mixedcos

Percentage Accurate: 66.5% → 96.7%

Time: 4.7s

Alternatives: 14

Speedup: 4.2×

• 32.0% 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: 96.7% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

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)) / (((s * c) * x) ** 2.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, c_, s_] := N[(N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] / N[Power[N[(N[(s * c), $MachinePrecision] * x), $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-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lift-pow.f64 N/A

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

   8. unpow2 N/A

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

   9. associate-*l* N/A

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

   10. associate-*r* N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. lift-pow.f64 N/A

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

   16. unpow2 N/A

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

   17. lower-*.f64 N/A

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

   18. *-commutative N/A

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

   19. lower-*.f64 77.8

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

3. Applied rewrites 77.8%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*l* N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. associate-*l* N/A

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

   10. lift-*.f64 N/A

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

   11. associate-*l* N/A

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

   12. lift-*.f64 N/A

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

   13. pow2 N/A

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

   14. lift-pow.f64 N/A

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

   15. lift-*.f64 N/A

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

   16. lift-*.f64 N/A

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

   17. lift-pow.f64 N/A

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

   18. pow2 N/A

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

   19. lift-*.f64 N/A

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

   20. *-commutative N/A

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

   21. associate-*l* N/A

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

   22. lift-*.f64 N/A

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

   23. associate-*l* N/A

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

5. Applied rewrites 96.7%

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

Alternative 2: 93.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(x + x\right)\\ \mathbf{if}\;x \leq 4.8:\\ \;\;\;\;\frac{t\_0}{\left(c \cdot \left(c \cdot \left(s \cdot x\right)\right)\right) \cdot \left(s \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t\_0}{\left(\left(c \cdot s\right) \cdot s\right) \cdot \left(c \cdot x\right)}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (cos (+ x x))))
   (if (<= x 4.8)
     (/ t_0 (* (* c (* c (* s x))) (* s x)))
     (/ (/ t_0 (* (* (* c s) s) (* c x))) x))))

C

double code(double x, double c, double s) {
	double t_0 = cos((x + x));
	double tmp;
	if (x <= 4.8) {
		tmp = t_0 / ((c * (c * (s * x))) * (s * x));
	} else {
		tmp = (t_0 / (((c * s) * s) * (c * x))) / x;
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos((x + x))
    if (x <= 4.8d0) then
        tmp = t_0 / ((c * (c * (s * x))) * (s * x))
    else
        tmp = (t_0 / (((c * s) * s) * (c * x))) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double c, double s) {
	double t_0 = Math.cos((x + x));
	double tmp;
	if (x <= 4.8) {
		tmp = t_0 / ((c * (c * (s * x))) * (s * x));
	} else {
		tmp = (t_0 / (((c * s) * s) * (c * x))) / x;
	}
	return tmp;
}

Python

def code(x, c, s):
	t_0 = math.cos((x + x))
	tmp = 0
	if x <= 4.8:
		tmp = t_0 / ((c * (c * (s * x))) * (s * x))
	else:
		tmp = (t_0 / (((c * s) * s) * (c * x))) / x
	return tmp

Julia

function code(x, c, s)
	t_0 = cos(Float64(x + x))
	tmp = 0.0
	if (x <= 4.8)
		tmp = Float64(t_0 / Float64(Float64(c * Float64(c * Float64(s * x))) * Float64(s * x)));
	else
		tmp = Float64(Float64(t_0 / Float64(Float64(Float64(c * s) * s) * Float64(c * x))) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	t_0 = cos((x + x));
	tmp = 0.0;
	if (x <= 4.8)
		tmp = t_0 / ((c * (c * (s * x))) * (s * x));
	else
		tmp = (t_0 / (((c * s) * s) * (c * x))) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := Block[{t$95$0 = N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 4.8], N[(t$95$0 / N[(N[(c * N[(c * N[(s * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(s * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 / N[(N[(N[(c * s), $MachinePrecision] * s), $MachinePrecision] * N[(c * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 4.79999999999999982

   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-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lift-pow.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 77.8

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

   3. Applied rewrites 77.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 92.5

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

   5. Applied rewrites 92.5%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 92.5

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

   7. Applied rewrites 92.5%

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

   if 4.79999999999999982 < 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-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lift-pow.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 77.8

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

   3. Applied rewrites 77.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lift-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-pow.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lift-pow.f64 N/A

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

      18. pow2 N/A

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. associate-*l* N/A

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

      22. lift-*.f64 N/A

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

      23. associate-*l* N/A

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

   5. Applied rewrites 96.7%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow-prod-down N/A

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

      5. pow2 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

   7. Applied rewrites 85.7%

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

Alternative 3: 92.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(x + x\right)\\ \mathbf{if}\;c \leq 2.5 \cdot 10^{-195}:\\ \;\;\;\;\frac{t\_0}{\left(\left(\left(s \cdot c\right) \cdot x\right) \cdot x\right) \cdot \left(s \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{\left(c \cdot \left(c \cdot \left(s \cdot x\right)\right)\right) \cdot \left(s \cdot x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (cos (+ x x))))
   (if (<= c 2.5e-195)
     (/ t_0 (* (* (* (* s c) x) x) (* s c)))
     (/ t_0 (* (* c (* c (* s x))) (* s x))))))

C

double code(double x, double c, double s) {
	double t_0 = cos((x + x));
	double tmp;
	if (c <= 2.5e-195) {
		tmp = t_0 / ((((s * c) * x) * x) * (s * c));
	} else {
		tmp = t_0 / ((c * (c * (s * x))) * (s * x));
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos((x + x))
    if (c <= 2.5d-195) then
        tmp = t_0 / ((((s * c) * x) * x) * (s * c))
    else
        tmp = t_0 / ((c * (c * (s * x))) * (s * x))
    end if
    code = tmp
end function

Java

public static double code(double x, double c, double s) {
	double t_0 = Math.cos((x + x));
	double tmp;
	if (c <= 2.5e-195) {
		tmp = t_0 / ((((s * c) * x) * x) * (s * c));
	} else {
		tmp = t_0 / ((c * (c * (s * x))) * (s * x));
	}
	return tmp;
}

Python

def code(x, c, s):
	t_0 = math.cos((x + x))
	tmp = 0
	if c <= 2.5e-195:
		tmp = t_0 / ((((s * c) * x) * x) * (s * c))
	else:
		tmp = t_0 / ((c * (c * (s * x))) * (s * x))
	return tmp

Julia

function code(x, c, s)
	t_0 = cos(Float64(x + x))
	tmp = 0.0
	if (c <= 2.5e-195)
		tmp = Float64(t_0 / Float64(Float64(Float64(Float64(s * c) * x) * x) * Float64(s * c)));
	else
		tmp = Float64(t_0 / Float64(Float64(c * Float64(c * Float64(s * x))) * Float64(s * x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	t_0 = cos((x + x));
	tmp = 0.0;
	if (c <= 2.5e-195)
		tmp = t_0 / ((((s * c) * x) * x) * (s * c));
	else
		tmp = t_0 / ((c * (c * (s * x))) * (s * x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := Block[{t$95$0 = N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[c, 2.5e-195], N[(t$95$0 / N[(N[(N[(N[(s * c), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * N[(s * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(N[(c * N[(c * N[(s * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(s * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if c < 2.50000000000000004e-195

   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-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lift-pow.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 77.8

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

   3. Applied rewrites 77.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 92.5

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

   5. Applied rewrites 92.5%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 92.5

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*l* N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. associate-*r* N/A

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

      20. lower-*.f64 N/A

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

      21. lower-*.f64 92.8

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

   7. Applied rewrites 92.8%

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

   if 2.50000000000000004e-195 < c

   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-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lift-pow.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 77.8

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

   3. Applied rewrites 77.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 92.5

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

   5. Applied rewrites 92.5%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 92.5

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

   7. Applied rewrites 92.5%

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

Alternative 4: 90.8% accurate, 1.5× speedup

Math

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

FPCore

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

C

double code(double x, double c, double s) {
	return cos((x + x)) / ((c * (c * (s * x))) * (s * 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((x + x)) / ((c * (c * (s * x))) * (s * x))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, c_, s_] := N[(N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] / N[(N[(c * N[(c * N[(s * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(s * x), $MachinePrecision]), $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-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lift-pow.f64 N/A

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

   8. unpow2 N/A

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

   9. associate-*l* N/A

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

   10. associate-*r* N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. lift-pow.f64 N/A

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

   16. unpow2 N/A

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

   17. lower-*.f64 N/A

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

   18. *-commutative N/A

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

   19. lower-*.f64 77.8

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

3. Applied rewrites 77.8%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lift-*.f64 N/A

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

   7. associate-*l* N/A

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

   8. lower-*.f64 N/A

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

   9. lower-*.f64 92.5

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

5. Applied rewrites 92.5%

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

   1. lift-*.f64 N/A

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

   2. count-2-rev N/A

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

   3. lower-+.f64 92.5

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

7. Applied rewrites 92.5%

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

Alternative 5: 76.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 8.2 \cdot 10^{-150}:\\ \;\;\;\;\frac{1 + -2 \cdot {x}^{2}}{\left(c \cdot \left(c \cdot \left(s \cdot x\right)\right)\right) \cdot \left(s \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(-1\right) \cdot 1}{\left(\left(s \cdot c\right) \cdot x\right) \cdot \left(s \cdot c\right)}}{-x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (if (<= x 8.2e-150)
   (/ (+ 1.0 (* -2.0 (pow x 2.0))) (* (* c (* c (* s x))) (* s x)))
   (/ (/ (* (- 1.0) 1.0) (* (* (* s c) x) (* s c))) (- x))))

C

double code(double x, double c, double s) {
	double tmp;
	if (x <= 8.2e-150) {
		tmp = (1.0 + (-2.0 * pow(x, 2.0))) / ((c * (c * (s * x))) * (s * x));
	} else {
		tmp = ((-1.0 * 1.0) / (((s * c) * x) * (s * c))) / -x;
	}
	return tmp;
}

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
    real(8) :: tmp
    if (x <= 8.2d-150) then
        tmp = (1.0d0 + ((-2.0d0) * (x ** 2.0d0))) / ((c * (c * (s * x))) * (s * x))
    else
        tmp = ((-1.0d0 * 1.0d0) / (((s * c) * x) * (s * c))) / -x
    end if
    code = tmp
end function

Java

public static double code(double x, double c, double s) {
	double tmp;
	if (x <= 8.2e-150) {
		tmp = (1.0 + (-2.0 * Math.pow(x, 2.0))) / ((c * (c * (s * x))) * (s * x));
	} else {
		tmp = ((-1.0 * 1.0) / (((s * c) * x) * (s * c))) / -x;
	}
	return tmp;
}

Python

def code(x, c, s):
	tmp = 0
	if x <= 8.2e-150:
		tmp = (1.0 + (-2.0 * math.pow(x, 2.0))) / ((c * (c * (s * x))) * (s * x))
	else:
		tmp = ((-1.0 * 1.0) / (((s * c) * x) * (s * c))) / -x
	return tmp

Julia

function code(x, c, s)
	tmp = 0.0
	if (x <= 8.2e-150)
		tmp = Float64(Float64(1.0 + Float64(-2.0 * (x ^ 2.0))) / Float64(Float64(c * Float64(c * Float64(s * x))) * Float64(s * x)));
	else
		tmp = Float64(Float64(Float64(Float64(-1.0) * 1.0) / Float64(Float64(Float64(s * c) * x) * Float64(s * c))) / Float64(-x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	tmp = 0.0;
	if (x <= 8.2e-150)
		tmp = (1.0 + (-2.0 * (x ^ 2.0))) / ((c * (c * (s * x))) * (s * x));
	else
		tmp = ((-1.0 * 1.0) / (((s * c) * x) * (s * c))) / -x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := If[LessEqual[x, 8.2e-150], N[(N[(1.0 + N[(-2.0 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(c * N[(c * N[(s * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(s * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[((-1.0) * 1.0), $MachinePrecision] / N[(N[(N[(s * c), $MachinePrecision] * x), $MachinePrecision] * N[(s * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / (-x)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 8.1999999999999997e-150

   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-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lift-pow.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 77.8

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

   3. Applied rewrites 77.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 92.5

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

   5. Applied rewrites 92.5%

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

   6. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 59.4

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

   8. Applied rewrites 59.4%

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

   if 8.1999999999999997e-150 < 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. Taylor expanded in x around 0

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

   4. Applied rewrites 58.5%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. associate-*r* N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 58.6

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

   6. Applied rewrites 58.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 59.8

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

   8. Applied rewrites 59.8%

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

   10. Applied rewrites 76.1%

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

Alternative 6: 74.9% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(s \cdot c\right) \cdot x\\ \mathbf{if}\;x \leq 1200000000:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(-2, x \cdot x, 1\right)}{t\_0}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 \cdot 1}{c \cdot x} \cdot \frac{1}{s}}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (* (* s c) x)))
   (if (<= x 1200000000.0)
     (/ (/ (fma -2.0 (* x x) 1.0) t_0) t_0)
     (/ (* (/ (* 1.0 1.0) (* c x)) (/ 1.0 s)) t_0))))

C

double code(double x, double c, double s) {
	double t_0 = (s * c) * x;
	double tmp;
	if (x <= 1200000000.0) {
		tmp = (fma(-2.0, (x * x), 1.0) / t_0) / t_0;
	} else {
		tmp = (((1.0 * 1.0) / (c * x)) * (1.0 / s)) / t_0;
	}
	return tmp;
}

Julia

function code(x, c, s)
	t_0 = Float64(Float64(s * c) * x)
	tmp = 0.0
	if (x <= 1200000000.0)
		tmp = Float64(Float64(fma(-2.0, Float64(x * x), 1.0) / t_0) / t_0);
	else
		tmp = Float64(Float64(Float64(Float64(1.0 * 1.0) / Float64(c * x)) * Float64(1.0 / s)) / t_0);
	end
	return tmp
end

Wolfram

code[x_, c_, s_] := Block[{t$95$0 = N[(N[(s * c), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, 1200000000.0], N[(N[(N[(-2.0 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[(N[(1.0 * 1.0), $MachinePrecision] / N[(c * x), $MachinePrecision]), $MachinePrecision] * N[(1.0 / s), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.2e9

   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-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lift-pow.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 77.8

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

   3. Applied rewrites 77.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lift-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-pow.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lift-pow.f64 N/A

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

      18. pow2 N/A

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. associate-*l* N/A

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

      22. lift-*.f64 N/A

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

      23. associate-*l* N/A

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

   5. Applied rewrites 96.7%

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

   6. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 61.4

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

   8. Applied rewrites 61.4%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 59.0

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 59.0

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

   10. Applied rewrites 59.0%

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

   if 1.2e9 < 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. Taylor expanded in x around 0

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

   4. Applied rewrites 58.5%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. associate-*r* N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 58.6

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

   6. Applied rewrites 58.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 59.8

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

   8. Applied rewrites 59.8%

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

   10. Applied rewrites 76.7%

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

Alternative 7: 74.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq 4 \cdot 10^{-90}:\\ \;\;\;\;\frac{\frac{\left(-1\right) \cdot 1}{\left(\left(s \cdot c\right) \cdot x\right) \cdot \left(s \cdot c\right)}}{-x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{-c} \cdot 1}{\left(-c\right) \cdot \left(\left(s \cdot x\right) \cdot \left(s \cdot x\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (if (<= c 4e-90)
   (/ (/ (* (- 1.0) 1.0) (* (* (* s c) x) (* s c))) (- x))
   (/ (* (/ 1.0 (- c)) 1.0) (* (- c) (* (* s x) (* s x))))))

C

double code(double x, double c, double s) {
	double tmp;
	if (c <= 4e-90) {
		tmp = ((-1.0 * 1.0) / (((s * c) * x) * (s * c))) / -x;
	} else {
		tmp = ((1.0 / -c) * 1.0) / (-c * ((s * x) * (s * x)));
	}
	return tmp;
}

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
    real(8) :: tmp
    if (c <= 4d-90) then
        tmp = ((-1.0d0 * 1.0d0) / (((s * c) * x) * (s * c))) / -x
    else
        tmp = ((1.0d0 / -c) * 1.0d0) / (-c * ((s * x) * (s * x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double c, double s) {
	double tmp;
	if (c <= 4e-90) {
		tmp = ((-1.0 * 1.0) / (((s * c) * x) * (s * c))) / -x;
	} else {
		tmp = ((1.0 / -c) * 1.0) / (-c * ((s * x) * (s * x)));
	}
	return tmp;
}

Python

def code(x, c, s):
	tmp = 0
	if c <= 4e-90:
		tmp = ((-1.0 * 1.0) / (((s * c) * x) * (s * c))) / -x
	else:
		tmp = ((1.0 / -c) * 1.0) / (-c * ((s * x) * (s * x)))
	return tmp

Julia

function code(x, c, s)
	tmp = 0.0
	if (c <= 4e-90)
		tmp = Float64(Float64(Float64(Float64(-1.0) * 1.0) / Float64(Float64(Float64(s * c) * x) * Float64(s * c))) / Float64(-x));
	else
		tmp = Float64(Float64(Float64(1.0 / Float64(-c)) * 1.0) / Float64(Float64(-c) * Float64(Float64(s * x) * Float64(s * x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	tmp = 0.0;
	if (c <= 4e-90)
		tmp = ((-1.0 * 1.0) / (((s * c) * x) * (s * c))) / -x;
	else
		tmp = ((1.0 / -c) * 1.0) / (-c * ((s * x) * (s * x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := If[LessEqual[c, 4e-90], N[(N[(N[((-1.0) * 1.0), $MachinePrecision] / N[(N[(N[(s * c), $MachinePrecision] * x), $MachinePrecision] * N[(s * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / (-x)), $MachinePrecision], N[(N[(N[(1.0 / (-c)), $MachinePrecision] * 1.0), $MachinePrecision] / N[((-c) * N[(N[(s * x), $MachinePrecision] * N[(s * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < 3.99999999999999998e-90

   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

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

   4. Applied rewrites 58.5%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. associate-*r* N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 58.6

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

   6. Applied rewrites 58.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 59.8

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

   8. Applied rewrites 59.8%

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

   10. Applied rewrites 76.1%

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

   if 3.99999999999999998e-90 < c

   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

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

   4. Applied rewrites 58.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. mult-flip N/A

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

      8. lift-*.f64 N/A

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

      9. sqr-neg-rev N/A

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

      10. associate-/r* N/A

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

      11. frac-times N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-*.f64 N/A

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

   6. Applied rewrites 64.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. unswap-sqr N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 72.5

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

   8. Applied rewrites 72.5%

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

Alternative 8: 73.2% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq 4 \cdot 10^{-90}:\\ \;\;\;\;\frac{1 \cdot \frac{1}{x}}{\left(\left(s \cdot c\right) \cdot x\right) \cdot \left(s \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{-c} \cdot 1}{\left(-c\right) \cdot \left(\left(s \cdot x\right) \cdot \left(s \cdot x\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (if (<= c 4e-90)
   (/ (* 1.0 (/ 1.0 x)) (* (* (* s c) x) (* s c)))
   (/ (* (/ 1.0 (- c)) 1.0) (* (- c) (* (* s x) (* s x))))))

C

double code(double x, double c, double s) {
	double tmp;
	if (c <= 4e-90) {
		tmp = (1.0 * (1.0 / x)) / (((s * c) * x) * (s * c));
	} else {
		tmp = ((1.0 / -c) * 1.0) / (-c * ((s * x) * (s * x)));
	}
	return tmp;
}

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
    real(8) :: tmp
    if (c <= 4d-90) then
        tmp = (1.0d0 * (1.0d0 / x)) / (((s * c) * x) * (s * c))
    else
        tmp = ((1.0d0 / -c) * 1.0d0) / (-c * ((s * x) * (s * x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double c, double s) {
	double tmp;
	if (c <= 4e-90) {
		tmp = (1.0 * (1.0 / x)) / (((s * c) * x) * (s * c));
	} else {
		tmp = ((1.0 / -c) * 1.0) / (-c * ((s * x) * (s * x)));
	}
	return tmp;
}

Python

def code(x, c, s):
	tmp = 0
	if c <= 4e-90:
		tmp = (1.0 * (1.0 / x)) / (((s * c) * x) * (s * c))
	else:
		tmp = ((1.0 / -c) * 1.0) / (-c * ((s * x) * (s * x)))
	return tmp

Julia

function code(x, c, s)
	tmp = 0.0
	if (c <= 4e-90)
		tmp = Float64(Float64(1.0 * Float64(1.0 / x)) / Float64(Float64(Float64(s * c) * x) * Float64(s * c)));
	else
		tmp = Float64(Float64(Float64(1.0 / Float64(-c)) * 1.0) / Float64(Float64(-c) * Float64(Float64(s * x) * Float64(s * x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	tmp = 0.0;
	if (c <= 4e-90)
		tmp = (1.0 * (1.0 / x)) / (((s * c) * x) * (s * c));
	else
		tmp = ((1.0 / -c) * 1.0) / (-c * ((s * x) * (s * x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := If[LessEqual[c, 4e-90], N[(N[(1.0 * N[(1.0 / x), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(s * c), $MachinePrecision] * x), $MachinePrecision] * N[(s * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / (-c)), $MachinePrecision] * 1.0), $MachinePrecision] / N[((-c) * N[(N[(s * x), $MachinePrecision] * N[(s * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < 3.99999999999999998e-90

   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

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

   4. Applied rewrites 58.5%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. associate-*r* N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 58.6

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

   6. Applied rewrites 58.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 59.8

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

   8. Applied rewrites 59.8%

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

   10. Applied rewrites 76.2%

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

   if 3.99999999999999998e-90 < c

   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

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

   4. Applied rewrites 58.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. mult-flip N/A

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

      8. lift-*.f64 N/A

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

      9. sqr-neg-rev N/A

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

      10. associate-/r* N/A

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

      11. frac-times N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-*.f64 N/A

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

   6. Applied rewrites 64.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. unswap-sqr N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 72.5

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

   8. Applied rewrites 72.5%

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

Alternative 9: 72.9% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq 7.2 \cdot 10^{-53}:\\ \;\;\;\;\frac{1 \cdot \frac{1}{x}}{\left(\left(s \cdot c\right) \cdot x\right) \cdot \left(s \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{c \cdot c}}{x}}{\left(s \cdot x\right) \cdot s}\\ \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (if (<= c 7.2e-53)
   (/ (* 1.0 (/ 1.0 x)) (* (* (* s c) x) (* s c)))
   (/ (/ (/ 1.0 (* c c)) x) (* (* s x) s))))

C

double code(double x, double c, double s) {
	double tmp;
	if (c <= 7.2e-53) {
		tmp = (1.0 * (1.0 / x)) / (((s * c) * x) * (s * c));
	} else {
		tmp = ((1.0 / (c * c)) / x) / ((s * x) * s);
	}
	return tmp;
}

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
    real(8) :: tmp
    if (c <= 7.2d-53) then
        tmp = (1.0d0 * (1.0d0 / x)) / (((s * c) * x) * (s * c))
    else
        tmp = ((1.0d0 / (c * c)) / x) / ((s * x) * s)
    end if
    code = tmp
end function

Java

public static double code(double x, double c, double s) {
	double tmp;
	if (c <= 7.2e-53) {
		tmp = (1.0 * (1.0 / x)) / (((s * c) * x) * (s * c));
	} else {
		tmp = ((1.0 / (c * c)) / x) / ((s * x) * s);
	}
	return tmp;
}

Python

def code(x, c, s):
	tmp = 0
	if c <= 7.2e-53:
		tmp = (1.0 * (1.0 / x)) / (((s * c) * x) * (s * c))
	else:
		tmp = ((1.0 / (c * c)) / x) / ((s * x) * s)
	return tmp

Julia

function code(x, c, s)
	tmp = 0.0
	if (c <= 7.2e-53)
		tmp = Float64(Float64(1.0 * Float64(1.0 / x)) / Float64(Float64(Float64(s * c) * x) * Float64(s * c)));
	else
		tmp = Float64(Float64(Float64(1.0 / Float64(c * c)) / x) / Float64(Float64(s * x) * s));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	tmp = 0.0;
	if (c <= 7.2e-53)
		tmp = (1.0 * (1.0 / x)) / (((s * c) * x) * (s * c));
	else
		tmp = ((1.0 / (c * c)) / x) / ((s * x) * s);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := If[LessEqual[c, 7.2e-53], N[(N[(1.0 * N[(1.0 / x), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(s * c), $MachinePrecision] * x), $MachinePrecision] * N[(s * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / N[(c * c), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / N[(N[(s * x), $MachinePrecision] * s), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < 7.1999999999999998e-53

   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

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

   4. Applied rewrites 58.5%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. associate-*r* N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 58.6

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

   6. Applied rewrites 58.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 59.8

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

   8. Applied rewrites 59.8%

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

   10. Applied rewrites 76.2%

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

   if 7.1999999999999998e-53 < c

   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

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

   4. Applied rewrites 58.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. lift-*.f64 N/A

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

      14. associate-/r* N/A

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

      15. lower-/.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-/.f64 59.0

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

   6. Applied rewrites 59.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 63.9

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

   8. Applied rewrites 63.9%

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

Alternative 10: 72.8% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq 3 \cdot 10^{-19}:\\ \;\;\;\;\frac{1}{\left(\left(\left(s \cdot c\right) \cdot x\right) \cdot x\right) \cdot \left(s \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{c \cdot c}}{x}}{\left(s \cdot x\right) \cdot s}\\ \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (if (<= c 3e-19)
   (/ 1.0 (* (* (* (* s c) x) x) (* s c)))
   (/ (/ (/ 1.0 (* c c)) x) (* (* s x) s))))

C

double code(double x, double c, double s) {
	double tmp;
	if (c <= 3e-19) {
		tmp = 1.0 / ((((s * c) * x) * x) * (s * c));
	} else {
		tmp = ((1.0 / (c * c)) / x) / ((s * x) * s);
	}
	return tmp;
}

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
    real(8) :: tmp
    if (c <= 3d-19) then
        tmp = 1.0d0 / ((((s * c) * x) * x) * (s * c))
    else
        tmp = ((1.0d0 / (c * c)) / x) / ((s * x) * s)
    end if
    code = tmp
end function

Java

public static double code(double x, double c, double s) {
	double tmp;
	if (c <= 3e-19) {
		tmp = 1.0 / ((((s * c) * x) * x) * (s * c));
	} else {
		tmp = ((1.0 / (c * c)) / x) / ((s * x) * s);
	}
	return tmp;
}

Python

def code(x, c, s):
	tmp = 0
	if c <= 3e-19:
		tmp = 1.0 / ((((s * c) * x) * x) * (s * c))
	else:
		tmp = ((1.0 / (c * c)) / x) / ((s * x) * s)
	return tmp

Julia

function code(x, c, s)
	tmp = 0.0
	if (c <= 3e-19)
		tmp = Float64(1.0 / Float64(Float64(Float64(Float64(s * c) * x) * x) * Float64(s * c)));
	else
		tmp = Float64(Float64(Float64(1.0 / Float64(c * c)) / x) / Float64(Float64(s * x) * s));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	tmp = 0.0;
	if (c <= 3e-19)
		tmp = 1.0 / ((((s * c) * x) * x) * (s * c));
	else
		tmp = ((1.0 / (c * c)) / x) / ((s * x) * s);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := If[LessEqual[c, 3e-19], N[(1.0 / N[(N[(N[(N[(s * c), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * N[(s * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / N[(c * c), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / N[(N[(s * x), $MachinePrecision] * s), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < 2.99999999999999993e-19

   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

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

   4. Applied rewrites 58.5%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. associate-*r* N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 58.6

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

   6. Applied rewrites 58.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 59.8

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

   8. Applied rewrites 59.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lift-*.f64 N/A

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

      8. unswap-sqr N/A

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

      9. swap-sqr N/A

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

      10. associate-*l* N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*l* N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. associate-*r* N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 76.1

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

   10. Applied rewrites 76.1%

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

   if 2.99999999999999993e-19 < c

   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

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

   4. Applied rewrites 58.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. lift-*.f64 N/A

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

      14. associate-/r* N/A

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

      15. lower-/.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-/.f64 59.0

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

   6. Applied rewrites 59.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 63.9

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

   8. Applied rewrites 63.9%

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

Alternative 11: 68.5% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq 4.8 \cdot 10^{-16}:\\ \;\;\;\;\frac{1}{\left(\left(\left(s \cdot c\right) \cdot x\right) \cdot x\right) \cdot \left(s \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x \cdot \left(\left(c \cdot c\right) \cdot \left(s \cdot x\right)\right)}}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (if (<= c 4.8e-16)
   (/ 1.0 (* (* (* (* s c) x) x) (* s c)))
   (/ (/ 1.0 (* x (* (* c c) (* s x)))) s)))

C

double code(double x, double c, double s) {
	double tmp;
	if (c <= 4.8e-16) {
		tmp = 1.0 / ((((s * c) * x) * x) * (s * c));
	} else {
		tmp = (1.0 / (x * ((c * c) * (s * x)))) / s;
	}
	return tmp;
}

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
    real(8) :: tmp
    if (c <= 4.8d-16) then
        tmp = 1.0d0 / ((((s * c) * x) * x) * (s * c))
    else
        tmp = (1.0d0 / (x * ((c * c) * (s * x)))) / s
    end if
    code = tmp
end function

Java

public static double code(double x, double c, double s) {
	double tmp;
	if (c <= 4.8e-16) {
		tmp = 1.0 / ((((s * c) * x) * x) * (s * c));
	} else {
		tmp = (1.0 / (x * ((c * c) * (s * x)))) / s;
	}
	return tmp;
}

Python

def code(x, c, s):
	tmp = 0
	if c <= 4.8e-16:
		tmp = 1.0 / ((((s * c) * x) * x) * (s * c))
	else:
		tmp = (1.0 / (x * ((c * c) * (s * x)))) / s
	return tmp

Julia

function code(x, c, s)
	tmp = 0.0
	if (c <= 4.8e-16)
		tmp = Float64(1.0 / Float64(Float64(Float64(Float64(s * c) * x) * x) * Float64(s * c)));
	else
		tmp = Float64(Float64(1.0 / Float64(x * Float64(Float64(c * c) * Float64(s * x)))) / s);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	tmp = 0.0;
	if (c <= 4.8e-16)
		tmp = 1.0 / ((((s * c) * x) * x) * (s * c));
	else
		tmp = (1.0 / (x * ((c * c) * (s * x)))) / s;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := If[LessEqual[c, 4.8e-16], N[(1.0 / N[(N[(N[(N[(s * c), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * N[(s * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(x * N[(N[(c * c), $MachinePrecision] * N[(s * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / s), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < 4.8000000000000001e-16

   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

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

   4. Applied rewrites 58.5%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. associate-*r* N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 58.6

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

   6. Applied rewrites 58.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 59.8

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

   8. Applied rewrites 59.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lift-*.f64 N/A

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

      8. unswap-sqr N/A

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

      9. swap-sqr N/A

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

      10. associate-*l* N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*l* N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. associate-*r* N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 76.1

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

   10. Applied rewrites 76.1%

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

   if 4.8000000000000001e-16 < c

   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

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. lift-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

   3. Applied rewrites 77.0%

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

   4. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 60.3

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

   6. Applied rewrites 60.3%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. *-commutative N/A

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

      13. associate-*l* N/A

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

      14. *-commutative N/A

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

      15. associate-*l* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 66.2

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

   8. Applied rewrites 66.2%

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

Alternative 12: 67.9% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\left(\left(\left(s \cdot c\right) \cdot x\right) \cdot x\right) \cdot \left(s \cdot c\right)} \end{array} \]

FPCore

(FPCore (x c s) :precision binary64 (/ 1.0 (* (* (* (* s c) x) x) (* s c))))

C

double code(double x, double c, double s) {
	return 1.0 / ((((s * c) * x) * x) * (s * c));
}

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 = 1.0d0 / ((((s * c) * x) * x) * (s * c))
end function

Java

public static double code(double x, double c, double s) {
	return 1.0 / ((((s * c) * x) * x) * (s * c));
}

Python

def code(x, c, s):
	return 1.0 / ((((s * c) * x) * x) * (s * c))

Julia

function code(x, c, s)
	return Float64(1.0 / Float64(Float64(Float64(Float64(s * c) * x) * x) * Float64(s * c)))
end

MATLAB

function tmp = code(x, c, s)
	tmp = 1.0 / ((((s * c) * x) * x) * (s * c));
end

Wolfram

code[x_, c_, s_] := N[(1.0 / N[(N[(N[(N[(s * c), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * N[(s * c), $MachinePrecision]), $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

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

4. Applied rewrites 58.5%

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

   1. lift-*.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. pow2 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-pow.f64 N/A

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

   8. pow2 N/A

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

   9. lift-*.f64 N/A

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

   10. associate-*l* N/A

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

   11. lift-*.f64 N/A

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

   12. associate-*l* N/A

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

   13. lift-*.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. *-commutative N/A

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

   16. associate-*r* N/A

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

   17. lower-*.f64 N/A

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

   18. lower-*.f64 58.6

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

6. Applied rewrites 58.6%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*l* N/A

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

   7. lower-*.f64 N/A

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

   8. lower-*.f64 59.8

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

8. Applied rewrites 59.8%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*r* N/A

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

   7. lift-*.f64 N/A

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

   8. unswap-sqr N/A

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

   9. swap-sqr N/A

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

   10. associate-*l* N/A

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

   11. lift-*.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. associate-*l* N/A

       \[\leadsto \frac{1}{\left(\left(s \cdot c\right) \cdot x\right) \cdot \color{blue}{\left(\left(s \cdot c\right) \cdot x\right)}} \]

   14. lift-*.f64 N/A

       \[\leadsto \frac{1}{\left(\left(s \cdot c\right) \cdot x\right) \cdot \left(\color{blue}{\left(s \cdot c\right)} \cdot x\right)} \]

   15. *-commutative N/A

       \[\leadsto \frac{1}{\left(\left(s \cdot c\right) \cdot x\right) \cdot \color{blue}{\left(x \cdot \left(s \cdot c\right)\right)}} \]

   16. associate-*r* N/A

       \[\leadsto \frac{1}{\color{blue}{\left(\left(\left(s \cdot c\right) \cdot x\right) \cdot x\right) \cdot \left(s \cdot c\right)}} \]

   17. lower-*.f64 N/A

       \[\leadsto \frac{1}{\color{blue}{\left(\left(\left(s \cdot c\right) \cdot x\right) \cdot x\right) \cdot \left(s \cdot c\right)}} \]

   18. lower-*.f64 76.1

       \[\leadsto \frac{1}{\color{blue}{\left(\left(\left(s \cdot c\right) \cdot x\right) \cdot x\right)} \cdot \left(s \cdot c\right)} \]

10. Applied rewrites 76.1%

    \[\leadsto \color{blue}{\frac{1}{\left(\left(\left(s \cdot c\right) \cdot x\right) \cdot x\right) \cdot \left(s \cdot c\right)}} \]
11. Add Preprocessing

Alternative 13: 66.2% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\left(\left(c \cdot x\right) \cdot \left(c \cdot x\right)\right) \cdot \left(s \cdot s\right)} \end{array} \]

FPCore

(FPCore (x c s) :precision binary64 (/ 1.0 (* (* (* c x) (* c x)) (* s s))))

C

double code(double x, double c, double s) {
	return 1.0 / (((c * x) * (c * x)) * (s * s));
}

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 = 1.0d0 / (((c * x) * (c * x)) * (s * s))
end function

Java

public static double code(double x, double c, double s) {
	return 1.0 / (((c * x) * (c * x)) * (s * s));
}

Python

def code(x, c, s):
	return 1.0 / (((c * x) * (c * x)) * (s * s))

Julia

function code(x, c, s)
	return Float64(1.0 / Float64(Float64(Float64(c * x) * Float64(c * x)) * Float64(s * s)))
end

MATLAB

function tmp = code(x, c, s)
	tmp = 1.0 / (((c * x) * (c * x)) * (s * s));
end

Wolfram

code[x_, c_, s_] := N[(1.0 / N[(N[(N[(c * x), $MachinePrecision] * N[(c * x), $MachinePrecision]), $MachinePrecision] * N[(s * s), $MachinePrecision]), $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

   \[\leadsto \frac{\color{blue}{1}}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
3. Step-by-step derivation

4. Applied rewrites 58.5%

   \[\leadsto \frac{\color{blue}{1}}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{1}{\color{blue}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)}} \]

   2. lift-pow.f64 N/A

      \[\leadsto \frac{1}{\color{blue}{{c}^{2}} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]

   3. pow2 N/A

      \[\leadsto \frac{1}{\color{blue}{\left(c \cdot c\right)} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{1}{\color{blue}{\left(c \cdot c\right)} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{1}{\left(c \cdot c\right) \cdot \color{blue}{\left(\left(x \cdot {s}^{2}\right) \cdot x\right)}} \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{1}{\left(c \cdot c\right) \cdot \left(\color{blue}{\left(x \cdot {s}^{2}\right)} \cdot x\right)} \]

   7. lift-pow.f64 N/A

      \[\leadsto \frac{1}{\left(c \cdot c\right) \cdot \left(\left(x \cdot \color{blue}{{s}^{2}}\right) \cdot x\right)} \]

   8. pow2 N/A

      \[\leadsto \frac{1}{\left(c \cdot c\right) \cdot \left(\left(x \cdot \color{blue}{\left(s \cdot s\right)}\right) \cdot x\right)} \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{1}{\left(c \cdot c\right) \cdot \left(\left(x \cdot \color{blue}{\left(s \cdot s\right)}\right) \cdot x\right)} \]

   10. associate-*l* N/A

       \[\leadsto \frac{1}{\left(c \cdot c\right) \cdot \color{blue}{\left(x \cdot \left(\left(s \cdot s\right) \cdot x\right)\right)}} \]

   11. lift-*.f64 N/A

       \[\leadsto \frac{1}{\left(c \cdot c\right) \cdot \left(x \cdot \color{blue}{\left(\left(s \cdot s\right) \cdot x\right)}\right)} \]

   12. associate-*l* N/A

       \[\leadsto \frac{1}{\color{blue}{\left(\left(c \cdot c\right) \cdot x\right) \cdot \left(\left(s \cdot s\right) \cdot x\right)}} \]

   13. lift-*.f64 N/A

       \[\leadsto \frac{1}{\color{blue}{\left(\left(c \cdot c\right) \cdot x\right)} \cdot \left(\left(s \cdot s\right) \cdot x\right)} \]

   14. lift-*.f64 N/A

       \[\leadsto \frac{1}{\left(\left(c \cdot c\right) \cdot x\right) \cdot \color{blue}{\left(\left(s \cdot s\right) \cdot x\right)}} \]

   15. *-commutative N/A

       \[\leadsto \frac{1}{\left(\left(c \cdot c\right) \cdot x\right) \cdot \color{blue}{\left(x \cdot \left(s \cdot s\right)\right)}} \]

   16. associate-*r* N/A

       \[\leadsto \frac{1}{\color{blue}{\left(\left(\left(c \cdot c\right) \cdot x\right) \cdot x\right) \cdot \left(s \cdot s\right)}} \]

   17. lower-*.f64 N/A

       \[\leadsto \frac{1}{\color{blue}{\left(\left(\left(c \cdot c\right) \cdot x\right) \cdot x\right) \cdot \left(s \cdot s\right)}} \]

   18. lower-*.f64 58.6

       \[\leadsto \frac{1}{\color{blue}{\left(\left(\left(c \cdot c\right) \cdot x\right) \cdot x\right)} \cdot \left(s \cdot s\right)} \]

6. Applied rewrites 58.6%

   \[\leadsto \color{blue}{\frac{1}{\left(\left(\left(c \cdot c\right) \cdot x\right) \cdot x\right) \cdot \left(s \cdot s\right)}} \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{1}{\color{blue}{\left(\left(\left(c \cdot c\right) \cdot x\right) \cdot x\right)} \cdot \left(s \cdot s\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{1}{\left(\color{blue}{\left(\left(c \cdot c\right) \cdot x\right)} \cdot x\right) \cdot \left(s \cdot s\right)} \]

   3. associate-*l* N/A

      \[\leadsto \frac{1}{\color{blue}{\left(\left(c \cdot c\right) \cdot \left(x \cdot x\right)\right)} \cdot \left(s \cdot s\right)} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{1}{\left(\color{blue}{\left(c \cdot c\right)} \cdot \left(x \cdot x\right)\right) \cdot \left(s \cdot s\right)} \]

   5. unswap-sqr N/A

      \[\leadsto \frac{1}{\color{blue}{\left(\left(c \cdot x\right) \cdot \left(c \cdot x\right)\right)} \cdot \left(s \cdot s\right)} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{1}{\color{blue}{\left(\left(c \cdot x\right) \cdot \left(c \cdot x\right)\right)} \cdot \left(s \cdot s\right)} \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{1}{\left(\color{blue}{\left(c \cdot x\right)} \cdot \left(c \cdot x\right)\right) \cdot \left(s \cdot s\right)} \]

   8. lower-*.f64 66.2

      \[\leadsto \frac{1}{\left(\left(c \cdot x\right) \cdot \color{blue}{\left(c \cdot x\right)}\right) \cdot \left(s \cdot s\right)} \]

8. Applied rewrites 66.2%

   \[\leadsto \frac{1}{\color{blue}{\left(\left(c \cdot x\right) \cdot \left(c \cdot x\right)\right)} \cdot \left(s \cdot s\right)} \]
9. Add Preprocessing

Alternative 14: 59.8% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\left(c \cdot \left(c \cdot \left(x \cdot x\right)\right)\right) \cdot \left(s \cdot s\right)} \end{array} \]

FPCore

(FPCore (x c s) :precision binary64 (/ 1.0 (* (* c (* c (* x x))) (* s s))))

C

double code(double x, double c, double s) {
	return 1.0 / ((c * (c * (x * x))) * (s * s));
}

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 = 1.0d0 / ((c * (c * (x * x))) * (s * s))
end function

Java

public static double code(double x, double c, double s) {
	return 1.0 / ((c * (c * (x * x))) * (s * s));
}

Python

def code(x, c, s):
	return 1.0 / ((c * (c * (x * x))) * (s * s))

Julia

function code(x, c, s)
	return Float64(1.0 / Float64(Float64(c * Float64(c * Float64(x * x))) * Float64(s * s)))
end

MATLAB

function tmp = code(x, c, s)
	tmp = 1.0 / ((c * (c * (x * x))) * (s * s));
end

Wolfram

code[x_, c_, s_] := N[(1.0 / N[(N[(c * N[(c * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(s * s), $MachinePrecision]), $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

   \[\leadsto \frac{\color{blue}{1}}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
3. Step-by-step derivation

4. Applied rewrites 58.5%

   \[\leadsto \frac{\color{blue}{1}}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{1}{\color{blue}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)}} \]

   2. lift-pow.f64 N/A

      \[\leadsto \frac{1}{\color{blue}{{c}^{2}} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]

   3. pow2 N/A

      \[\leadsto \frac{1}{\color{blue}{\left(c \cdot c\right)} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{1}{\color{blue}{\left(c \cdot c\right)} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{1}{\left(c \cdot c\right) \cdot \color{blue}{\left(\left(x \cdot {s}^{2}\right) \cdot x\right)}} \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{1}{\left(c \cdot c\right) \cdot \left(\color{blue}{\left(x \cdot {s}^{2}\right)} \cdot x\right)} \]

   7. lift-pow.f64 N/A

      \[\leadsto \frac{1}{\left(c \cdot c\right) \cdot \left(\left(x \cdot \color{blue}{{s}^{2}}\right) \cdot x\right)} \]

   8. pow2 N/A

      \[\leadsto \frac{1}{\left(c \cdot c\right) \cdot \left(\left(x \cdot \color{blue}{\left(s \cdot s\right)}\right) \cdot x\right)} \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{1}{\left(c \cdot c\right) \cdot \left(\left(x \cdot \color{blue}{\left(s \cdot s\right)}\right) \cdot x\right)} \]

   10. associate-*l* N/A

       \[\leadsto \frac{1}{\left(c \cdot c\right) \cdot \color{blue}{\left(x \cdot \left(\left(s \cdot s\right) \cdot x\right)\right)}} \]

   11. lift-*.f64 N/A

       \[\leadsto \frac{1}{\left(c \cdot c\right) \cdot \left(x \cdot \color{blue}{\left(\left(s \cdot s\right) \cdot x\right)}\right)} \]

   12. associate-*l* N/A

       \[\leadsto \frac{1}{\color{blue}{\left(\left(c \cdot c\right) \cdot x\right) \cdot \left(\left(s \cdot s\right) \cdot x\right)}} \]

   13. lift-*.f64 N/A

       \[\leadsto \frac{1}{\color{blue}{\left(\left(c \cdot c\right) \cdot x\right)} \cdot \left(\left(s \cdot s\right) \cdot x\right)} \]

   14. lift-*.f64 N/A

       \[\leadsto \frac{1}{\left(\left(c \cdot c\right) \cdot x\right) \cdot \color{blue}{\left(\left(s \cdot s\right) \cdot x\right)}} \]

   15. *-commutative N/A

       \[\leadsto \frac{1}{\left(\left(c \cdot c\right) \cdot x\right) \cdot \color{blue}{\left(x \cdot \left(s \cdot s\right)\right)}} \]

   16. associate-*r* N/A

       \[\leadsto \frac{1}{\color{blue}{\left(\left(\left(c \cdot c\right) \cdot x\right) \cdot x\right) \cdot \left(s \cdot s\right)}} \]

   17. lower-*.f64 N/A

       \[\leadsto \frac{1}{\color{blue}{\left(\left(\left(c \cdot c\right) \cdot x\right) \cdot x\right) \cdot \left(s \cdot s\right)}} \]

   18. lower-*.f64 58.6

       \[\leadsto \frac{1}{\color{blue}{\left(\left(\left(c \cdot c\right) \cdot x\right) \cdot x\right)} \cdot \left(s \cdot s\right)} \]

6. Applied rewrites 58.6%

   \[\leadsto \color{blue}{\frac{1}{\left(\left(\left(c \cdot c\right) \cdot x\right) \cdot x\right) \cdot \left(s \cdot s\right)}} \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{1}{\color{blue}{\left(\left(\left(c \cdot c\right) \cdot x\right) \cdot x\right)} \cdot \left(s \cdot s\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{1}{\left(\color{blue}{\left(\left(c \cdot c\right) \cdot x\right)} \cdot x\right) \cdot \left(s \cdot s\right)} \]

   3. associate-*l* N/A

      \[\leadsto \frac{1}{\color{blue}{\left(\left(c \cdot c\right) \cdot \left(x \cdot x\right)\right)} \cdot \left(s \cdot s\right)} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{1}{\left(\color{blue}{\left(c \cdot c\right)} \cdot \left(x \cdot x\right)\right) \cdot \left(s \cdot s\right)} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{1}{\left(\left(c \cdot c\right) \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left(s \cdot s\right)} \]

   6. associate-*l* N/A

      \[\leadsto \frac{1}{\color{blue}{\left(c \cdot \left(c \cdot \left(x \cdot x\right)\right)\right)} \cdot \left(s \cdot s\right)} \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{1}{\color{blue}{\left(c \cdot \left(c \cdot \left(x \cdot x\right)\right)\right)} \cdot \left(s \cdot s\right)} \]

   8. lower-*.f64 59.8

      \[\leadsto \frac{1}{\left(c \cdot \color{blue}{\left(c \cdot \left(x \cdot x\right)\right)}\right) \cdot \left(s \cdot s\right)} \]

8. Applied rewrites 59.8%

   \[\leadsto \frac{1}{\color{blue}{\left(c \cdot \left(c \cdot \left(x \cdot x\right)\right)\right)} \cdot \left(s \cdot s\right)} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2025162 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))