mixedcos

Percentage Accurate: 66.5% → 97.2%

Time: 2.6s

Alternatives: 7

Speedup: 4.2×

• 31.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 66.5% 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: 97.2% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

c_m =     private
NOTE: x, c_m, and s 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_m, s)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s
    real(8) :: t_0
    t_0 = (c_m * x) * s
    code = cos((x + x)) / (t_0 * t_0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 66.5%

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

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

   7. associate-*r* N/A

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

   8. unpow2 N/A

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

   9. *-commutative N/A

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

   10. pow-prod-down N/A

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

   11. pow-prod-down N/A

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

   12. lower-pow.f64 N/A

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

   13. lower-*.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 97.0

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

3. Applied rewrites 97.0%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 97.2

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

5. Applied rewrites 97.2%

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

   1. lift-*.f64 N/A

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

   2. count-2-rev N/A

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

   3. lower-+.f64 97.2

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

   4. lift-pow.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 97.2

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 97.2

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 97.2

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 97.2

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

   16. lift-*.f64 N/A

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

   17. *-commutative N/A

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

   18. lower-*.f64 97.2

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

7. Applied rewrites 97.2%

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

Alternative 2: 82.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} c_m = \left|c\right| \\ [x, c_m, s] = \mathsf{sort}([x, c_m, s])\\ \\ \begin{array}{l} \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{{c\_m}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \leq -2 \cdot 10^{-15}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2, x \cdot x, 1\right)}{\left(\left(\left(c\_m \cdot c\_m\right) \cdot x\right) \cdot \left(s \cdot s\right)\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\left(s \cdot x\right) \cdot c\_m\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

c_m = (fabs.f64 c)
NOTE: x, c_m, and s should be sorted in increasing order before calling this function.
(FPCore (x c_m s)
 :precision binary64
 (if (<= (/ (cos (* 2.0 x)) (* (pow c_m 2.0) (* (* x (pow s 2.0)) x))) -2e-15)
   (/ (fma -2.0 (* x x) 1.0) (* (* (* (* c_m c_m) x) (* s s)) x))
   (/ 1.0 (pow (* (* s x) c_m) 2.0))))

C

c_m = fabs(c);
assert(x < c_m && c_m < s);
double code(double x, double c_m, double s) {
	double tmp;
	if ((cos((2.0 * x)) / (pow(c_m, 2.0) * ((x * pow(s, 2.0)) * x))) <= -2e-15) {
		tmp = fma(-2.0, (x * x), 1.0) / ((((c_m * c_m) * x) * (s * s)) * x);
	} else {
		tmp = 1.0 / pow(((s * x) * c_m), 2.0);
	}
	return tmp;
}

Julia

c_m = abs(c)
x, c_m, s = sort([x, c_m, s])
function code(x, c_m, s)
	tmp = 0.0
	if (Float64(cos(Float64(2.0 * x)) / Float64((c_m ^ 2.0) * Float64(Float64(x * (s ^ 2.0)) * x))) <= -2e-15)
		tmp = Float64(fma(-2.0, Float64(x * x), 1.0) / Float64(Float64(Float64(Float64(c_m * c_m) * x) * Float64(s * s)) * x));
	else
		tmp = Float64(1.0 / (Float64(Float64(s * x) * c_m) ^ 2.0));
	end
	return tmp
end

Wolfram

c_m = N[Abs[c], $MachinePrecision]
NOTE: x, c_m, and s should be sorted in increasing order before calling this function.
code[x_, c$95$m_, s_] := If[LessEqual[N[(N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] / N[(N[Power[c$95$m, 2.0], $MachinePrecision] * N[(N[(x * N[Power[s, 2.0], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2e-15], N[(N[(-2.0 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] / N[(N[(N[(N[(c$95$m * c$95$m), $MachinePrecision] * x), $MachinePrecision] * N[(s * s), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Power[N[(N[(s * x), $MachinePrecision] * c$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x))) < -2.0000000000000002e-15

   1. Initial program 65.8%

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

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

      7. lower-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 63.3

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

   3. Applied rewrites 63.3%

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

   4. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 52.7

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

   6. Applied rewrites 52.7%

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

   if -2.0000000000000002e-15 < (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x)))

   1. Initial program 66.5%

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

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

      7. associate-*r* N/A

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

      8. unpow2 N/A

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

      9. *-commutative N/A

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

      10. pow-prod-down N/A

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

      11. pow-prod-down N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-*.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 96.8

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

   3. Applied rewrites 96.8%

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

   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

   6. Applied rewrites 84.7%

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

Alternative 3: 78.9% accurate, 3.1× speedup

Math

\[\begin{array}{l} c_m = \left|c\right| \\ [x, c_m, s] = \mathsf{sort}([x, c_m, s])\\ \\ \frac{1}{{\left(\left(s \cdot x\right) \cdot c\_m\right)}^{2}} \end{array} \]

FPCore

c_m = (fabs.f64 c)
NOTE: x, c_m, and s should be sorted in increasing order before calling this function.
(FPCore (x c_m s) :precision binary64 (/ 1.0 (pow (* (* s x) c_m) 2.0)))

C

c_m = fabs(c);
assert(x < c_m && c_m < s);
double code(double x, double c_m, double s) {
	return 1.0 / pow(((s * x) * c_m), 2.0);
}

Fortran

c_m =     private
NOTE: x, c_m, and s 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_m, s)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s
    code = 1.0d0 / (((s * x) * c_m) ** 2.0d0)
end function

Java

c_m = Math.abs(c);
assert x < c_m && c_m < s;
public static double code(double x, double c_m, double s) {
	return 1.0 / Math.pow(((s * x) * c_m), 2.0);
}

Python

c_m = math.fabs(c)
[x, c_m, s] = sort([x, c_m, s])
def code(x, c_m, s):
	return 1.0 / math.pow(((s * x) * c_m), 2.0)

Julia

c_m = abs(c)
x, c_m, s = sort([x, c_m, s])
function code(x, c_m, s)
	return Float64(1.0 / (Float64(Float64(s * x) * c_m) ^ 2.0))
end

MATLAB

c_m = abs(c);
x, c_m, s = num2cell(sort([x, c_m, s])){:}
function tmp = code(x, c_m, s)
	tmp = 1.0 / (((s * x) * c_m) ^ 2.0);
end

Wolfram

c_m = N[Abs[c], $MachinePrecision]
NOTE: x, c_m, and s should be sorted in increasing order before calling this function.
code[x_, c$95$m_, s_] := N[(1.0 / N[Power[N[(N[(s * x), $MachinePrecision] * c$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 66.5%

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

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

   7. associate-*r* N/A

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

   8. unpow2 N/A

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

   9. *-commutative N/A

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

   10. pow-prod-down N/A

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

   11. pow-prod-down N/A

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

   12. lower-pow.f64 N/A

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

   13. lower-*.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 97.0

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

3. Applied rewrites 97.0%

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

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

6. Applied rewrites 78.9%

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

Alternative 4: 78.0% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

Fortran

c_m =     private
NOTE: x, c_m, and s 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_m, s)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s
    real(8) :: t_0
    t_0 = (c_m * x) * s
    code = 1.0d0 / (t_0 * t_0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 66.5%

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

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

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

   3. unpow2 N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. unpow2 N/A

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

   8. associate-*r* N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 64.1

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

4. Applied rewrites 64.1%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. pow2 N/A

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

   7. associate-*l* N/A

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

   8. pow2 N/A

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

   9. associate-*r* N/A

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

   10. *-commutative N/A

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

   11. pow2 N/A

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

   12. lower-*.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. pow2 N/A

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

   16. lift-*.f64 60.1

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

6. Applied rewrites 60.1%

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

7. Taylor expanded in x around 0

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

   1. pow2 N/A

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

   2. associate-/r* N/A

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

   3. count-2-rev N/A

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

   4. pow2 N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. associate-/l/ N/A

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

   9. lift-*.f64 N/A

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

   10. lift-*.f64 N/A

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

9. Applied rewrites 78.0%

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

Alternative 5: 74.3% accurate, 0.8× speedup

Math

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

FPCore

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

C

c_m = fabs(c);
assert(x < c_m && c_m < s);
double code(double x, double c_m, double s) {
	double tmp;
	if ((cos((2.0 * x)) / (pow(c_m, 2.0) * ((x * pow(s, 2.0)) * x))) <= ((double) INFINITY)) {
		tmp = 1.0 / ((((s * (s * x)) * x) * c_m) * c_m);
	} else {
		tmp = 1.0 / ((s * ((s * c_m) * (x * x))) * c_m);
	}
	return tmp;
}

Java

c_m = Math.abs(c);
assert x < c_m && c_m < s;
public static double code(double x, double c_m, double s) {
	double tmp;
	if ((Math.cos((2.0 * x)) / (Math.pow(c_m, 2.0) * ((x * Math.pow(s, 2.0)) * x))) <= Double.POSITIVE_INFINITY) {
		tmp = 1.0 / ((((s * (s * x)) * x) * c_m) * c_m);
	} else {
		tmp = 1.0 / ((s * ((s * c_m) * (x * x))) * c_m);
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x))) < +inf.0

   1. Initial program 81.0%

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

   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

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 73.2

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

   4. Applied rewrites 73.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 79.1

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

   6. Applied rewrites 79.1%

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

   if +inf.0 < (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x)))

   1. Initial program 0.0%

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

   2. 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

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 22.0

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

   4. Applied rewrites 22.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. associate-*l* N/A

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

      8. pow2 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. pow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 24.2

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

   6. Applied rewrites 24.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 46.9

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

   8. Applied rewrites 46.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 52.4

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

   10. Applied rewrites 52.4%

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

Alternative 6: 70.2% accurate, 3.5× speedup

Math

\[\begin{array}{l} c_m = \left|c\right| \\ [x, c_m, s] = \mathsf{sort}([x, c_m, s])\\ \\ \begin{array}{l} \mathbf{if}\;s \leq 6.8 \cdot 10^{+160}:\\ \;\;\;\;\frac{1}{\left(\left(\left(s \cdot s\right) \cdot x\right) \cdot \left(x \cdot c\_m\right)\right) \cdot c\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(s \cdot \left(\left(s \cdot c\_m\right) \cdot \left(x \cdot x\right)\right)\right) \cdot c\_m}\\ \end{array} \end{array} \]

FPCore

c_m = (fabs.f64 c)
NOTE: x, c_m, and s should be sorted in increasing order before calling this function.
(FPCore (x c_m s)
 :precision binary64
 (if (<= s 6.8e+160)
   (/ 1.0 (* (* (* (* s s) x) (* x c_m)) c_m))
   (/ 1.0 (* (* s (* (* s c_m) (* x x))) c_m))))

C

c_m = fabs(c);
assert(x < c_m && c_m < s);
double code(double x, double c_m, double s) {
	double tmp;
	if (s <= 6.8e+160) {
		tmp = 1.0 / ((((s * s) * x) * (x * c_m)) * c_m);
	} else {
		tmp = 1.0 / ((s * ((s * c_m) * (x * x))) * c_m);
	}
	return tmp;
}

Fortran

c_m =     private
NOTE: x, c_m, and s 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_m, s)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s
    real(8) :: tmp
    if (s <= 6.8d+160) then
        tmp = 1.0d0 / ((((s * s) * x) * (x * c_m)) * c_m)
    else
        tmp = 1.0d0 / ((s * ((s * c_m) * (x * x))) * c_m)
    end if
    code = tmp
end function

Java

c_m = Math.abs(c);
assert x < c_m && c_m < s;
public static double code(double x, double c_m, double s) {
	double tmp;
	if (s <= 6.8e+160) {
		tmp = 1.0 / ((((s * s) * x) * (x * c_m)) * c_m);
	} else {
		tmp = 1.0 / ((s * ((s * c_m) * (x * x))) * c_m);
	}
	return tmp;
}

Python

c_m = math.fabs(c)
[x, c_m, s] = sort([x, c_m, s])
def code(x, c_m, s):
	tmp = 0
	if s <= 6.8e+160:
		tmp = 1.0 / ((((s * s) * x) * (x * c_m)) * c_m)
	else:
		tmp = 1.0 / ((s * ((s * c_m) * (x * x))) * c_m)
	return tmp

Julia

c_m = abs(c)
x, c_m, s = sort([x, c_m, s])
function code(x, c_m, s)
	tmp = 0.0
	if (s <= 6.8e+160)
		tmp = Float64(1.0 / Float64(Float64(Float64(Float64(s * s) * x) * Float64(x * c_m)) * c_m));
	else
		tmp = Float64(1.0 / Float64(Float64(s * Float64(Float64(s * c_m) * Float64(x * x))) * c_m));
	end
	return tmp
end

MATLAB

c_m = abs(c);
x, c_m, s = num2cell(sort([x, c_m, s])){:}
function tmp_2 = code(x, c_m, s)
	tmp = 0.0;
	if (s <= 6.8e+160)
		tmp = 1.0 / ((((s * s) * x) * (x * c_m)) * c_m);
	else
		tmp = 1.0 / ((s * ((s * c_m) * (x * x))) * c_m);
	end
	tmp_2 = tmp;
end

Wolfram

c_m = N[Abs[c], $MachinePrecision]
NOTE: x, c_m, and s should be sorted in increasing order before calling this function.
code[x_, c$95$m_, s_] := If[LessEqual[s, 6.8e+160], N[(1.0 / N[(N[(N[(N[(s * s), $MachinePrecision] * x), $MachinePrecision] * N[(x * c$95$m), $MachinePrecision]), $MachinePrecision] * c$95$m), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(s * N[(N[(s * c$95$m), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * c$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if s < 6.80000000000000061e160

   1. Initial program 73.6%

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

   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

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 66.1

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

   4. Applied rewrites 66.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-*.f64 68.1

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

   6. Applied rewrites 68.1%

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

   if 6.80000000000000061e160 < s

   1. Initial program 52.3%

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

   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

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 60.0

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

   4. Applied rewrites 60.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. associate-*l* N/A

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

      8. pow2 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. pow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 55.4

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

   6. Applied rewrites 55.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 66.9

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

   8. Applied rewrites 66.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 74.4

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

   10. Applied rewrites 74.4%

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

Alternative 7: 68.4% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

Fortran

c_m =     private
NOTE: x, c_m, and s 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_m, s)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s
    code = 1.0d0 / ((s * ((s * c_m) * (x * x))) * c_m)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

c_m = N[Abs[c], $MachinePrecision]
NOTE: x, c_m, and s should be sorted in increasing order before calling this function.
code[x_, c$95$m_, s_] := N[(1.0 / N[(N[(s * N[(N[(s * c$95$m), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * c$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 66.5%

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

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

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

   3. unpow2 N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. unpow2 N/A

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

   8. associate-*r* N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 64.1

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

4. Applied rewrites 64.1%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. pow2 N/A

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

   7. associate-*l* N/A

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

   8. pow2 N/A

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

   9. associate-*r* N/A

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

   10. *-commutative N/A

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

   11. pow2 N/A

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

   12. lower-*.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. pow2 N/A

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

   16. lift-*.f64 60.1

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

6. Applied rewrites 60.1%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 65.2

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

8. Applied rewrites 65.2%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. pow2 N/A

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

   6. associate-*l* N/A

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

   7. lower-*.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. pow2 N/A

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

   11. lift-*.f64 68.4

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

10. Applied rewrites 68.4%

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

Reproduce

herbie shell --seed 2025120 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))