mixedcos

Percentage Accurate: 67.0% → 99.1%

Time: 3.3s

Alternatives: 10

Speedup: 4.2×

• 32.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 67.0% 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: 99.1% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

x, c, s = sort([x, c, s])
function code(x, c, s)
	t_0 = cos(Float64(x + x))
	t_1 = Float64(Float64(s * x) * c)
	t_2 = Float64(Float64(s * c) * x)
	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(t_0 / t_1) / t_1);
	else
		tmp = Float64(Float64(t_0 / t_2) / t_2);
	end
	return tmp
end

MATLAB

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

Wolfram

NOTE: x, c, and s should be sorted in increasing order before calling this function.
code[x_, c_, s_] := Block[{t$95$0 = N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(s * x), $MachinePrecision] * c), $MachinePrecision]}, Block[{t$95$2 = N[(N[(s * c), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[N[(N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] / N[(N[Power[c, 2.0], $MachinePrecision] * N[(N[(x * N[Power[s, 2.0], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(t$95$0 / t$95$1), $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[(t$95$0 / t$95$2), $MachinePrecision] / t$95$2), $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 81.2%

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

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

      10. pow-prod-down N/A

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

      11. pow-prod-down N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 96.9

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

   3. Applied rewrites 96.9%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 96.9

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 96.9

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 96.9

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

   5. Applied rewrites 96.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. count-2-rev N/A

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

      10. lower-+.f64 97.2

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 97.2

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

      17. lift-*.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. associate-*r* N/A

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

      20. *-commutative N/A

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

      21. lower-*.f64 N/A

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

      22. lower-*.f64 99.7

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

   7. Applied rewrites 99.7%

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

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

   1. Initial program 0.0%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. unpow2 N/A

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

      9. associate-*r* N/A

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

      10. pow-prod-down N/A

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

      11. pow-prod-down N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 96.3

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

   3. Applied rewrites 96.3%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 96.3

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 96.3

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 96.3

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

   5. Applied rewrites 96.3%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 96.3

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

   7. Applied rewrites 96.3%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. count-2-rev N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. count-2-rev N/A

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

      10. lift-cos.f64 N/A

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

      11. lift-+.f64 96.7

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 96.7

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 96.7

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

   9. Applied rewrites 96.7%

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

Alternative 2: 97.1% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: x, c, and s should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, c, 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 * c) * x
    code = (cos((x + x)) / t_0) / t_0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.0%

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

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

   10. pow-prod-down N/A

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

   11. pow-prod-down N/A

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

   12. lower-pow.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 96.8

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

3. Applied rewrites 96.8%

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

   1. lift-pow.f64 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f64 96.8

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 96.8

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 96.8

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

5. Applied rewrites 96.8%

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

   1. lift-*.f64 N/A

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

   2. count-2-rev N/A

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

   3. lower-+.f64 96.8

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

7. Applied rewrites 96.8%

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

   1. lift-/.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. count-2-rev N/A

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

   4. lift-cos.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-/r* N/A

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

   7. lower-/.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. count-2-rev N/A

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

   10. lift-cos.f64 N/A

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

   11. lift-+.f64 97.1

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

   12. lift-*.f64 N/A

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

   13. *-commutative N/A

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

   14. lift-*.f64 97.1

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

   15. lift-*.f64 N/A

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

   16. *-commutative N/A

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

   17. lift-*.f64 97.1

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

9. Applied rewrites 97.1%

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

Alternative 3: 96.8% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: x, c, and s should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, c, 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

assert x < c && c < s;
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

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.0%

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

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

   10. pow-prod-down N/A

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

   11. pow-prod-down N/A

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

   12. lower-pow.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 96.8

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

3. Applied rewrites 96.8%

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

   1. lift-pow.f64 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f64 96.8

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 96.8

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 96.8

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

5. Applied rewrites 96.8%

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

   1. lift-*.f64 N/A

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

   2. count-2-rev N/A

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

   3. lower-+.f64 96.8

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

7. Applied rewrites 96.8%

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

Alternative 4: 83.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, c, s] = \mathsf{sort}([x, c, s])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 5.5 \cdot 10^{-200}:\\ \;\;\;\;\frac{1}{\left({\left(s \cdot x\right)}^{2} \cdot c\right) \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x + x\right)}{\left(s \cdot c\right) \cdot \left(x \cdot \left(\left(s \cdot c\right) \cdot x\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x, c, and s should be sorted in increasing order before calling this function.
(FPCore (x c s)
 :precision binary64
 (if (<= x 5.5e-200)
   (/ 1.0 (* (* (pow (* s x) 2.0) c) c))
   (/ (cos (+ x x)) (* (* s c) (* x (* (* s c) x))))))

C

assert(x < c && c < s);
double code(double x, double c, double s) {
	double tmp;
	if (x <= 5.5e-200) {
		tmp = 1.0 / ((pow((s * x), 2.0) * c) * c);
	} else {
		tmp = cos((x + x)) / ((s * c) * (x * ((s * c) * x)));
	}
	return tmp;
}

Fortran

NOTE: x, c, and s should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, c, 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 <= 5.5d-200) then
        tmp = 1.0d0 / ((((s * x) ** 2.0d0) * c) * c)
    else
        tmp = cos((x + x)) / ((s * c) * (x * ((s * c) * x)))
    end if
    code = tmp
end function

Java

assert x < c && c < s;
public static double code(double x, double c, double s) {
	double tmp;
	if (x <= 5.5e-200) {
		tmp = 1.0 / ((Math.pow((s * x), 2.0) * c) * c);
	} else {
		tmp = Math.cos((x + x)) / ((s * c) * (x * ((s * c) * x)));
	}
	return tmp;
}

Python

[x, c, s] = sort([x, c, s])
def code(x, c, s):
	tmp = 0
	if x <= 5.5e-200:
		tmp = 1.0 / ((math.pow((s * x), 2.0) * c) * c)
	else:
		tmp = math.cos((x + x)) / ((s * c) * (x * ((s * c) * x)))
	return tmp

Julia

x, c, s = sort([x, c, s])
function code(x, c, s)
	tmp = 0.0
	if (x <= 5.5e-200)
		tmp = Float64(1.0 / Float64(Float64((Float64(s * x) ^ 2.0) * c) * c));
	else
		tmp = Float64(cos(Float64(x + x)) / Float64(Float64(s * c) * Float64(x * Float64(Float64(s * c) * x))));
	end
	return tmp
end

MATLAB

x, c, s = num2cell(sort([x, c, s])){:}
function tmp_2 = code(x, c, s)
	tmp = 0.0;
	if (x <= 5.5e-200)
		tmp = 1.0 / ((((s * x) ^ 2.0) * c) * c);
	else
		tmp = cos((x + x)) / ((s * c) * (x * ((s * c) * x)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, c, and s should be sorted in increasing order before calling this function.
code[x_, c_, s_] := If[LessEqual[x, 5.5e-200], N[(1.0 / N[(N[(N[Power[N[(s * x), $MachinePrecision], 2.0], $MachinePrecision] * c), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] / N[(N[(s * c), $MachinePrecision] * N[(x * N[(N[(s * c), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.4999999999999996e-200

   1. Initial program 66.0%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 65.1

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

   4. Applied rewrites 65.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. associate-*l* N/A

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

      6. pow2 N/A

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

      7. pow-prod-down N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-*.f64 75.6

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

   6. Applied rewrites 75.6%

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

   if 5.4999999999999996e-200 < x

   1. Initial program 68.4%

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

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

      10. pow-prod-down N/A

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

      11. pow-prod-down N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 97.9

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

   3. Applied rewrites 97.9%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 97.9

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 97.9

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 97.9

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

   5. Applied rewrites 97.9%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 97.9

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

   7. Applied rewrites 97.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lower-*.f64 94.7

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 94.7

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

   9. Applied rewrites 94.7%

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

Alternative 5: 82.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, c, s] = \mathsf{sort}([x, c, s])\\ \\ \begin{array}{l} t_0 := \left(c \cdot s\right) \cdot x\\ t_1 := t\_0 \cdot t\_0\\ \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \leq -4 \cdot 10^{+105}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2, x \cdot x, 1\right)}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t\_1}\\ \end{array} \end{array} \]

FPCore

NOTE: x, c, and s should be sorted in increasing order before calling this function.
(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (* (* c s) x)) (t_1 (* t_0 t_0)))
   (if (<= (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))) -4e+105)
     (/ (fma -2.0 (* x x) 1.0) t_1)
     (/ 1.0 t_1))))

C

assert(x < c && c < s);
double code(double x, double c, double s) {
	double t_0 = (c * s) * x;
	double t_1 = t_0 * t_0;
	double tmp;
	if ((cos((2.0 * x)) / (pow(c, 2.0) * ((x * pow(s, 2.0)) * x))) <= -4e+105) {
		tmp = fma(-2.0, (x * x), 1.0) / t_1;
	} else {
		tmp = 1.0 / t_1;
	}
	return tmp;
}

Julia

x, c, s = sort([x, c, s])
function code(x, c, s)
	t_0 = Float64(Float64(c * s) * x)
	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))) <= -4e+105)
		tmp = Float64(fma(-2.0, Float64(x * x), 1.0) / t_1);
	else
		tmp = Float64(1.0 / t_1);
	end
	return tmp
end

Wolfram

NOTE: x, c, and s should be sorted in increasing order before calling this function.
code[x_, c_, s_] := Block[{t$95$0 = N[(N[(c * s), $MachinePrecision] * x), $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], -4e+105], N[(N[(-2.0 * N[(x * x), $MachinePrecision] + 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))) < -3.9999999999999998e105

   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. associate-*r* N/A

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

      10. pow-prod-down N/A

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

      11. pow-prod-down N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 92.6

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

   3. Applied rewrites 92.6%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 92.6

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 92.6

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 92.6

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

   5. Applied rewrites 92.6%

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

   6. Taylor expanded in x around 0

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

      1. count-2-rev N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 58.0

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

   8. Applied rewrites 58.0%

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

   if -3.9999999999999998e105 < (/.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 67.2%

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

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

      10. pow-prod-down N/A

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

      11. pow-prod-down N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 97.1

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

   3. Applied rewrites 97.1%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 97.1

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 97.1

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 97.1

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

   5. Applied rewrites 97.1%

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

   6. Taylor expanded in x around 0

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

      1. count-2-rev 84.0

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

   8. Applied rewrites 84.0%

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

Alternative 6: 78.5% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: x, c, and s should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, c, 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 = 1.0d0 / (t_0 * t_0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.0%

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

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

   10. pow-prod-down N/A

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

   11. pow-prod-down N/A

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

   12. lower-pow.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 96.8

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

3. Applied rewrites 96.8%

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

   1. lift-pow.f64 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f64 96.8

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 96.8

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 96.8

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

5. Applied rewrites 96.8%

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

6. Taylor expanded in x around 0

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

   1. count-2-rev 78.5

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

8. Applied rewrites 78.5%

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

Alternative 7: 72.7% accurate, 0.8× speedup

Math

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

FPCore

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

C

assert(x < c && c < s);
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 = 1.0 / ((((s * (s * x)) * x) * c) * c);
	} else {
		tmp = 1.0 / (((s * c) * (s * (x * x))) * c);
	}
	return tmp;
}

Java

assert x < c && c < s;
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 = 1.0 / ((((s * (s * x)) * x) * c) * c);
	} else {
		tmp = 1.0 / (((s * c) * (s * (x * x))) * c);
	}
	return tmp;
}

Python

[x, c, s] = sort([x, c, s])
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 = 1.0 / ((((s * (s * x)) * x) * c) * c)
	else:
		tmp = 1.0 / (((s * c) * (s * (x * x))) * c)
	return tmp

Julia

x, c, s = sort([x, c, s])
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(1.0 / Float64(Float64(Float64(Float64(s * Float64(s * x)) * x) * c) * c));
	else
		tmp = Float64(1.0 / Float64(Float64(Float64(s * c) * Float64(s * Float64(x * x))) * c));
	end
	return tmp
end

MATLAB

x, c, s = num2cell(sort([x, c, s])){:}
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 = 1.0 / ((((s * (s * x)) * x) * c) * c);
	else
		tmp = 1.0 / (((s * c) * (s * (x * x))) * c);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, c, and s should be sorted in increasing order before calling this function.
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[(1.0 / N[(N[(N[(N[(s * N[(s * x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] * c), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[(s * c), $MachinePrecision] * N[(s * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 81.2%

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

   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. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 73.2

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

   4. Applied rewrites 73.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 77.7

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

   6. Applied rewrites 77.7%

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

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

   1. Initial program 0.0%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 22.0

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

   4. Applied rewrites 22.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. associate-*l* N/A

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

      8. pow2 N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 25.4

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

   6. Applied rewrites 25.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 48.4

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

   8. Applied rewrites 48.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 49.2

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

   10. Applied rewrites 49.2%

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

Alternative 8: 69.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} [x, c, s] = \mathsf{sort}([x, c, s])\\ \\ \begin{array}{l} \mathbf{if}\;{c}^{2} \leq 5 \cdot 10^{-99}:\\ \;\;\;\;\frac{1}{\left(\left(s \cdot c\right) \cdot \left(s \cdot \left(x \cdot x\right)\right)\right) \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(\left(s \cdot s\right) \cdot x\right) \cdot \left(x \cdot c\right)\right) \cdot c}\\ \end{array} \end{array} \]

FPCore

NOTE: x, c, and s should be sorted in increasing order before calling this function.
(FPCore (x c s)
 :precision binary64
 (if (<= (pow c 2.0) 5e-99)
   (/ 1.0 (* (* (* s c) (* s (* x x))) c))
   (/ 1.0 (* (* (* (* s s) x) (* x c)) c))))

C

assert(x < c && c < s);
double code(double x, double c, double s) {
	double tmp;
	if (pow(c, 2.0) <= 5e-99) {
		tmp = 1.0 / (((s * c) * (s * (x * x))) * c);
	} else {
		tmp = 1.0 / ((((s * s) * x) * (x * c)) * c);
	}
	return tmp;
}

Fortran

NOTE: x, c, and s should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, c, 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) <= 5d-99) then
        tmp = 1.0d0 / (((s * c) * (s * (x * x))) * c)
    else
        tmp = 1.0d0 / ((((s * s) * x) * (x * c)) * c)
    end if
    code = tmp
end function

Java

assert x < c && c < s;
public static double code(double x, double c, double s) {
	double tmp;
	if (Math.pow(c, 2.0) <= 5e-99) {
		tmp = 1.0 / (((s * c) * (s * (x * x))) * c);
	} else {
		tmp = 1.0 / ((((s * s) * x) * (x * c)) * c);
	}
	return tmp;
}

Python

[x, c, s] = sort([x, c, s])
def code(x, c, s):
	tmp = 0
	if math.pow(c, 2.0) <= 5e-99:
		tmp = 1.0 / (((s * c) * (s * (x * x))) * c)
	else:
		tmp = 1.0 / ((((s * s) * x) * (x * c)) * c)
	return tmp

Julia

x, c, s = sort([x, c, s])
function code(x, c, s)
	tmp = 0.0
	if ((c ^ 2.0) <= 5e-99)
		tmp = Float64(1.0 / Float64(Float64(Float64(s * c) * Float64(s * Float64(x * x))) * c));
	else
		tmp = Float64(1.0 / Float64(Float64(Float64(Float64(s * s) * x) * Float64(x * c)) * c));
	end
	return tmp
end

MATLAB

x, c, s = num2cell(sort([x, c, s])){:}
function tmp_2 = code(x, c, s)
	tmp = 0.0;
	if ((c ^ 2.0) <= 5e-99)
		tmp = 1.0 / (((s * c) * (s * (x * x))) * c);
	else
		tmp = 1.0 / ((((s * s) * x) * (x * c)) * c);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, c, and s should be sorted in increasing order before calling this function.
code[x_, c_, s_] := If[LessEqual[N[Power[c, 2.0], $MachinePrecision], 5e-99], N[(1.0 / N[(N[(N[(s * c), $MachinePrecision] * N[(s * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[(N[(s * s), $MachinePrecision] * x), $MachinePrecision] * N[(x * c), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 c #s(literal 2 binary64)) < 4.99999999999999969e-99

   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. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 56.9

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

   4. Applied rewrites 56.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. associate-*l* N/A

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

      8. pow2 N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 54.2

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

   6. Applied rewrites 54.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 61.1

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

   8. Applied rewrites 61.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 63.9

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

   10. Applied rewrites 63.9%

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

   if 4.99999999999999969e-99 < (pow.f64 c #s(literal 2 binary64))

   1. Initial program 68.5%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 69.6

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

   4. Applied rewrites 69.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-*.f64 73.2

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

   6. Applied rewrites 73.2%

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

Alternative 9: 67.3% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: x, c, and s should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.0%

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

2. Taylor expanded in x around 0

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

   1. lower-/.f64 N/A

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

   2. *-commutative N/A

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

   3. unpow2 N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. unpow2 N/A

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

   8. associate-*r* N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 64.3

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

4. Applied rewrites 64.3%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. pow2 N/A

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

   7. associate-*l* N/A

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

   8. pow2 N/A

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

   9. associate-*r* N/A

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

   10. lower-*.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. pow2 N/A

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

   13. lift-*.f64 N/A

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

   14. pow2 N/A

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

   15. lift-*.f64 60.4

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

6. Applied rewrites 60.4%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lift-*.f64 65.4

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

8. Applied rewrites 65.4%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. pow2 N/A

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

   6. associate-*l* N/A

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

   7. lower-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. pow2 N/A

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

   11. lift-*.f64 67.3

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

10. Applied rewrites 67.3%

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

Alternative 10: 60.4% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: x, c, and s should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.0%

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

2. Taylor expanded in x around 0

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

   1. lower-/.f64 N/A

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

   2. *-commutative N/A

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

   3. unpow2 N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. unpow2 N/A

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

   8. associate-*r* N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 64.3

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

4. Applied rewrites 64.3%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. pow2 N/A

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

   7. associate-*l* N/A

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

   8. pow2 N/A

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

   9. associate-*r* N/A

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

   10. lower-*.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. pow2 N/A

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

   13. lift-*.f64 N/A

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

   14. pow2 N/A

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

   15. lift-*.f64 60.4

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

6. Applied rewrites 60.4%

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

Reproduce

herbie shell --seed 2025114 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))