_m);
double code(double x, double c, double s_m) {
	double tmp;
	if (x <= 2.8e-38) {
		tmp = 1.0 / pow(((s_m * x) * c), 2.0);
	} else {
		tmp = cos((x + x)) / (s_m * ((c * s_m) * ((c * x) * x)));
	}
	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) :: tmp
    if (x <= 2.8d-38) then
        tmp = 1.0d0 / (((s_m * x) * c) ** 2.0d0)
    else
        tmp = cos((x + x)) / (s_m * ((c * s_m) * ((c * x) * x)))
    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 tmp;
	if (x <= 2.8e-38) {
		tmp = 1.0 / Math.pow(((s_m * x) * c), 2.0);
	} else {
		tmp = Math.cos((x + x)) / (s_m * ((c * s_m) * ((c * x) * x)));
	}
	return tmp;
}

Python

s_m = math.fabs(s)
[x, c, s_m] = sort([x, c, s_m])
def code(x, c, s_m):
	tmp = 0
	if x <= 2.8e-38:
		tmp = 1.0 / math.pow(((s_m * x) * c), 2.0)
	else:
		tmp = math.cos((x + x)) / (s_m * ((c * s_m) * ((c * x) * x)))
	return tmp

Julia

s_m = abs(s)
x, c, s_m = sort([x, c, s_m])
function code(x, c, s_m)
	tmp = 0.0
	if (x <= 2.8e-38)
		tmp = Float64(1.0 / (Float64(Float64(s_m * x) * c) ^ 2.0));
	else
		tmp = Float64(cos(Float64(x + x)) / Float64(s_m * Float64(Float64(c * s_m) * Float64(Float64(c * x) * x))));
	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)
	tmp = 0.0;
	if (x <= 2.8e-38)
		tmp = 1.0 / (((s_m * x) * c) ^ 2.0);
	else
		tmp = cos((x + x)) / (s_m * ((c * s_m) * ((c * x) * x)));
	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_] := If[LessEqual[x, 2.8e-38], N[(1.0 / N[Power[N[(N[(s$95$m * x), $MachinePrecision] * c), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] / N[(s$95$m * N[(N[(c * s$95$m), $MachinePrecision] * N[(N[(c * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.8e-38

   1. Initial program 65.6%

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

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. unpow2 N/A

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

      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. pow-prod-down N/A

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

      12. pow-prod-down N/A

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

      13. lower-pow.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 96.6

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

   3. Applied rewrites 96.6%

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

   4. Taylor expanded in x around 0

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

      1. count-2-rev 79.1

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

   6. Applied rewrites 79.1%

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

   if 2.8e-38 < x

   1. Initial program 65.6%

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

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. unpow2 N/A

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

      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. pow-prod-down N/A

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

      12. pow-prod-down N/A

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

      13. lower-pow.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 96.6

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

   3. Applied rewrites 96.6%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lift-+.f64 96.6

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. unpow2 N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 92.8

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

   5. Applied rewrites 92.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. swap-sqr N/A

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

      9. unpow2 N/A

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

      10. pow-prod-down N/A

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

      11. associate-*l* N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*r* N/A

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

      15. *-commutative N/A

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

      16. associate-*l* N/A

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

      17. associate-*l* N/A

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

      18. *-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. associate-*l* N/A

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

      21. *-commutative N/A

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

      22. associate-*l* N/A

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

   7. Applied rewrites 85.5%

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

Alternative 4: 82.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ [x, c, s_m] = \mathsf{sort}([x, c, s_m])\\ \\ \begin{array}{l} t_0 := \left(s\_m \cdot x\right) \cdot c\\ \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s\_m}^{2}\right) \cdot x\right)} \leq -1 \cdot 10^{-140}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot x, -2, 1\right)}{\left(t\_0 \cdot c\right) \cdot \left(s\_m \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{t\_0}^{2}}\\ \end{array} \end{array} \]

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)))
   (if (<=
        (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s_m 2.0)) x)))
        -1e-140)
     (/ (fma (* x x) -2.0 1.0) (* (* t_0 c) (* s_m x)))
     (/ 1.0 (pow t_0 2.0)))))

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 tmp;
	if ((cos((2.0 * x)) / (pow(c, 2.0) * ((x * pow(s_m, 2.0)) * x))) <= -1e-140) {
		tmp = fma((x * x), -2.0, 1.0) / ((t_0 * c) * (s_m * x));
	} else {
		tmp = 1.0 / pow(t_0, 2.0);
	}
	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(s_m * x) * c)
	tmp = 0.0
	if (Float64(cos(Float64(2.0 * x)) / Float64((c ^ 2.0) * Float64(Float64(x * (s_m ^ 2.0)) * x))) <= -1e-140)
		tmp = Float64(fma(Float64(x * x), -2.0, 1.0) / Float64(Float64(t_0 * c) * Float64(s_m * x)));
	else
		tmp = Float64(1.0 / (t_0 ^ 2.0));
	end
	return 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]}, 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], -1e-140], N[(N[(N[(x * x), $MachinePrecision] * -2.0 + 1.0), $MachinePrecision] / N[(N[(t$95$0 * c), $MachinePrecision] * N[(s$95$m * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x))) < -9.9999999999999998e-141

   1. Initial program 65.6%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 45.8

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

   4. Applied rewrites 45.8%

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

      1. count-2-rev 45.8

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 45.8

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

      9. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, -2, 1\right)}{\mathsf{Rewrite=>}\left(lift-*.f64, \left({c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, -2, 1\right)}{\mathsf{Rewrite=>}\left(lift-pow.f64, \left({c}^{2}\right)\right) \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, -2, 1\right)}{{c}^{2} \cdot \mathsf{Rewrite=>}\left(lift-*.f64, \left(\left(x \cdot {s}^{2}\right) \cdot x\right)\right)} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, -2, 1\right)}{{c}^{2} \cdot \left(\mathsf{Rewrite=>}\left(lift-*.f64, \left(x \cdot {s}^{2}\right)\right) \cdot x\right)} \]

      13. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, -2, 1\right)}{{c}^{2} \cdot \left(\left(x \cdot \mathsf{Rewrite=>}\left(lift-pow.f64, \left({s}^{2}\right)\right)\right) \cdot x\right)} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, -2, 1\right)}{{c}^{2} \cdot \left(\left(x \cdot \mathsf{Rewrite<=}\left(pow2, \left(s \cdot s\right)\right)\right) \cdot x\right)} \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, -2, 1\right)}{{c}^{2} \cdot \left(\mathsf{Rewrite<=}\left(*-commutative, \left(\left(s \cdot s\right) \cdot x\right)\right) \cdot x\right)} \]

      16. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, -2, 1\right)}{{c}^{2} \cdot \mathsf{Rewrite=>}\left(associate-*l*, \left(\left(s \cdot s\right) \cdot \left(x \cdot x\right)\right)\right)} \]

      17. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, -2, 1\right)}{{c}^{2} \cdot \left(\left(s \cdot s\right) \cdot \mathsf{Rewrite=>}\left(pow2, \left({x}^{2}\right)\right)\right)} \]

      18. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, -2, 1\right)}{\mathsf{Rewrite<=}\left(pow2, \left(c \cdot c\right)\right) \cdot \left(\left(s \cdot s\right) \cdot {x}^{2}\right)} \]

      19. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, -2, 1\right)}{\left(c \cdot c\right) \cdot \left(\left(s \cdot s\right) \cdot \mathsf{Rewrite<=}\left(pow2, \left(x \cdot x\right)\right)\right)} \]

      20. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, -2, 1\right)}{\left(c \cdot c\right) \cdot \mathsf{Rewrite=>}\left(unswap-sqr, \left(\left(s \cdot x\right) \cdot \left(s \cdot x\right)\right)\right)} \]

   6. Applied rewrites 60.8%

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

   if -9.9999999999999998e-141 < (/.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 65.6%

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

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. unpow2 N/A

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

      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. pow-prod-down N/A

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

      12. pow-prod-down N/A

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

      13. lower-pow.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 96.6

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

   3. Applied rewrites 96.6%

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

   4. Taylor expanded in x around 0

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

      1. count-2-rev 79.1

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

   6. Applied rewrites 79.1%

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

Alternative 5: 82.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ [x, c, s_m] = \mathsf{sort}([x, c, s_m])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s\_m}^{2}\right) \cdot x\right)} \leq -1 \cdot 10^{-140}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2, x \cdot x, 1\right)}{\left(\left(c \cdot c\right) \cdot x\right) \cdot \left(\left(s\_m \cdot s\_m\right) \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\left(s\_m \cdot x\right) \cdot c\right)}^{2}}\\ \end{array} \end{array} \]

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
 (if (<= (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s_m 2.0)) x))) -1e-140)
   (/ (fma -2.0 (* x x) 1.0) (* (* (* c c) x) (* (* s_m s_m) x)))
   (/ 1.0 (pow (* (* s_m x) c) 2.0))))

C

s_m = fabs(s);
assert(x < c && c < s_m);
double code(double x, double c, double s_m) {
	double tmp;
	if ((cos((2.0 * x)) / (pow(c, 2.0) * ((x * pow(s_m, 2.0)) * x))) <= -1e-140) {
		tmp = fma(-2.0, (x * x), 1.0) / (((c * c) * x) * ((s_m * s_m) * x));
	} else {
		tmp = 1.0 / pow(((s_m * x) * c), 2.0);
	}
	return tmp;
}

Julia

s_m = abs(s)
x, c, s_m = sort([x, c, s_m])
function code(x, c, s_m)
	tmp = 0.0
	if (Float64(cos(Float64(2.0 * x)) / Float64((c ^ 2.0) * Float64(Float64(x * (s_m ^ 2.0)) * x))) <= -1e-140)
		tmp = Float64(fma(-2.0, Float64(x * x), 1.0) / Float64(Float64(Float64(c * c) * x) * Float64(Float64(s_m * s_m) * x)));
	else
		tmp = Float64(1.0 / (Float64(Float64(s_m * x) * c) ^ 2.0));
	end
	return 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_] := 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], -1e-140], N[(N[(-2.0 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] / N[(N[(N[(c * c), $MachinePrecision] * x), $MachinePrecision] * N[(N[(s$95$m * s$95$m), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Power[N[(N[(s$95$m * x), $MachinePrecision] * c), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x))) < -9.9999999999999998e-141

   1. Initial program 65.6%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 65.6

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 67.1

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

   3. Applied rewrites 67.1%

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

   4. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 46.1

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

   6. Applied rewrites 46.1%

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

   if -9.9999999999999998e-141 < (/.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 65.6%

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

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. unpow2 N/A

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

      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. pow-prod-down N/A

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

      12. pow-prod-down N/A

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

      13. lower-pow.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 96.6

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

   3. Applied rewrites 96.6%

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

   4. Taylor expanded in x around 0

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

      1. count-2-rev 79.1

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

   6. Applied rewrites 79.1%

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

Alternative 6: 82.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ [x, c, s_m] = \mathsf{sort}([x, c, s_m])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s\_m}^{2}\right) \cdot x\right)} \leq -1 \cdot 10^{-140}:\\ \;\;\;\;\frac{\frac{\left(x \cdot x\right) \cdot -2}{\left(\left(\left(c \cdot c\right) \cdot x\right) \cdot s\_m\right) \cdot s\_m}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\left(s\_m \cdot x\right) \cdot c\right)}^{2}}\\ \end{array} \end{array} \]

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
 (if (<= (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s_m 2.0)) x))) -1e-140)
   (/ (/ (* (* x x) -2.0) (* (* (* (* c c) x) s_m) s_m)) x)
   (/ 1.0 (pow (* (* s_m x) c) 2.0))))

C

s_m = fabs(s);
assert(x < c && c < s_m);
double code(double x, double c, double s_m) {
	double tmp;
	if ((cos((2.0 * x)) / (pow(c, 2.0) * ((x * pow(s_m, 2.0)) * x))) <= -1e-140) {
		tmp = (((x * x) * -2.0) / ((((c * c) * x) * s_m) * s_m)) / x;
	} else {
		tmp = 1.0 / pow(((s_m * x) * c), 2.0);
	}
	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) :: tmp
    if ((cos((2.0d0 * x)) / ((c ** 2.0d0) * ((x * (s_m ** 2.0d0)) * x))) <= (-1d-140)) then
        tmp = (((x * x) * (-2.0d0)) / ((((c * c) * x) * s_m) * s_m)) / x
    else
        tmp = 1.0d0 / (((s_m * x) * c) ** 2.0d0)
    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 tmp;
	if ((Math.cos((2.0 * x)) / (Math.pow(c, 2.0) * ((x * Math.pow(s_m, 2.0)) * x))) <= -1e-140) {
		tmp = (((x * x) * -2.0) / ((((c * c) * x) * s_m) * s_m)) / x;
	} else {
		tmp = 1.0 / Math.pow(((s_m * x) * c), 2.0);
	}
	return tmp;
}

Python

s_m = math.fabs(s)
[x, c, s_m] = sort([x, c, s_m])
def code(x, c, s_m):
	tmp = 0
	if (math.cos((2.0 * x)) / (math.pow(c, 2.0) * ((x * math.pow(s_m, 2.0)) * x))) <= -1e-140:
		tmp = (((x * x) * -2.0) / ((((c * c) * x) * s_m) * s_m)) / x
	else:
		tmp = 1.0 / math.pow(((s_m * x) * c), 2.0)
	return tmp

Julia

s_m = abs(s)
x, c, s_m = sort([x, c, s_m])
function code(x, c, s_m)
	tmp = 0.0
	if (Float64(cos(Float64(2.0 * x)) / Float64((c ^ 2.0) * Float64(Float64(x * (s_m ^ 2.0)) * x))) <= -1e-140)
		tmp = Float64(Float64(Float64(Float64(x * x) * -2.0) / Float64(Float64(Float64(Float64(c * c) * x) * s_m) * s_m)) / x);
	else
		tmp = Float64(1.0 / (Float64(Float64(s_m * x) * c) ^ 2.0));
	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)
	tmp = 0.0;
	if ((cos((2.0 * x)) / ((c ^ 2.0) * ((x * (s_m ^ 2.0)) * x))) <= -1e-140)
		tmp = (((x * x) * -2.0) / ((((c * c) * x) * s_m) * s_m)) / x;
	else
		tmp = 1.0 / (((s_m * x) * c) ^ 2.0);
	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_] := 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], -1e-140], N[(N[(N[(N[(x * x), $MachinePrecision] * -2.0), $MachinePrecision] / N[(N[(N[(N[(c * c), $MachinePrecision] * x), $MachinePrecision] * s$95$m), $MachinePrecision] * s$95$m), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(1.0 / N[Power[N[(N[(s$95$m * x), $MachinePrecision] * c), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x))) < -9.9999999999999998e-141

   1. Initial program 65.6%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 45.8

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

   4. Applied rewrites 45.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

   6. Applied rewrites 51.3%

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

   7. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 18.6

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

   9. Applied rewrites 18.6%

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

   if -9.9999999999999998e-141 < (/.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 65.6%

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

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. unpow2 N/A

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

      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. pow-prod-down N/A

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

      12. pow-prod-down N/A

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

      13. lower-pow.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 96.6

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

   3. Applied rewrites 96.6%

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

   4. Taylor expanded in x around 0

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

      1. count-2-rev 79.1

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

   6. Applied rewrites 79.1%

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

Alternative 7: 82.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ [x, c, s_m] = \mathsf{sort}([x, c, s_m])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s\_m}^{2}\right) \cdot x\right)} \leq -1 \cdot 10^{-140}:\\ \;\;\;\;\frac{\frac{\left(x \cdot x\right) \cdot -2}{\left(\left(\left(c \cdot c\right) \cdot x\right) \cdot s\_m\right) \cdot s\_m}}{x}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(s\_m \cdot x\right) \cdot c\right)}^{-2}\\ \end{array} \end{array} \]

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
 (if (<= (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s_m 2.0)) x))) -1e-140)
   (/ (/ (* (* x x) -2.0) (* (* (* (* c c) x) s_m) s_m)) x)
   (pow (* (* s_m x) c) -2.0)))

C

s_m = fabs(s);
assert(x < c && c < s_m);
double code(double x, double c, double s_m) {
	double tmp;
	if ((cos((2.0 * x)) / (pow(c, 2.0) * ((x * pow(s_m, 2.0)) * x))) <= -1e-140) {
		tmp = (((x * x) * -2.0) / ((((c * c) * x) * s_m) * s_m)) / x;
	} else {
		tmp = pow(((s_m * x) * c), -2.0);
	}
	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) :: tmp
    if ((cos((2.0d0 * x)) / ((c ** 2.0d0) * ((x * (s_m ** 2.0d0)) * x))) <= (-1d-140)) then
        tmp = (((x * x) * (-2.0d0)) / ((((c * c) * x) * s_m) * s_m)) / x
    else
        tmp = ((s_m * x) * c) ** (-2.0d0)
    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 tmp;
	if ((Math.cos((2.0 * x)) / (Math.pow(c, 2.0) * ((x * Math.pow(s_m, 2.0)) * x))) <= -1e-140) {
		tmp = (((x * x) * -2.0) / ((((c * c) * x) * s_m) * s_m)) / x;
	} else {
		tmp = Math.pow(((s_m * x) * c), -2.0);
	}
	return tmp;
}

Python

s_m = math.fabs(s)
[x, c, s_m] = sort([x, c, s_m])
def code(x, c, s_m):
	tmp = 0
	if (math.cos((2.0 * x)) / (math.pow(c, 2.0) * ((x * math.pow(s_m, 2.0)) * x))) <= -1e-140:
		tmp = (((x * x) * -2.0) / ((((c * c) * x) * s_m) * s_m)) / x
	else:
		tmp = math.pow(((s_m * x) * c), -2.0)
	return tmp

Julia

s_m = abs(s)
x, c, s_m = sort([x, c, s_m])
function code(x, c, s_m)
	tmp = 0.0
	if (Float64(cos(Float64(2.0 * x)) / Float64((c ^ 2.0) * Float64(Float64(x * (s_m ^ 2.0)) * x))) <= -1e-140)
		tmp = Float64(Float64(Float64(Float64(x * x) * -2.0) / Float64(Float64(Float64(Float64(c * c) * x) * s_m) * s_m)) / x);
	else
		tmp = Float64(Float64(s_m * x) * c) ^ -2.0;
	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)
	tmp = 0.0;
	if ((cos((2.0 * x)) / ((c ^ 2.0) * ((x * (s_m ^ 2.0)) * x))) <= -1e-140)
		tmp = (((x * x) * -2.0) / ((((c * c) * x) * s_m) * s_m)) / x;
	else
		tmp = ((s_m * x) * c) ^ -2.0;
	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_] := 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], -1e-140], N[(N[(N[(N[(x * x), $MachinePrecision] * -2.0), $MachinePrecision] / N[(N[(N[(N[(c * c), $MachinePrecision] * x), $MachinePrecision] * s$95$m), $MachinePrecision] * s$95$m), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[Power[N[(N[(s$95$m * x), $MachinePrecision] * c), $MachinePrecision], -2.0], $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))) < -9.9999999999999998e-141

   1. Initial program 65.6%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 45.8

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

   4. Applied rewrites 45.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

   6. Applied rewrites 51.3%

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

   7. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 18.6

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

   9. Applied rewrites 18.6%

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

   if -9.9999999999999998e-141 < (/.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 65.6%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 59.0

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

   4. Applied rewrites 59.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. associate-*l* N/A

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

      10. pow2 N/A

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

      11. lift-pow.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*r* N/A

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

      14. lift-*.f64 N/A

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

      15. associate-/r* N/A

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

      16. lift-*.f64 N/A

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

   6. Applied rewrites 79.3%

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

Alternative 8: 79.3% accurate, 3.5× speedup

Math

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

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 (pow (* (* s_m x) c) -2.0))

C

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

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
    code = ((s_m * x) * c) ** (-2.0d0)
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) {
	return Math.pow(((s_m * x) * c), -2.0);
}

Python

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

Julia

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

MATLAB

s_m = abs(s);
x, c, s_m = num2cell(sort([x, c, s_m])){:}
function tmp = code(x, c, s_m)
	tmp = ((s_m * x) * c) ^ -2.0;
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_] := N[Power[N[(N[(s$95$m * x), $MachinePrecision] * c), $MachinePrecision], -2.0], $MachinePrecision]

Derivation

1. Initial program 65.6%

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

2. Taylor expanded in x around 0

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

   1. lower-/.f64 N/A

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

   2. *-commutative N/A

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

   3. unpow2 N/A

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

   4. associate-*l* N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f64 59.0

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

4. Applied rewrites 59.0%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*r* N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. pow2 N/A

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

   9. associate-*l* N/A

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

   10. pow2 N/A

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

   11. lift-pow.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. associate-*r* N/A

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

   14. lift-*.f64 N/A

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

   15. associate-/r* N/A

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

   16. lift-*.f64 N/A

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

6. Applied rewrites 79.3%

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

Alternative 9: 79.3% accurate, 4.1× speedup

Math

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

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 (* (* s_m x) c))) (/ (/ 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 = (s_m * x) * c;
	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 = (s_m * x) * c
    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 = (s_m * x) * c;
	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 = (s_m * x) * c
	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(s_m * x) * c)
	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 = (s_m * x) * c;
	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[(s$95$m * x), $MachinePrecision] * c), $MachinePrecision]}, N[(N[(1.0 / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]]

Derivation

1. Initial program 65.6%

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

2. Taylor expanded in x around 0

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

   1. lower-/.f64 N/A

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

   2. *-commutative N/A

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

   3. unpow2 N/A

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

   4. associate-*l* N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f64 59.0

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

4. Applied rewrites 59.0%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*r* N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. pow2 N/A

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

   9. associate-*l* N/A

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

   10. pow2 N/A

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

   11. lift-pow.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. associate-*r* N/A

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

   14. lift-*.f64 N/A

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

   15. associate-/r* N/A

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

   16. lift-*.f64 N/A

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

   17. *-commutative N/A

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

   18. lift-*.f64 N/A

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

   19. lift-pow.f64 N/A

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

   20. pow2 N/A

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

6. Applied rewrites 79.3%

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

Alternative 10: 77.7% accurate, 4.2× speedup

Math

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

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
 (/ 1.0 (* (* (* (* s_m x) c) c) (* s_m x))))

C

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

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
    code = 1.0d0 / ((((s_m * x) * c) * c) * (s_m * x))
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) {
	return 1.0 / ((((s_m * x) * c) * c) * (s_m * x));
}

Python

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

Julia

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

MATLAB

s_m = abs(s);
x, c, s_m = num2cell(sort([x, c, s_m])){:}
function tmp = code(x, c, s_m)
	tmp = 1.0 / ((((s_m * x) * c) * c) * (s_m * x));
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_] := N[(1.0 / N[(N[(N[(N[(s$95$m * x), $MachinePrecision] * c), $MachinePrecision] * c), $MachinePrecision] * N[(s$95$m * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 65.6%

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

2. Taylor expanded in x around 0

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

   1. lower-/.f64 N/A

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

   2. *-commutative N/A

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

   3. unpow2 N/A

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

   4. associate-*l* N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f64 59.0

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

4. Applied rewrites 59.0%

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

6. Applied rewrites 77.4%

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

Alternative 11: 77.4% accurate, 4.2× speedup

Math

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

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
 (/ 1.0 (* c (* (* (* s_m x) c) (* s_m x)))))

C

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

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
    code = 1.0d0 / (c * (((s_m * x) * c) * (s_m * x)))
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) {
	return 1.0 / (c * (((s_m * x) * c) * (s_m * x)));
}

Python

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

Julia

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

MATLAB

s_m = abs(s);
x, c, s_m = num2cell(sort([x, c, s_m])){:}
function tmp = code(x, c, s_m)
	tmp = 1.0 / (c * (((s_m * x) * c) * (s_m * x)));
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_] := N[(1.0 / N[(c * N[(N[(N[(s$95$m * x), $MachinePrecision] * c), $MachinePrecision] * N[(s$95$m * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 65.6%

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

2. Taylor expanded in x around 0

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

   1. lower-/.f64 N/A

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

   2. *-commutative N/A

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

   3. unpow2 N/A

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

   4. associate-*l* N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f64 59.0

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

4. Applied rewrites 59.0%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lift-*.f64 65.4

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

6. Applied rewrites 65.4%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. unpow2 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. associate-*l* N/A

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

   10. associate-*l* N/A

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

   11. lower-*.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. unpow2 N/A

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

   15. associate-*l* N/A

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

   16. *-commutative N/A

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

   17. lift-*.f64 N/A

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

   18. lift-*.f64 72.2

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

8. Applied rewrites 72.2%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. associate-*r* N/A

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

   8. *-commutative N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. lift-*.f64 77.7

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

10. Applied rewrites 77.7%

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

Alternative 12: 72.2% accurate, 4.2× speedup

Math

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

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
 (/ 1.0 (* c (* (* c x) (* (* s_m x) s_m)))))

C

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

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
    code = 1.0d0 / (c * ((c * x) * ((s_m * x) * s_m)))
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) {
	return 1.0 / (c * ((c * x) * ((s_m * x) * s_m)));
}

Python

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

Julia

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

MATLAB

s_m = abs(s);
x, c, s_m = num2cell(sort([x, c, s_m])){:}
function tmp = code(x, c, s_m)
	tmp = 1.0 / (c * ((c * x) * ((s_m * x) * s_m)));
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_] := N[(1.0 / N[(c * N[(N[(c * x), $MachinePrecision] * N[(N[(s$95$m * x), $MachinePrecision] * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 65.6%

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

2. Taylor expanded in x around 0

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

   1. lower-/.f64 N/A

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

   2. *-commutative N/A

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

   3. unpow2 N/A

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

   4. associate-*l* N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f64 59.0

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

4. Applied rewrites 59.0%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lift-*.f64 65.4

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

6. Applied rewrites 65.4%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. unpow2 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. associate-*l* N/A

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

   10. associate-*l* N/A

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

   11. lower-*.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. unpow2 N/A

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

   15. associate-*l* N/A

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

   16. *-commutative N/A

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

   17. lift-*.f64 N/A

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

   18. lift-*.f64 72.2

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

8. Applied rewrites 72.2%

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

Reproduce

herbie shell --seed 2025135 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))