mixedcos

Percentage Accurate: 67.1% → 97.5%

Time: 3.5s

Alternatives: 13

Speedup: 4.2×

• 31.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 67.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.5% accurate, 1.5× speedup

Math

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

FPCore

(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (* (* c s) x))) (/ (/ (cos (+ x x)) t_0) t_0)))

C

double code(double x, double c, double s) {
	double t_0 = (c * s) * x;
	return (cos((x + x)) / t_0) / t_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
    real(8) :: t_0
    t_0 = (c * s) * x
    code = (cos((x + x)) / t_0) / t_0
end function

Java

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

Python

def code(x, c, s):
	t_0 = (c * s) * x
	return (math.cos((x + x)) / t_0) / t_0

Julia

function code(x, c, s)
	t_0 = Float64(Float64(c * s) * x)
	return Float64(Float64(cos(Float64(x + x)) / t_0) / t_0)
end

MATLAB

function tmp = code(x, c, s)
	t_0 = (c * s) * x;
	tmp = (cos((x + x)) / t_0) / t_0;
end

Wolfram

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

Derivation

1. Initial program 67.1%

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

   1. lift-*.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-pow.f64 N/A

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

   6. *-commutative N/A

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

   7. associate-*r* N/A

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

   8. unpow2 N/A

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

   9. *-commutative N/A

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

   10. pow-prod-down N/A

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

   11. pow-prod-down N/A

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

   12. lower-pow.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. lower-*.f64 97.2

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

3. Applied rewrites 97.2%

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

   1. lift-pow.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 97.2

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

5. Applied rewrites 97.2%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-/r* N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lift-cos.f64 N/A

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

   9. count-2-rev N/A

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

   10. lift-+.f64 97.5

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

   11. lift-*.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. *-commutative N/A

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

   14. associate-*r* N/A

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

   15. lower-*.f64 N/A

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

   16. lower-*.f64 95.5

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

   17. lift-*.f64 N/A

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

   18. lift-*.f64 N/A

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

   19. *-commutative N/A

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

   20. associate-*r* N/A

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

   21. lower-*.f64 N/A

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

   22. lower-*.f64 97.5

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

7. Applied rewrites 97.5%

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

Alternative 2: 97.2% accurate, 1.5× speedup

Math

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

FPCore

(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (* (* s x) c))) (/ (cos (+ x x)) (* t_0 t_0))))

C

double code(double x, double c, double s) {
	double t_0 = (s * x) * c;
	return cos((x + x)) / (t_0 * t_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
    real(8) :: t_0
    t_0 = (s * x) * c
    code = cos((x + x)) / (t_0 * t_0)
end function

Java

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

Python

def code(x, c, s):
	t_0 = (s * x) * c
	return math.cos((x + x)) / (t_0 * t_0)

Julia

function code(x, c, s)
	t_0 = Float64(Float64(s * x) * c)
	return Float64(cos(Float64(x + x)) / Float64(t_0 * t_0))
end

MATLAB

function tmp = code(x, c, s)
	t_0 = (s * x) * c;
	tmp = cos((x + x)) / (t_0 * t_0);
end

Wolfram

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

Derivation

1. Initial program 67.1%

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

   1. lift-*.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-pow.f64 N/A

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

   6. *-commutative N/A

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

   7. associate-*r* N/A

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

   8. unpow2 N/A

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

   9. *-commutative N/A

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

   10. pow-prod-down N/A

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

   11. pow-prod-down N/A

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

   12. lower-pow.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. lower-*.f64 97.2

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

3. Applied rewrites 97.2%

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

   1. lift-pow.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 97.2

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

5. Applied rewrites 97.2%

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

   1. lift-*.f64 N/A

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

   2. count-2-rev N/A

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

   3. lift-+.f64 97.2

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

7. Applied rewrites 97.2%

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

Alternative 3: 81.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(s \cdot x\right) \cdot c\\ \mathbf{if}\;s \leq 1.8 \cdot 10^{+84}:\\ \;\;\;\;\frac{\cos \left(x + x\right)}{\left(\left(c \cdot c\right) \cdot x\right) \cdot \left(\left(s \cdot s\right) \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t\_0 \cdot t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (* (* s x) c)))
   (if (<= s 1.8e+84)
     (/ (cos (+ x x)) (* (* (* c c) x) (* (* s s) x)))
     (/ 1.0 (* t_0 t_0)))))

C

double code(double x, double c, double s) {
	double t_0 = (s * x) * c;
	double tmp;
	if (s <= 1.8e+84) {
		tmp = cos((x + x)) / (((c * c) * x) * ((s * s) * x));
	} else {
		tmp = 1.0 / (t_0 * t_0);
	}
	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 = (s * x) * c
    if (s <= 1.8d+84) then
        tmp = cos((x + x)) / (((c * c) * x) * ((s * s) * x))
    else
        tmp = 1.0d0 / (t_0 * t_0)
    end if
    code = tmp
end function

Java

public static double code(double x, double c, double s) {
	double t_0 = (s * x) * c;
	double tmp;
	if (s <= 1.8e+84) {
		tmp = Math.cos((x + x)) / (((c * c) * x) * ((s * s) * x));
	} else {
		tmp = 1.0 / (t_0 * t_0);
	}
	return tmp;
}

Python

def code(x, c, s):
	t_0 = (s * x) * c
	tmp = 0
	if s <= 1.8e+84:
		tmp = math.cos((x + x)) / (((c * c) * x) * ((s * s) * x))
	else:
		tmp = 1.0 / (t_0 * t_0)
	return tmp

Julia

function code(x, c, s)
	t_0 = Float64(Float64(s * x) * c)
	tmp = 0.0
	if (s <= 1.8e+84)
		tmp = Float64(cos(Float64(x + x)) / Float64(Float64(Float64(c * c) * x) * Float64(Float64(s * s) * x)));
	else
		tmp = Float64(1.0 / Float64(t_0 * t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	t_0 = (s * x) * c;
	tmp = 0.0;
	if (s <= 1.8e+84)
		tmp = cos((x + x)) / (((c * c) * x) * ((s * s) * x));
	else
		tmp = 1.0 / (t_0 * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := Block[{t$95$0 = N[(N[(s * x), $MachinePrecision] * c), $MachinePrecision]}, If[LessEqual[s, 1.8e+84], N[(N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] / N[(N[(N[(c * c), $MachinePrecision] * x), $MachinePrecision] * N[(N[(s * s), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if s < 1.8e84

   1. Initial program 67.9%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 67.9

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 69.7

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

   3. Applied rewrites 69.7%

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

   if 1.8e84 < s

   1. Initial program 63.9%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. unpow2 N/A

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

      9. *-commutative N/A

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

      10. pow-prod-down N/A

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

      11. pow-prod-down N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 96.5

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

   3. Applied rewrites 96.5%

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

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 96.5

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

   5. Applied rewrites 96.5%

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

   6. Taylor expanded in x around 0

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

      1. count-2-rev 91.5

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

   8. Applied rewrites 91.5%

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

Alternative 4: 78.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(s \cdot x\right) \cdot c\\ t_1 := t\_0 \cdot t\_0\\ \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \leq -2 \cdot 10^{-152}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot x, -2, 1\right)}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (* (* s x) c)) (t_1 (* t_0 t_0)))
   (if (<= (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))) -2e-152)
     (/ (fma (* x x) -2.0 1.0) t_1)
     (/ 1.0 t_1))))

C

double code(double x, double c, double s) {
	double t_0 = (s * x) * c;
	double t_1 = t_0 * t_0;
	double tmp;
	if ((cos((2.0 * x)) / (pow(c, 2.0) * ((x * pow(s, 2.0)) * x))) <= -2e-152) {
		tmp = fma((x * x), -2.0, 1.0) / t_1;
	} else {
		tmp = 1.0 / t_1;
	}
	return tmp;
}

Julia

function code(x, c, s)
	t_0 = Float64(Float64(s * x) * c)
	t_1 = Float64(t_0 * t_0)
	tmp = 0.0
	if (Float64(cos(Float64(2.0 * x)) / Float64((c ^ 2.0) * Float64(Float64(x * (s ^ 2.0)) * x))) <= -2e-152)
		tmp = Float64(fma(Float64(x * x), -2.0, 1.0) / t_1);
	else
		tmp = Float64(1.0 / t_1);
	end
	return tmp
end

Wolfram

code[x_, c_, s_] := Block[{t$95$0 = N[(N[(s * x), $MachinePrecision] * c), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * t$95$0), $MachinePrecision]}, If[LessEqual[N[(N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] / N[(N[Power[c, 2.0], $MachinePrecision] * N[(N[(x * N[Power[s, 2.0], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2e-152], N[(N[(N[(x * x), $MachinePrecision] * -2.0 + 1.0), $MachinePrecision] / t$95$1), $MachinePrecision], N[(1.0 / t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x))) < -2.00000000000000013e-152

   1. Initial program 64.1%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. unpow2 N/A

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

      9. *-commutative N/A

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

      10. pow-prod-down N/A

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

      11. pow-prod-down N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 99.3

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

   3. Applied rewrites 99.3%

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

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 99.3

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in x around 0

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

      1. count-2-rev N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 45.0

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

   8. Applied rewrites 45.0%

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

   if -2.00000000000000013e-152 < (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x)))

   1. Initial program 67.4%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. unpow2 N/A

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

      9. *-commutative N/A

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

      10. pow-prod-down N/A

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

      11. pow-prod-down N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 97.0

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

   3. Applied rewrites 97.0%

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

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 97.0

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

   5. Applied rewrites 97.0%

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

   6. Taylor expanded in x around 0

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

      1. count-2-rev 85.2

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

   8. Applied rewrites 85.2%

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

Alternative 5: 73.7% accurate, 4.2× speedup

Math

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

FPCore

(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (* (* s x) c))) (/ 1.0 (* t_0 t_0))))

C

double code(double x, double c, double s) {
	double t_0 = (s * x) * c;
	return 1.0 / (t_0 * t_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
    real(8) :: t_0
    t_0 = (s * x) * c
    code = 1.0d0 / (t_0 * t_0)
end function

Java

public static double code(double x, double c, double s) {
	double t_0 = (s * x) * c;
	return 1.0 / (t_0 * t_0);
}

Python

def code(x, c, s):
	t_0 = (s * x) * c
	return 1.0 / (t_0 * t_0)

Julia

function code(x, c, s)
	t_0 = Float64(Float64(s * x) * c)
	return Float64(1.0 / Float64(t_0 * t_0))
end

MATLAB

function tmp = code(x, c, s)
	t_0 = (s * x) * c;
	tmp = 1.0 / (t_0 * t_0);
end

Wolfram

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

Derivation

1. Initial program 67.1%

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

   1. lift-*.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-pow.f64 N/A

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

   6. *-commutative N/A

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

   7. associate-*r* N/A

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

   8. unpow2 N/A

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

   9. *-commutative N/A

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

   10. pow-prod-down N/A

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

   11. pow-prod-down N/A

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

   12. lower-pow.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. lower-*.f64 97.2

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

3. Applied rewrites 97.2%

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

   1. lift-pow.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 97.2

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

5. Applied rewrites 97.2%

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

6. Taylor expanded in x around 0

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

   1. count-2-rev 78.4

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

8. Applied rewrites 78.4%

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

Alternative 6: 69.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{c}^{2} \leq 2 \cdot 10^{-315}:\\ \;\;\;\;\frac{1}{\left(\left(\left(c \cdot \left(c \cdot x\right)\right) \cdot s\right) \cdot s\right) \cdot x}\\ \mathbf{elif}\;{c}^{2} \leq 0.02:\\ \;\;\;\;\frac{1}{\left(\left(\left(\left(c \cdot c\right) \cdot s\right) \cdot x\right) \cdot s\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(\left(c \cdot c\right) \cdot x\right) \cdot s\right) \cdot \left(s \cdot x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (if (<= (pow c 2.0) 2e-315)
   (/ 1.0 (* (* (* (* c (* c x)) s) s) x))
   (if (<= (pow c 2.0) 0.02)
     (/ 1.0 (* (* (* (* (* c c) s) x) s) x))
     (/ 1.0 (* (* (* (* c c) x) s) (* s x))))))

C

double code(double x, double c, double s) {
	double tmp;
	if (pow(c, 2.0) <= 2e-315) {
		tmp = 1.0 / ((((c * (c * x)) * s) * s) * x);
	} else if (pow(c, 2.0) <= 0.02) {
		tmp = 1.0 / (((((c * c) * s) * x) * s) * x);
	} else {
		tmp = 1.0 / ((((c * c) * x) * s) * (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 ** 2.0d0) <= 2d-315) then
        tmp = 1.0d0 / ((((c * (c * x)) * s) * s) * x)
    else if ((c ** 2.0d0) <= 0.02d0) then
        tmp = 1.0d0 / (((((c * c) * s) * x) * s) * x)
    else
        tmp = 1.0d0 / ((((c * c) * x) * s) * (s * x))
    end if
    code = tmp
end function

Java

public static double code(double x, double c, double s) {
	double tmp;
	if (Math.pow(c, 2.0) <= 2e-315) {
		tmp = 1.0 / ((((c * (c * x)) * s) * s) * x);
	} else if (Math.pow(c, 2.0) <= 0.02) {
		tmp = 1.0 / (((((c * c) * s) * x) * s) * x);
	} else {
		tmp = 1.0 / ((((c * c) * x) * s) * (s * x));
	}
	return tmp;
}

Python

def code(x, c, s):
	tmp = 0
	if math.pow(c, 2.0) <= 2e-315:
		tmp = 1.0 / ((((c * (c * x)) * s) * s) * x)
	elif math.pow(c, 2.0) <= 0.02:
		tmp = 1.0 / (((((c * c) * s) * x) * s) * x)
	else:
		tmp = 1.0 / ((((c * c) * x) * s) * (s * x))
	return tmp

Julia

function code(x, c, s)
	tmp = 0.0
	if ((c ^ 2.0) <= 2e-315)
		tmp = Float64(1.0 / Float64(Float64(Float64(Float64(c * Float64(c * x)) * s) * s) * x));
	elseif ((c ^ 2.0) <= 0.02)
		tmp = Float64(1.0 / Float64(Float64(Float64(Float64(Float64(c * c) * s) * x) * s) * x));
	else
		tmp = Float64(1.0 / Float64(Float64(Float64(Float64(c * c) * x) * s) * Float64(s * x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	tmp = 0.0;
	if ((c ^ 2.0) <= 2e-315)
		tmp = 1.0 / ((((c * (c * x)) * s) * s) * x);
	elseif ((c ^ 2.0) <= 0.02)
		tmp = 1.0 / (((((c * c) * s) * x) * s) * x);
	else
		tmp = 1.0 / ((((c * c) * x) * s) * (s * x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := If[LessEqual[N[Power[c, 2.0], $MachinePrecision], 2e-315], N[(1.0 / N[(N[(N[(N[(c * N[(c * x), $MachinePrecision]), $MachinePrecision] * s), $MachinePrecision] * s), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[c, 2.0], $MachinePrecision], 0.02], N[(1.0 / N[(N[(N[(N[(N[(c * c), $MachinePrecision] * s), $MachinePrecision] * x), $MachinePrecision] * s), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[(N[(c * c), $MachinePrecision] * x), $MachinePrecision] * s), $MachinePrecision] * N[(s * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 c #s(literal 2 binary64)) < 2.0000000019e-315

   1. Initial program 56.7%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 47.4

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

   4. Applied rewrites 47.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. pow2 N/A

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

      4. lower-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 47.5

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*r* N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 53.8

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

   6. Applied rewrites 53.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 61.5

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

   8. Applied rewrites 61.5%

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

   if 2.0000000019e-315 < (pow.f64 c #s(literal 2 binary64)) < 0.0200000000000000004

   1. Initial program 76.9%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 61.2

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

   4. Applied rewrites 61.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. pow2 N/A

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

      4. lower-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 61.3

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*r* N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 66.3

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

   6. Applied rewrites 66.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 71.8

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

   8. Applied rewrites 71.8%

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

   if 0.0200000000000000004 < (pow.f64 c #s(literal 2 binary64))

   1. Initial program 67.2%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 65.1

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

   4. Applied rewrites 65.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. pow2 N/A

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

      4. lower-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 65.3

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*r* N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 70.0

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

   6. Applied rewrites 70.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 72.6

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

   8. Applied rewrites 72.6%

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

Alternative 7: 67.6% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{-27}:\\ \;\;\;\;\frac{1}{\left(\left(\left(c \cdot c\right) \cdot x\right) \cdot s\right) \cdot \left(s \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(\left(c \cdot \left(c \cdot x\right)\right) \cdot s\right) \cdot s\right) \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (if (<= x 2e-27)
   (/ 1.0 (* (* (* (* c c) x) s) (* s x)))
   (/ 1.0 (* (* (* (* c (* c x)) s) s) x))))

C

double code(double x, double c, double s) {
	double tmp;
	if (x <= 2e-27) {
		tmp = 1.0 / ((((c * c) * x) * s) * (s * x));
	} else {
		tmp = 1.0 / ((((c * (c * x)) * s) * 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 (x <= 2d-27) then
        tmp = 1.0d0 / ((((c * c) * x) * s) * (s * x))
    else
        tmp = 1.0d0 / ((((c * (c * x)) * s) * s) * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double c, double s) {
	double tmp;
	if (x <= 2e-27) {
		tmp = 1.0 / ((((c * c) * x) * s) * (s * x));
	} else {
		tmp = 1.0 / ((((c * (c * x)) * s) * s) * x);
	}
	return tmp;
}

Python

def code(x, c, s):
	tmp = 0
	if x <= 2e-27:
		tmp = 1.0 / ((((c * c) * x) * s) * (s * x))
	else:
		tmp = 1.0 / ((((c * (c * x)) * s) * s) * x)
	return tmp

Julia

function code(x, c, s)
	tmp = 0.0
	if (x <= 2e-27)
		tmp = Float64(1.0 / Float64(Float64(Float64(Float64(c * c) * x) * s) * Float64(s * x)));
	else
		tmp = Float64(1.0 / Float64(Float64(Float64(Float64(c * Float64(c * x)) * s) * s) * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	tmp = 0.0;
	if (x <= 2e-27)
		tmp = 1.0 / ((((c * c) * x) * s) * (s * x));
	else
		tmp = 1.0 / ((((c * (c * x)) * s) * s) * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := If[LessEqual[x, 2e-27], N[(1.0 / N[(N[(N[(N[(c * c), $MachinePrecision] * x), $MachinePrecision] * s), $MachinePrecision] * N[(s * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[(N[(c * N[(c * x), $MachinePrecision]), $MachinePrecision] * s), $MachinePrecision] * s), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.0000000000000001e-27

   1. Initial program 67.1%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 62.5

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

   4. Applied rewrites 62.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. pow2 N/A

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

      4. lower-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 62.6

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*r* N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 68.4

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

   6. Applied rewrites 68.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 70.5

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

   8. Applied rewrites 70.5%

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

   if 2.0000000000000001e-27 < x

   1. Initial program 67.3%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 52.9

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

   4. Applied rewrites 52.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. pow2 N/A

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

      4. lower-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 53.0

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*r* N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 56.5

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

   6. Applied rewrites 56.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 60.0

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

   8. Applied rewrites 60.0%

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

Alternative 8: 66.7% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.75 \cdot 10^{+45}:\\ \;\;\;\;\frac{1}{\left(\left(\left(c \cdot c\right) \cdot x\right) \cdot s\right) \cdot \left(s \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(c \cdot \left(c \cdot x\right)\right) \cdot \left(\left(s \cdot s\right) \cdot x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (if (<= x 2.75e+45)
   (/ 1.0 (* (* (* (* c c) x) s) (* s x)))
   (/ 1.0 (* (* c (* c x)) (* (* s s) x)))))

C

double code(double x, double c, double s) {
	double tmp;
	if (x <= 2.75e+45) {
		tmp = 1.0 / ((((c * c) * x) * s) * (s * x));
	} else {
		tmp = 1.0 / ((c * (c * x)) * ((s * 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 (x <= 2.75d+45) then
        tmp = 1.0d0 / ((((c * c) * x) * s) * (s * x))
    else
        tmp = 1.0d0 / ((c * (c * x)) * ((s * s) * x))
    end if
    code = tmp
end function

Java

public static double code(double x, double c, double s) {
	double tmp;
	if (x <= 2.75e+45) {
		tmp = 1.0 / ((((c * c) * x) * s) * (s * x));
	} else {
		tmp = 1.0 / ((c * (c * x)) * ((s * s) * x));
	}
	return tmp;
}

Python

def code(x, c, s):
	tmp = 0
	if x <= 2.75e+45:
		tmp = 1.0 / ((((c * c) * x) * s) * (s * x))
	else:
		tmp = 1.0 / ((c * (c * x)) * ((s * s) * x))
	return tmp

Julia

function code(x, c, s)
	tmp = 0.0
	if (x <= 2.75e+45)
		tmp = Float64(1.0 / Float64(Float64(Float64(Float64(c * c) * x) * s) * Float64(s * x)));
	else
		tmp = Float64(1.0 / Float64(Float64(c * Float64(c * x)) * Float64(Float64(s * s) * x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	tmp = 0.0;
	if (x <= 2.75e+45)
		tmp = 1.0 / ((((c * c) * x) * s) * (s * x));
	else
		tmp = 1.0 / ((c * (c * x)) * ((s * s) * x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := If[LessEqual[x, 2.75e+45], N[(1.0 / N[(N[(N[(N[(c * c), $MachinePrecision] * x), $MachinePrecision] * s), $MachinePrecision] * N[(s * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(c * N[(c * x), $MachinePrecision]), $MachinePrecision] * N[(N[(s * s), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.75e45

   1. Initial program 67.8%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 61.9

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

   4. Applied rewrites 61.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. pow2 N/A

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

      4. lower-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 62.1

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*r* N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 67.5

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

   6. Applied rewrites 67.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 69.5

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

   8. Applied rewrites 69.5%

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

   if 2.75e45 < x

   1. Initial program 64.9%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 52.3

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

   4. Applied rewrites 52.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 56.8

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

   6. Applied rewrites 56.8%

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

Alternative 9: 64.7% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 7 \cdot 10^{-29}:\\ \;\;\;\;\frac{1}{\left(\left(c \cdot c\right) \cdot x\right) \cdot \left(s \cdot \left(s \cdot x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(c \cdot \left(c \cdot x\right)\right) \cdot \left(\left(s \cdot s\right) \cdot x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (if (<= x 7e-29)
   (/ 1.0 (* (* (* c c) x) (* s (* s x))))
   (/ 1.0 (* (* c (* c x)) (* (* s s) x)))))

C

double code(double x, double c, double s) {
	double tmp;
	if (x <= 7e-29) {
		tmp = 1.0 / (((c * c) * x) * (s * (s * x)));
	} else {
		tmp = 1.0 / ((c * (c * x)) * ((s * 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 (x <= 7d-29) then
        tmp = 1.0d0 / (((c * c) * x) * (s * (s * x)))
    else
        tmp = 1.0d0 / ((c * (c * x)) * ((s * s) * x))
    end if
    code = tmp
end function

Java

public static double code(double x, double c, double s) {
	double tmp;
	if (x <= 7e-29) {
		tmp = 1.0 / (((c * c) * x) * (s * (s * x)));
	} else {
		tmp = 1.0 / ((c * (c * x)) * ((s * s) * x));
	}
	return tmp;
}

Python

def code(x, c, s):
	tmp = 0
	if x <= 7e-29:
		tmp = 1.0 / (((c * c) * x) * (s * (s * x)))
	else:
		tmp = 1.0 / ((c * (c * x)) * ((s * s) * x))
	return tmp

Julia

function code(x, c, s)
	tmp = 0.0
	if (x <= 7e-29)
		tmp = Float64(1.0 / Float64(Float64(Float64(c * c) * x) * Float64(s * Float64(s * x))));
	else
		tmp = Float64(1.0 / Float64(Float64(c * Float64(c * x)) * Float64(Float64(s * s) * x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	tmp = 0.0;
	if (x <= 7e-29)
		tmp = 1.0 / (((c * c) * x) * (s * (s * x)));
	else
		tmp = 1.0 / ((c * (c * x)) * ((s * s) * x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := If[LessEqual[x, 7e-29], N[(1.0 / N[(N[(N[(c * c), $MachinePrecision] * x), $MachinePrecision] * N[(s * N[(s * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(c * N[(c * x), $MachinePrecision]), $MachinePrecision] * N[(N[(s * s), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 6.9999999999999995e-29

   1. Initial program 67.1%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 62.5

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

   4. Applied rewrites 62.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 68.0

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

   6. Applied rewrites 68.0%

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

   if 6.9999999999999995e-29 < x

   1. Initial program 67.3%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. associate-*r* N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 53.0

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

   4. Applied rewrites 53.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 56.5

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

   6. Applied rewrites 56.5%

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

Alternative 10: 64.7% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\left(c \cdot \left(c \cdot x\right)\right) \cdot \left(\left(s \cdot s\right) \cdot x\right)} \end{array} \]

FPCore

(FPCore (x c s) :precision binary64 (/ 1.0 (* (* c (* c x)) (* (* s s) x))))

C

double code(double x, double c, double s) {
	return 1.0 / ((c * (c * x)) * ((s * 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 = 1.0d0 / ((c * (c * x)) * ((s * s) * x))
end function

Java

public static double code(double x, double c, double s) {
	return 1.0 / ((c * (c * x)) * ((s * s) * x));
}

Python

def code(x, c, s):
	return 1.0 / ((c * (c * x)) * ((s * s) * x))

Julia

function code(x, c, s)
	return Float64(1.0 / Float64(Float64(c * Float64(c * x)) * Float64(Float64(s * s) * x)))
end

MATLAB

function tmp = code(x, c, s)
	tmp = 1.0 / ((c * (c * x)) * ((s * s) * x));
end

Wolfram

code[x_, c_, s_] := N[(1.0 / N[(N[(c * N[(c * x), $MachinePrecision]), $MachinePrecision] * N[(N[(s * s), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 67.1%

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

2. Taylor expanded in x around 0

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

   1. lower-/.f64 N/A

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

   2. associate-*r* N/A

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

   3. unpow2 N/A

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

   4. associate-*r* N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

   7. associate-*r* N/A

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

   8. *-commutative N/A

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

   9. associate-*r* N/A

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

   10. lower-*.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 N/A

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

   16. unpow2 N/A

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

   17. lower-*.f64 59.8

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

4. Applied rewrites 59.8%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 64.7

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

6. Applied rewrites 64.7%

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

Alternative 11: 47.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \leq \infty:\\ \;\;\;\;\left(x \cdot \frac{x}{\left(s \cdot s\right) \cdot \left(c \cdot c\right)}\right) \cdot 0.6666666666666666\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \frac{x}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c}\right) \cdot 0.6666666666666666\\ \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (if (<= (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))) INFINITY)
   (* (* x (/ x (* (* s s) (* c c)))) 0.6666666666666666)
   (* (* x (/ x (* (* (* s s) c) c))) 0.6666666666666666)))

C

double code(double x, double c, double s) {
	double tmp;
	if ((cos((2.0 * x)) / (pow(c, 2.0) * ((x * pow(s, 2.0)) * x))) <= ((double) INFINITY)) {
		tmp = (x * (x / ((s * s) * (c * c)))) * 0.6666666666666666;
	} else {
		tmp = (x * (x / (((s * s) * c) * c))) * 0.6666666666666666;
	}
	return tmp;
}

Java

public static double code(double x, double c, double s) {
	double tmp;
	if ((Math.cos((2.0 * x)) / (Math.pow(c, 2.0) * ((x * Math.pow(s, 2.0)) * x))) <= Double.POSITIVE_INFINITY) {
		tmp = (x * (x / ((s * s) * (c * c)))) * 0.6666666666666666;
	} else {
		tmp = (x * (x / (((s * s) * c) * c))) * 0.6666666666666666;
	}
	return tmp;
}

Python

def code(x, c, s):
	tmp = 0
	if (math.cos((2.0 * x)) / (math.pow(c, 2.0) * ((x * math.pow(s, 2.0)) * x))) <= math.inf:
		tmp = (x * (x / ((s * s) * (c * c)))) * 0.6666666666666666
	else:
		tmp = (x * (x / (((s * s) * c) * c))) * 0.6666666666666666
	return tmp

Julia

function code(x, c, s)
	tmp = 0.0
	if (Float64(cos(Float64(2.0 * x)) / Float64((c ^ 2.0) * Float64(Float64(x * (s ^ 2.0)) * x))) <= Inf)
		tmp = Float64(Float64(x * Float64(x / Float64(Float64(s * s) * Float64(c * c)))) * 0.6666666666666666);
	else
		tmp = Float64(Float64(x * Float64(x / Float64(Float64(Float64(s * s) * c) * c))) * 0.6666666666666666);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	tmp = 0.0;
	if ((cos((2.0 * x)) / ((c ^ 2.0) * ((x * (s ^ 2.0)) * x))) <= Inf)
		tmp = (x * (x / ((s * s) * (c * c)))) * 0.6666666666666666;
	else
		tmp = (x * (x / (((s * s) * c) * c))) * 0.6666666666666666;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := If[LessEqual[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], Infinity], N[(N[(x * N[(x / N[(N[(s * s), $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.6666666666666666), $MachinePrecision], N[(N[(x * N[(x / N[(N[(N[(s * s), $MachinePrecision] * c), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.6666666666666666), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 81.1%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 30.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.6666666666666666, x \cdot x, -2\right)}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c} \cdot x, x, \frac{1}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c}\right)}{x \cdot x}} \]

   5. Taylor expanded in x around inf

      \[\leadsto \frac{2}{3} \cdot \color{blue}{\frac{{x}^{2}}{{c}^{2} \cdot {s}^{2}}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{{x}^{2}}{{c}^{2} \cdot {s}^{2}} \cdot \frac{2}{3} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{{x}^{2}}{{c}^{2} \cdot {s}^{2}} \cdot \frac{2}{3} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{{x}^{2}}{{c}^{2} \cdot {s}^{2}} \cdot \frac{2}{3} \]

      4. pow2 N/A

         \[\leadsto \frac{x \cdot x}{{c}^{2} \cdot {s}^{2}} \cdot \frac{2}{3} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot x}{{c}^{2} \cdot {s}^{2}} \cdot \frac{2}{3} \]

      6. *-commutative N/A

         \[\leadsto \frac{x \cdot x}{{s}^{2} \cdot {c}^{2}} \cdot \frac{2}{3} \]

      7. pow2 N/A

         \[\leadsto \frac{x \cdot x}{{s}^{2} \cdot \left(c \cdot c\right)} \cdot \frac{2}{3} \]

      8. associate-*l* N/A

         \[\leadsto \frac{x \cdot x}{\left({s}^{2} \cdot c\right) \cdot c} \cdot \frac{2}{3} \]

      9. pow2 N/A

         \[\leadsto \frac{x \cdot x}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c} \cdot \frac{2}{3} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{x \cdot x}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c} \cdot \frac{2}{3} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{x \cdot x}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c} \cdot \frac{2}{3} \]

      12. lift-*.f64 30.9

          \[\leadsto \frac{x \cdot x}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c} \cdot 0.6666666666666666 \]

   7. Applied rewrites 30.9%

      \[\leadsto \frac{x \cdot x}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c} \cdot \color{blue}{0.6666666666666666} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot x}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c} \cdot \frac{2}{3} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{x \cdot x}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c} \cdot \frac{2}{3} \]

      3. associate-/l* N/A

         \[\leadsto \left(x \cdot \frac{x}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c}\right) \cdot \frac{2}{3} \]

      4. lower-*.f64 N/A

         \[\leadsto \left(x \cdot \frac{x}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c}\right) \cdot \frac{2}{3} \]

      5. lower-/.f64 50.2

         \[\leadsto \left(x \cdot \frac{x}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c}\right) \cdot 0.6666666666666666 \]

   9. Applied rewrites 50.2%

      \[\leadsto \left(x \cdot \frac{x}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c}\right) \cdot 0.6666666666666666 \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \left(x \cdot \frac{x}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c}\right) \cdot \frac{2}{3} \]

       2. lift-*.f64 N/A

          \[\leadsto \left(x \cdot \frac{x}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c}\right) \cdot \frac{2}{3} \]

       3. lift-*.f64 N/A

          \[\leadsto \left(x \cdot \frac{x}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c}\right) \cdot \frac{2}{3} \]

       4. pow2 N/A

          \[\leadsto \left(x \cdot \frac{x}{\left({s}^{2} \cdot c\right) \cdot c}\right) \cdot \frac{2}{3} \]

       5. associate-*l* N/A

          \[\leadsto \left(x \cdot \frac{x}{{s}^{2} \cdot \left(c \cdot c\right)}\right) \cdot \frac{2}{3} \]

       6. pow2 N/A

          \[\leadsto \left(x \cdot \frac{x}{{s}^{2} \cdot {c}^{2}}\right) \cdot \frac{2}{3} \]

       7. lower-*.f64 N/A

          \[\leadsto \left(x \cdot \frac{x}{{s}^{2} \cdot {c}^{2}}\right) \cdot \frac{2}{3} \]

       8. pow2 N/A

          \[\leadsto \left(x \cdot \frac{x}{\left(s \cdot s\right) \cdot {c}^{2}}\right) \cdot \frac{2}{3} \]

       9. lift-*.f64 N/A

          \[\leadsto \left(x \cdot \frac{x}{\left(s \cdot s\right) \cdot {c}^{2}}\right) \cdot \frac{2}{3} \]

       10. pow2 N/A

           \[\leadsto \left(x \cdot \frac{x}{\left(s \cdot s\right) \cdot \left(c \cdot c\right)}\right) \cdot \frac{2}{3} \]

       11. lift-*.f64 53.2

           \[\leadsto \left(x \cdot \frac{x}{\left(s \cdot s\right) \cdot \left(c \cdot c\right)}\right) \cdot 0.6666666666666666 \]

   11. Applied rewrites 53.2%

       \[\leadsto \left(x \cdot \frac{x}{\left(s \cdot s\right) \cdot \left(c \cdot c\right)}\right) \cdot 0.6666666666666666 \]

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

   1. Initial program 0.0%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 7.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.6666666666666666, x \cdot x, -2\right)}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c} \cdot x, x, \frac{1}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c}\right)}{x \cdot x}} \]

   5. Taylor expanded in x around inf

      \[\leadsto \frac{2}{3} \cdot \color{blue}{\frac{{x}^{2}}{{c}^{2} \cdot {s}^{2}}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{{x}^{2}}{{c}^{2} \cdot {s}^{2}} \cdot \frac{2}{3} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{{x}^{2}}{{c}^{2} \cdot {s}^{2}} \cdot \frac{2}{3} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{{x}^{2}}{{c}^{2} \cdot {s}^{2}} \cdot \frac{2}{3} \]

      4. pow2 N/A

         \[\leadsto \frac{x \cdot x}{{c}^{2} \cdot {s}^{2}} \cdot \frac{2}{3} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot x}{{c}^{2} \cdot {s}^{2}} \cdot \frac{2}{3} \]

      6. *-commutative N/A

         \[\leadsto \frac{x \cdot x}{{s}^{2} \cdot {c}^{2}} \cdot \frac{2}{3} \]

      7. pow2 N/A

         \[\leadsto \frac{x \cdot x}{{s}^{2} \cdot \left(c \cdot c\right)} \cdot \frac{2}{3} \]

      8. associate-*l* N/A

         \[\leadsto \frac{x \cdot x}{\left({s}^{2} \cdot c\right) \cdot c} \cdot \frac{2}{3} \]

      9. pow2 N/A

         \[\leadsto \frac{x \cdot x}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c} \cdot \frac{2}{3} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{x \cdot x}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c} \cdot \frac{2}{3} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{x \cdot x}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c} \cdot \frac{2}{3} \]

      12. lift-*.f64 5.0

          \[\leadsto \frac{x \cdot x}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c} \cdot 0.6666666666666666 \]

   7. Applied rewrites 5.0%

      \[\leadsto \frac{x \cdot x}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c} \cdot \color{blue}{0.6666666666666666} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot x}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c} \cdot \frac{2}{3} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{x \cdot x}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c} \cdot \frac{2}{3} \]

      3. associate-/l* N/A

         \[\leadsto \left(x \cdot \frac{x}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c}\right) \cdot \frac{2}{3} \]

      4. lower-*.f64 N/A

         \[\leadsto \left(x \cdot \frac{x}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c}\right) \cdot \frac{2}{3} \]

      5. lower-/.f64 19.0

         \[\leadsto \left(x \cdot \frac{x}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c}\right) \cdot 0.6666666666666666 \]

   9. Applied rewrites 19.0%

      \[\leadsto \left(x \cdot \frac{x}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c}\right) \cdot 0.6666666666666666 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 45.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \leq \infty:\\ \;\;\;\;\left(x \cdot \frac{x}{\left(s \cdot s\right) \cdot \left(c \cdot c\right)}\right) \cdot 0.6666666666666666\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \frac{x}{\left(s \cdot \left(s \cdot c\right)\right) \cdot c}\right) \cdot 0.6666666666666666\\ \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (if (<= (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))) INFINITY)
   (* (* x (/ x (* (* s s) (* c c)))) 0.6666666666666666)
   (* (* x (/ x (* (* s (* s c)) c))) 0.6666666666666666)))

C

double code(double x, double c, double s) {
	double tmp;
	if ((cos((2.0 * x)) / (pow(c, 2.0) * ((x * pow(s, 2.0)) * x))) <= ((double) INFINITY)) {
		tmp = (x * (x / ((s * s) * (c * c)))) * 0.6666666666666666;
	} else {
		tmp = (x * (x / ((s * (s * c)) * c))) * 0.6666666666666666;
	}
	return tmp;
}

Java

public static double code(double x, double c, double s) {
	double tmp;
	if ((Math.cos((2.0 * x)) / (Math.pow(c, 2.0) * ((x * Math.pow(s, 2.0)) * x))) <= Double.POSITIVE_INFINITY) {
		tmp = (x * (x / ((s * s) * (c * c)))) * 0.6666666666666666;
	} else {
		tmp = (x * (x / ((s * (s * c)) * c))) * 0.6666666666666666;
	}
	return tmp;
}

Python

def code(x, c, s):
	tmp = 0
	if (math.cos((2.0 * x)) / (math.pow(c, 2.0) * ((x * math.pow(s, 2.0)) * x))) <= math.inf:
		tmp = (x * (x / ((s * s) * (c * c)))) * 0.6666666666666666
	else:
		tmp = (x * (x / ((s * (s * c)) * c))) * 0.6666666666666666
	return tmp

Julia

function code(x, c, s)
	tmp = 0.0
	if (Float64(cos(Float64(2.0 * x)) / Float64((c ^ 2.0) * Float64(Float64(x * (s ^ 2.0)) * x))) <= Inf)
		tmp = Float64(Float64(x * Float64(x / Float64(Float64(s * s) * Float64(c * c)))) * 0.6666666666666666);
	else
		tmp = Float64(Float64(x * Float64(x / Float64(Float64(s * Float64(s * c)) * c))) * 0.6666666666666666);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	tmp = 0.0;
	if ((cos((2.0 * x)) / ((c ^ 2.0) * ((x * (s ^ 2.0)) * x))) <= Inf)
		tmp = (x * (x / ((s * s) * (c * c)))) * 0.6666666666666666;
	else
		tmp = (x * (x / ((s * (s * c)) * c))) * 0.6666666666666666;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := If[LessEqual[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], Infinity], N[(N[(x * N[(x / N[(N[(s * s), $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.6666666666666666), $MachinePrecision], N[(N[(x * N[(x / N[(N[(s * N[(s * c), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.6666666666666666), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 81.1%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 30.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.6666666666666666, x \cdot x, -2\right)}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c} \cdot x, x, \frac{1}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c}\right)}{x \cdot x}} \]

   5. Taylor expanded in x around inf

      \[\leadsto \frac{2}{3} \cdot \color{blue}{\frac{{x}^{2}}{{c}^{2} \cdot {s}^{2}}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{{x}^{2}}{{c}^{2} \cdot {s}^{2}} \cdot \frac{2}{3} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{{x}^{2}}{{c}^{2} \cdot {s}^{2}} \cdot \frac{2}{3} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{{x}^{2}}{{c}^{2} \cdot {s}^{2}} \cdot \frac{2}{3} \]

      4. pow2 N/A

         \[\leadsto \frac{x \cdot x}{{c}^{2} \cdot {s}^{2}} \cdot \frac{2}{3} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot x}{{c}^{2} \cdot {s}^{2}} \cdot \frac{2}{3} \]

      6. *-commutative N/A

         \[\leadsto \frac{x \cdot x}{{s}^{2} \cdot {c}^{2}} \cdot \frac{2}{3} \]

      7. pow2 N/A

         \[\leadsto \frac{x \cdot x}{{s}^{2} \cdot \left(c \cdot c\right)} \cdot \frac{2}{3} \]

      8. associate-*l* N/A

         \[\leadsto \frac{x \cdot x}{\left({s}^{2} \cdot c\right) \cdot c} \cdot \frac{2}{3} \]

      9. pow2 N/A

         \[\leadsto \frac{x \cdot x}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c} \cdot \frac{2}{3} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{x \cdot x}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c} \cdot \frac{2}{3} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{x \cdot x}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c} \cdot \frac{2}{3} \]

      12. lift-*.f64 30.9

          \[\leadsto \frac{x \cdot x}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c} \cdot 0.6666666666666666 \]

   7. Applied rewrites 30.9%

      \[\leadsto \frac{x \cdot x}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c} \cdot \color{blue}{0.6666666666666666} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot x}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c} \cdot \frac{2}{3} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{x \cdot x}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c} \cdot \frac{2}{3} \]

      3. associate-/l* N/A

         \[\leadsto \left(x \cdot \frac{x}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c}\right) \cdot \frac{2}{3} \]

      4. lower-*.f64 N/A

         \[\leadsto \left(x \cdot \frac{x}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c}\right) \cdot \frac{2}{3} \]

      5. lower-/.f64 50.2

         \[\leadsto \left(x \cdot \frac{x}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c}\right) \cdot 0.6666666666666666 \]

   9. Applied rewrites 50.2%

      \[\leadsto \left(x \cdot \frac{x}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c}\right) \cdot 0.6666666666666666 \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \left(x \cdot \frac{x}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c}\right) \cdot \frac{2}{3} \]

       2. lift-*.f64 N/A

          \[\leadsto \left(x \cdot \frac{x}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c}\right) \cdot \frac{2}{3} \]

       3. lift-*.f64 N/A

          \[\leadsto \left(x \cdot \frac{x}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c}\right) \cdot \frac{2}{3} \]

       4. pow2 N/A

          \[\leadsto \left(x \cdot \frac{x}{\left({s}^{2} \cdot c\right) \cdot c}\right) \cdot \frac{2}{3} \]

       5. associate-*l* N/A

          \[\leadsto \left(x \cdot \frac{x}{{s}^{2} \cdot \left(c \cdot c\right)}\right) \cdot \frac{2}{3} \]

       6. pow2 N/A

          \[\leadsto \left(x \cdot \frac{x}{{s}^{2} \cdot {c}^{2}}\right) \cdot \frac{2}{3} \]

       7. lower-*.f64 N/A

          \[\leadsto \left(x \cdot \frac{x}{{s}^{2} \cdot {c}^{2}}\right) \cdot \frac{2}{3} \]

       8. pow2 N/A

          \[\leadsto \left(x \cdot \frac{x}{\left(s \cdot s\right) \cdot {c}^{2}}\right) \cdot \frac{2}{3} \]

       9. lift-*.f64 N/A

          \[\leadsto \left(x \cdot \frac{x}{\left(s \cdot s\right) \cdot {c}^{2}}\right) \cdot \frac{2}{3} \]

       10. pow2 N/A

           \[\leadsto \left(x \cdot \frac{x}{\left(s \cdot s\right) \cdot \left(c \cdot c\right)}\right) \cdot \frac{2}{3} \]

       11. lift-*.f64 53.2

           \[\leadsto \left(x \cdot \frac{x}{\left(s \cdot s\right) \cdot \left(c \cdot c\right)}\right) \cdot 0.6666666666666666 \]

   11. Applied rewrites 53.2%

       \[\leadsto \left(x \cdot \frac{x}{\left(s \cdot s\right) \cdot \left(c \cdot c\right)}\right) \cdot 0.6666666666666666 \]

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

   1. Initial program 0.0%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 7.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.6666666666666666, x \cdot x, -2\right)}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c} \cdot x, x, \frac{1}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c}\right)}{x \cdot x}} \]

   5. Taylor expanded in x around inf

      \[\leadsto \frac{2}{3} \cdot \color{blue}{\frac{{x}^{2}}{{c}^{2} \cdot {s}^{2}}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{{x}^{2}}{{c}^{2} \cdot {s}^{2}} \cdot \frac{2}{3} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{{x}^{2}}{{c}^{2} \cdot {s}^{2}} \cdot \frac{2}{3} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{{x}^{2}}{{c}^{2} \cdot {s}^{2}} \cdot \frac{2}{3} \]

      4. pow2 N/A

         \[\leadsto \frac{x \cdot x}{{c}^{2} \cdot {s}^{2}} \cdot \frac{2}{3} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot x}{{c}^{2} \cdot {s}^{2}} \cdot \frac{2}{3} \]

      6. *-commutative N/A

         \[\leadsto \frac{x \cdot x}{{s}^{2} \cdot {c}^{2}} \cdot \frac{2}{3} \]

      7. pow2 N/A

         \[\leadsto \frac{x \cdot x}{{s}^{2} \cdot \left(c \cdot c\right)} \cdot \frac{2}{3} \]

      8. associate-*l* N/A

         \[\leadsto \frac{x \cdot x}{\left({s}^{2} \cdot c\right) \cdot c} \cdot \frac{2}{3} \]

      9. pow2 N/A

         \[\leadsto \frac{x \cdot x}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c} \cdot \frac{2}{3} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{x \cdot x}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c} \cdot \frac{2}{3} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{x \cdot x}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c} \cdot \frac{2}{3} \]

      12. lift-*.f64 5.0

          \[\leadsto \frac{x \cdot x}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c} \cdot 0.6666666666666666 \]

   7. Applied rewrites 5.0%

      \[\leadsto \frac{x \cdot x}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c} \cdot \color{blue}{0.6666666666666666} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot x}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c} \cdot \frac{2}{3} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{x \cdot x}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c} \cdot \frac{2}{3} \]

      3. associate-/l* N/A

         \[\leadsto \left(x \cdot \frac{x}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c}\right) \cdot \frac{2}{3} \]

      4. lower-*.f64 N/A

         \[\leadsto \left(x \cdot \frac{x}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c}\right) \cdot \frac{2}{3} \]

      5. lower-/.f64 19.0

         \[\leadsto \left(x \cdot \frac{x}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c}\right) \cdot 0.6666666666666666 \]

   9. Applied rewrites 19.0%

      \[\leadsto \left(x \cdot \frac{x}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c}\right) \cdot 0.6666666666666666 \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \left(x \cdot \frac{x}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c}\right) \cdot \frac{2}{3} \]

       2. lift-*.f64 N/A

          \[\leadsto \left(x \cdot \frac{x}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c}\right) \cdot \frac{2}{3} \]

       3. associate-*l* N/A

          \[\leadsto \left(x \cdot \frac{x}{\left(s \cdot \left(s \cdot c\right)\right) \cdot c}\right) \cdot \frac{2}{3} \]

       4. *-commutative N/A

          \[\leadsto \left(x \cdot \frac{x}{\left(s \cdot \left(c \cdot s\right)\right) \cdot c}\right) \cdot \frac{2}{3} \]

       5. lower-*.f64 N/A

          \[\leadsto \left(x \cdot \frac{x}{\left(s \cdot \left(c \cdot s\right)\right) \cdot c}\right) \cdot \frac{2}{3} \]

       6. *-commutative N/A

          \[\leadsto \left(x \cdot \frac{x}{\left(s \cdot \left(s \cdot c\right)\right) \cdot c}\right) \cdot \frac{2}{3} \]

       7. lower-*.f64 7.9

          \[\leadsto \left(x \cdot \frac{x}{\left(s \cdot \left(s \cdot c\right)\right) \cdot c}\right) \cdot 0.6666666666666666 \]

   11. Applied rewrites 7.9%

       \[\leadsto \left(x \cdot \frac{x}{\left(s \cdot \left(s \cdot c\right)\right) \cdot c}\right) \cdot 0.6666666666666666 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 44.2% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \left(x \cdot \frac{x}{\left(s \cdot s\right) \cdot \left(c \cdot c\right)}\right) \cdot 0.6666666666666666 \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (* (* x (/ x (* (* s s) (* c c)))) 0.6666666666666666))

C

double code(double x, double c, double s) {
	return (x * (x / ((s * s) * (c * c)))) * 0.6666666666666666;
}

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 = (x * (x / ((s * s) * (c * c)))) * 0.6666666666666666d0
end function

Java

public static double code(double x, double c, double s) {
	return (x * (x / ((s * s) * (c * c)))) * 0.6666666666666666;
}

Python

def code(x, c, s):
	return (x * (x / ((s * s) * (c * c)))) * 0.6666666666666666

Julia

function code(x, c, s)
	return Float64(Float64(x * Float64(x / Float64(Float64(s * s) * Float64(c * c)))) * 0.6666666666666666)
end

MATLAB

function tmp = code(x, c, s)
	tmp = (x * (x / ((s * s) * (c * c)))) * 0.6666666666666666;
end

Wolfram

code[x_, c_, s_] := N[(N[(x * N[(x / N[(N[(s * s), $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.6666666666666666), $MachinePrecision]

Derivation

1. Initial program 67.1%

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

2. Taylor expanded in x around 0

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

   1. lower-/.f64 N/A

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

4. Applied rewrites 26.6%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.6666666666666666, x \cdot x, -2\right)}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c} \cdot x, x, \frac{1}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c}\right)}{x \cdot x}} \]

5. Taylor expanded in x around inf

   \[\leadsto \frac{2}{3} \cdot \color{blue}{\frac{{x}^{2}}{{c}^{2} \cdot {s}^{2}}} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{{x}^{2}}{{c}^{2} \cdot {s}^{2}} \cdot \frac{2}{3} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{{x}^{2}}{{c}^{2} \cdot {s}^{2}} \cdot \frac{2}{3} \]

   3. lower-/.f64 N/A

      \[\leadsto \frac{{x}^{2}}{{c}^{2} \cdot {s}^{2}} \cdot \frac{2}{3} \]

   4. pow2 N/A

      \[\leadsto \frac{x \cdot x}{{c}^{2} \cdot {s}^{2}} \cdot \frac{2}{3} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{x \cdot x}{{c}^{2} \cdot {s}^{2}} \cdot \frac{2}{3} \]

   6. *-commutative N/A

      \[\leadsto \frac{x \cdot x}{{s}^{2} \cdot {c}^{2}} \cdot \frac{2}{3} \]

   7. pow2 N/A

      \[\leadsto \frac{x \cdot x}{{s}^{2} \cdot \left(c \cdot c\right)} \cdot \frac{2}{3} \]

   8. associate-*l* N/A

      \[\leadsto \frac{x \cdot x}{\left({s}^{2} \cdot c\right) \cdot c} \cdot \frac{2}{3} \]

   9. pow2 N/A

      \[\leadsto \frac{x \cdot x}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c} \cdot \frac{2}{3} \]

   10. lift-*.f64 N/A

       \[\leadsto \frac{x \cdot x}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c} \cdot \frac{2}{3} \]

   11. lift-*.f64 N/A

       \[\leadsto \frac{x \cdot x}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c} \cdot \frac{2}{3} \]

   12. lift-*.f64 26.5

       \[\leadsto \frac{x \cdot x}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c} \cdot 0.6666666666666666 \]

7. Applied rewrites 26.5%

   \[\leadsto \frac{x \cdot x}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c} \cdot \color{blue}{0.6666666666666666} \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{x \cdot x}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c} \cdot \frac{2}{3} \]

   2. lift-/.f64 N/A

      \[\leadsto \frac{x \cdot x}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c} \cdot \frac{2}{3} \]

   3. associate-/l* N/A

      \[\leadsto \left(x \cdot \frac{x}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c}\right) \cdot \frac{2}{3} \]

   4. lower-*.f64 N/A

      \[\leadsto \left(x \cdot \frac{x}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c}\right) \cdot \frac{2}{3} \]

   5. lower-/.f64 44.8

      \[\leadsto \left(x \cdot \frac{x}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c}\right) \cdot 0.6666666666666666 \]

9. Applied rewrites 44.8%

   \[\leadsto \left(x \cdot \frac{x}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c}\right) \cdot 0.6666666666666666 \]
10. Step-by-step derivation

    1. lift-*.f64 N/A

       \[\leadsto \left(x \cdot \frac{x}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c}\right) \cdot \frac{2}{3} \]

    2. lift-*.f64 N/A

       \[\leadsto \left(x \cdot \frac{x}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c}\right) \cdot \frac{2}{3} \]

    3. lift-*.f64 N/A

       \[\leadsto \left(x \cdot \frac{x}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c}\right) \cdot \frac{2}{3} \]

    4. pow2 N/A

       \[\leadsto \left(x \cdot \frac{x}{\left({s}^{2} \cdot c\right) \cdot c}\right) \cdot \frac{2}{3} \]

    5. associate-*l* N/A

       \[\leadsto \left(x \cdot \frac{x}{{s}^{2} \cdot \left(c \cdot c\right)}\right) \cdot \frac{2}{3} \]

    6. pow2 N/A

       \[\leadsto \left(x \cdot \frac{x}{{s}^{2} \cdot {c}^{2}}\right) \cdot \frac{2}{3} \]

    7. lower-*.f64 N/A

       \[\leadsto \left(x \cdot \frac{x}{{s}^{2} \cdot {c}^{2}}\right) \cdot \frac{2}{3} \]

    8. pow2 N/A

       \[\leadsto \left(x \cdot \frac{x}{\left(s \cdot s\right) \cdot {c}^{2}}\right) \cdot \frac{2}{3} \]

    9. lift-*.f64 N/A

       \[\leadsto \left(x \cdot \frac{x}{\left(s \cdot s\right) \cdot {c}^{2}}\right) \cdot \frac{2}{3} \]

    10. pow2 N/A

        \[\leadsto \left(x \cdot \frac{x}{\left(s \cdot s\right) \cdot \left(c \cdot c\right)}\right) \cdot \frac{2}{3} \]

    11. lift-*.f64 44.2

        \[\leadsto \left(x \cdot \frac{x}{\left(s \cdot s\right) \cdot \left(c \cdot c\right)}\right) \cdot 0.6666666666666666 \]

11. Applied rewrites 44.2%

    \[\leadsto \left(x \cdot \frac{x}{\left(s \cdot s\right) \cdot \left(c \cdot c\right)}\right) \cdot 0.6666666666666666 \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2025112 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))