mixedcos

Percentage Accurate: 67.3% → 97.7%

Time: 4.6s

Alternatives: 10

Speedup: 2.0×

• 67.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

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: 67.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{max}\left(\left|c\right|, \left|s\right|\right)\\ t_1 := \mathsf{min}\left(\left|c\right|, \left|s\right|\right)\\ t_2 := \left(t\_1 \cdot t\_0\right) \cdot x\\ t_3 := t\_0 \cdot x\\ \mathbf{if}\;t\_1 \leq 1.8 \cdot 10^{-26}:\\ \;\;\;\;\frac{\cos \left(x + x\right)}{t\_2 \cdot t\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(t\_1 \cdot t\_3\right) \cdot t\_1} \cdot \frac{1}{t\_3}\\ \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (fmax (fabs c) (fabs s)))
        (t_1 (fmin (fabs c) (fabs s)))
        (t_2 (* (* t_1 t_0) x))
        (t_3 (* t_0 x)))
   (if (<= t_1 1.8e-26)
     (/ (cos (+ x x)) (* t_2 t_2))
     (* (/ 1.0 (* (* t_1 t_3) t_1)) (/ 1.0 t_3)))))

C

double code(double x, double c, double s) {
	double t_0 = fmax(fabs(c), fabs(s));
	double t_1 = fmin(fabs(c), fabs(s));
	double t_2 = (t_1 * t_0) * x;
	double t_3 = t_0 * x;
	double tmp;
	if (t_1 <= 1.8e-26) {
		tmp = cos((x + x)) / (t_2 * t_2);
	} else {
		tmp = (1.0 / ((t_1 * t_3) * t_1)) * (1.0 / t_3);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, c, s)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = fmax(abs(c), abs(s))
    t_1 = fmin(abs(c), abs(s))
    t_2 = (t_1 * t_0) * x
    t_3 = t_0 * x
    if (t_1 <= 1.8d-26) then
        tmp = cos((x + x)) / (t_2 * t_2)
    else
        tmp = (1.0d0 / ((t_1 * t_3) * t_1)) * (1.0d0 / t_3)
    end if
    code = tmp
end function

Java

public static double code(double x, double c, double s) {
	double t_0 = fmax(Math.abs(c), Math.abs(s));
	double t_1 = fmin(Math.abs(c), Math.abs(s));
	double t_2 = (t_1 * t_0) * x;
	double t_3 = t_0 * x;
	double tmp;
	if (t_1 <= 1.8e-26) {
		tmp = Math.cos((x + x)) / (t_2 * t_2);
	} else {
		tmp = (1.0 / ((t_1 * t_3) * t_1)) * (1.0 / t_3);
	}
	return tmp;
}

Python

def code(x, c, s):
	t_0 = fmax(math.fabs(c), math.fabs(s))
	t_1 = fmin(math.fabs(c), math.fabs(s))
	t_2 = (t_1 * t_0) * x
	t_3 = t_0 * x
	tmp = 0
	if t_1 <= 1.8e-26:
		tmp = math.cos((x + x)) / (t_2 * t_2)
	else:
		tmp = (1.0 / ((t_1 * t_3) * t_1)) * (1.0 / t_3)
	return tmp

Julia

function code(x, c, s)
	t_0 = fmax(abs(c), abs(s))
	t_1 = fmin(abs(c), abs(s))
	t_2 = Float64(Float64(t_1 * t_0) * x)
	t_3 = Float64(t_0 * x)
	tmp = 0.0
	if (t_1 <= 1.8e-26)
		tmp = Float64(cos(Float64(x + x)) / Float64(t_2 * t_2));
	else
		tmp = Float64(Float64(1.0 / Float64(Float64(t_1 * t_3) * t_1)) * Float64(1.0 / t_3));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	t_0 = max(abs(c), abs(s));
	t_1 = min(abs(c), abs(s));
	t_2 = (t_1 * t_0) * x;
	t_3 = t_0 * x;
	tmp = 0.0;
	if (t_1 <= 1.8e-26)
		tmp = cos((x + x)) / (t_2 * t_2);
	else
		tmp = (1.0 / ((t_1 * t_3) * t_1)) * (1.0 / t_3);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := Block[{t$95$0 = N[Max[N[Abs[c], $MachinePrecision], N[Abs[s], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Min[N[Abs[c], $MachinePrecision], N[Abs[s], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$1 * t$95$0), $MachinePrecision] * x), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$0 * x), $MachinePrecision]}, If[LessEqual[t$95$1, 1.8e-26], N[(N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] / N[(t$95$2 * t$95$2), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(N[(t$95$1 * t$95$3), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] * N[(1.0 / t$95$3), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if c < 1.8000000000000001e-26

   1. Initial program 67.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. unswap-sqr N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 77.6%

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

   3. Applied rewrites 77.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. pow2 N/A

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

      6. unswap-sqr N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 97.0%

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

   5. Applied rewrites 97.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 94.9%

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

   7. Applied rewrites 94.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 97.1%

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

   9. Applied rewrites 97.1%

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\left(\left(c \cdot s\right) \cdot x\right) \cdot \color{blue}{\left(\left(c \cdot s\right) \cdot x\right)}} \]
   10. 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 97.1%

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

   11. Applied rewrites 97.1%

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

   if 1.8000000000000001e-26 < c

   1. Initial program 67.3%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 59.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*l* N/A

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

      16. associate-*l* N/A

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

      17. *-commutative N/A

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

      18. associate-*r* N/A

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

      19. lower-*.f64 N/A

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

      20. lower-*.f64 N/A

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

      21. lift-pow.f64 N/A

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

      22. pow2 N/A

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

      23. lift-*.f64 N/A

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

      24. lift-*.f64 N/A

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

      25. *-commutative N/A

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

   6. Applied rewrites 60.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-pow.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-/r* N/A

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

      10. associate-/r* N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*l* N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*l* N/A

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

      17. lift-*.f64 N/A

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

      18. lift-*.f64 N/A

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

   8. Applied rewrites 76.0%

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

Alternative 2: 86.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{max}\left(\left|c\right|, s\right)\\ t_1 := \mathsf{min}\left(\left|c\right|, s\right)\\ t_2 := t\_1 \cdot \left(t\_0 \cdot \left|x\right|\right)\\ t_3 := \frac{1}{\left(-t\_0\right) \cdot \left(t\_1 \cdot \left|x\right|\right)}\\ \mathbf{if}\;\left|x\right| \leq 1.06 \cdot 10^{-13}:\\ \;\;\;\;\frac{\frac{1}{t\_2}}{t\_2}\\ \mathbf{elif}\;\left|x\right| \leq 9.8 \cdot 10^{+109}:\\ \;\;\;\;\frac{\cos \left(\left|x\right| + \left|x\right|\right)}{\left(t\_0 \cdot \left(t\_1 \cdot t\_1\right)\right) \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot t\_0\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_3 \cdot t\_3\\ \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (fmax (fabs c) s))
        (t_1 (fmin (fabs c) s))
        (t_2 (* t_1 (* t_0 (fabs x))))
        (t_3 (/ 1.0 (* (- t_0) (* t_1 (fabs x))))))
   (if (<= (fabs x) 1.06e-13)
     (/ (/ 1.0 t_2) t_2)
     (if (<= (fabs x) 9.8e+109)
       (/
        (cos (+ (fabs x) (fabs x)))
        (* (* t_0 (* t_1 t_1)) (* (* (fabs x) (fabs x)) t_0)))
       (* t_3 t_3)))))

C

double code(double x, double c, double s) {
	double t_0 = fmax(fabs(c), s);
	double t_1 = fmin(fabs(c), s);
	double t_2 = t_1 * (t_0 * fabs(x));
	double t_3 = 1.0 / (-t_0 * (t_1 * fabs(x)));
	double tmp;
	if (fabs(x) <= 1.06e-13) {
		tmp = (1.0 / t_2) / t_2;
	} else if (fabs(x) <= 9.8e+109) {
		tmp = cos((fabs(x) + fabs(x))) / ((t_0 * (t_1 * t_1)) * ((fabs(x) * fabs(x)) * t_0));
	} else {
		tmp = t_3 * t_3;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, c, s)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = fmax(abs(c), s)
    t_1 = fmin(abs(c), s)
    t_2 = t_1 * (t_0 * abs(x))
    t_3 = 1.0d0 / (-t_0 * (t_1 * abs(x)))
    if (abs(x) <= 1.06d-13) then
        tmp = (1.0d0 / t_2) / t_2
    else if (abs(x) <= 9.8d+109) then
        tmp = cos((abs(x) + abs(x))) / ((t_0 * (t_1 * t_1)) * ((abs(x) * abs(x)) * t_0))
    else
        tmp = t_3 * t_3
    end if
    code = tmp
end function

Java

public static double code(double x, double c, double s) {
	double t_0 = fmax(Math.abs(c), s);
	double t_1 = fmin(Math.abs(c), s);
	double t_2 = t_1 * (t_0 * Math.abs(x));
	double t_3 = 1.0 / (-t_0 * (t_1 * Math.abs(x)));
	double tmp;
	if (Math.abs(x) <= 1.06e-13) {
		tmp = (1.0 / t_2) / t_2;
	} else if (Math.abs(x) <= 9.8e+109) {
		tmp = Math.cos((Math.abs(x) + Math.abs(x))) / ((t_0 * (t_1 * t_1)) * ((Math.abs(x) * Math.abs(x)) * t_0));
	} else {
		tmp = t_3 * t_3;
	}
	return tmp;
}

Python

def code(x, c, s):
	t_0 = fmax(math.fabs(c), s)
	t_1 = fmin(math.fabs(c), s)
	t_2 = t_1 * (t_0 * math.fabs(x))
	t_3 = 1.0 / (-t_0 * (t_1 * math.fabs(x)))
	tmp = 0
	if math.fabs(x) <= 1.06e-13:
		tmp = (1.0 / t_2) / t_2
	elif math.fabs(x) <= 9.8e+109:
		tmp = math.cos((math.fabs(x) + math.fabs(x))) / ((t_0 * (t_1 * t_1)) * ((math.fabs(x) * math.fabs(x)) * t_0))
	else:
		tmp = t_3 * t_3
	return tmp

Julia

function code(x, c, s)
	t_0 = fmax(abs(c), s)
	t_1 = fmin(abs(c), s)
	t_2 = Float64(t_1 * Float64(t_0 * abs(x)))
	t_3 = Float64(1.0 / Float64(Float64(-t_0) * Float64(t_1 * abs(x))))
	tmp = 0.0
	if (abs(x) <= 1.06e-13)
		tmp = Float64(Float64(1.0 / t_2) / t_2);
	elseif (abs(x) <= 9.8e+109)
		tmp = Float64(cos(Float64(abs(x) + abs(x))) / Float64(Float64(t_0 * Float64(t_1 * t_1)) * Float64(Float64(abs(x) * abs(x)) * t_0)));
	else
		tmp = Float64(t_3 * t_3);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	t_0 = max(abs(c), s);
	t_1 = min(abs(c), s);
	t_2 = t_1 * (t_0 * abs(x));
	t_3 = 1.0 / (-t_0 * (t_1 * abs(x)));
	tmp = 0.0;
	if (abs(x) <= 1.06e-13)
		tmp = (1.0 / t_2) / t_2;
	elseif (abs(x) <= 9.8e+109)
		tmp = cos((abs(x) + abs(x))) / ((t_0 * (t_1 * t_1)) * ((abs(x) * abs(x)) * t_0));
	else
		tmp = t_3 * t_3;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := Block[{t$95$0 = N[Max[N[Abs[c], $MachinePrecision], s], $MachinePrecision]}, Block[{t$95$1 = N[Min[N[Abs[c], $MachinePrecision], s], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[(t$95$0 * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(1.0 / N[((-t$95$0) * N[(t$95$1 * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[x], $MachinePrecision], 1.06e-13], N[(N[(1.0 / t$95$2), $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[N[Abs[x], $MachinePrecision], 9.8e+109], N[(N[Cos[N[(N[Abs[x], $MachinePrecision] + N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(t$95$0 * N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$3 * t$95$3), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < 1.06e-13

   1. Initial program 67.3%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 59.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*l* N/A

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

      16. associate-*l* N/A

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

      17. *-commutative N/A

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

      18. associate-*r* N/A

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

      19. lower-*.f64 N/A

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

      20. lower-*.f64 N/A

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

      21. lift-pow.f64 N/A

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

      22. pow2 N/A

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

      23. lift-*.f64 N/A

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

      24. lift-*.f64 N/A

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

      25. *-commutative N/A

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

   6. Applied rewrites 60.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-pow.f64 N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. swap-sqr N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-pow.f64 N/A

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

      17. pow2 N/A

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

      18. swap-sqr N/A

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

      19. lift-*.f64 N/A

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

      20. lift-*.f64 N/A

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

   8. Applied rewrites 78.4%

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

   if 1.06e-13 < x < 9.8000000000000007e109

   1. Initial program 67.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. unswap-sqr N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 77.6%

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

   3. Applied rewrites 77.6%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lift-+.f64 77.6%

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-pow.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. associate-*r* N/A

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

      20. lower-*.f64 N/A

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

      21. lower-*.f64 68.3%

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

   5. Applied rewrites 68.3%

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

   if 9.8000000000000007e109 < x

   1. Initial program 67.3%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 59.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*l* N/A

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

      16. associate-*l* N/A

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

      17. *-commutative N/A

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

      18. associate-*r* N/A

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

      19. lower-*.f64 N/A

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

      20. lower-*.f64 N/A

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

      21. lift-pow.f64 N/A

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

      22. pow2 N/A

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

      23. lift-*.f64 N/A

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

      24. lift-*.f64 N/A

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

      25. *-commutative N/A

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

   6. Applied rewrites 60.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-pow.f64 N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. swap-sqr N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-pow.f64 N/A

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

      17. pow2 N/A

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

      18. swap-sqr N/A

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

      19. lift-*.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. sqr-neg-rev N/A

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

   8. Applied rewrites 78.6%

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

Alternative 3: 84.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(\left|c\right|, \left|s\right|\right)\\ t_1 := \mathsf{max}\left(\left|c\right|, \left|s\right|\right)\\ t_2 := t\_1 \cdot x\\ t_3 := t\_0 \cdot t\_2\\ \mathbf{if}\;{t\_1}^{2} \leq 2 \cdot 10^{+207}:\\ \;\;\;\;\frac{\cos \left(x + x\right)}{\left(\left(t\_0 \cdot t\_0\right) \cdot x\right) \cdot \left(t\_2 \cdot t\_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{t\_3}}{t\_3}\\ \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (fmin (fabs c) (fabs s)))
        (t_1 (fmax (fabs c) (fabs s)))
        (t_2 (* t_1 x))
        (t_3 (* t_0 t_2)))
   (if (<= (pow t_1 2.0) 2e+207)
     (/ (cos (+ x x)) (* (* (* t_0 t_0) x) (* t_2 t_1)))
     (/ (/ 1.0 t_3) t_3))))

C

double code(double x, double c, double s) {
	double t_0 = fmin(fabs(c), fabs(s));
	double t_1 = fmax(fabs(c), fabs(s));
	double t_2 = t_1 * x;
	double t_3 = t_0 * t_2;
	double tmp;
	if (pow(t_1, 2.0) <= 2e+207) {
		tmp = cos((x + x)) / (((t_0 * t_0) * x) * (t_2 * t_1));
	} else {
		tmp = (1.0 / t_3) / t_3;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, c, s)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = fmin(abs(c), abs(s))
    t_1 = fmax(abs(c), abs(s))
    t_2 = t_1 * x
    t_3 = t_0 * t_2
    if ((t_1 ** 2.0d0) <= 2d+207) then
        tmp = cos((x + x)) / (((t_0 * t_0) * x) * (t_2 * t_1))
    else
        tmp = (1.0d0 / t_3) / t_3
    end if
    code = tmp
end function

Java

public static double code(double x, double c, double s) {
	double t_0 = fmin(Math.abs(c), Math.abs(s));
	double t_1 = fmax(Math.abs(c), Math.abs(s));
	double t_2 = t_1 * x;
	double t_3 = t_0 * t_2;
	double tmp;
	if (Math.pow(t_1, 2.0) <= 2e+207) {
		tmp = Math.cos((x + x)) / (((t_0 * t_0) * x) * (t_2 * t_1));
	} else {
		tmp = (1.0 / t_3) / t_3;
	}
	return tmp;
}

Python

def code(x, c, s):
	t_0 = fmin(math.fabs(c), math.fabs(s))
	t_1 = fmax(math.fabs(c), math.fabs(s))
	t_2 = t_1 * x
	t_3 = t_0 * t_2
	tmp = 0
	if math.pow(t_1, 2.0) <= 2e+207:
		tmp = math.cos((x + x)) / (((t_0 * t_0) * x) * (t_2 * t_1))
	else:
		tmp = (1.0 / t_3) / t_3
	return tmp

Julia

function code(x, c, s)
	t_0 = fmin(abs(c), abs(s))
	t_1 = fmax(abs(c), abs(s))
	t_2 = Float64(t_1 * x)
	t_3 = Float64(t_0 * t_2)
	tmp = 0.0
	if ((t_1 ^ 2.0) <= 2e+207)
		tmp = Float64(cos(Float64(x + x)) / Float64(Float64(Float64(t_0 * t_0) * x) * Float64(t_2 * t_1)));
	else
		tmp = Float64(Float64(1.0 / t_3) / t_3);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	t_0 = min(abs(c), abs(s));
	t_1 = max(abs(c), abs(s));
	t_2 = t_1 * x;
	t_3 = t_0 * t_2;
	tmp = 0.0;
	if ((t_1 ^ 2.0) <= 2e+207)
		tmp = cos((x + x)) / (((t_0 * t_0) * x) * (t_2 * t_1));
	else
		tmp = (1.0 / t_3) / t_3;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := Block[{t$95$0 = N[Min[N[Abs[c], $MachinePrecision], N[Abs[s], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Max[N[Abs[c], $MachinePrecision], N[Abs[s], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * x), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$0 * t$95$2), $MachinePrecision]}, If[LessEqual[N[Power[t$95$1, 2.0], $MachinePrecision], 2e+207], N[(N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] / N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] * x), $MachinePrecision] * N[(t$95$2 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / t$95$3), $MachinePrecision] / t$95$3), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 s #s(literal 2 binary64)) < 2.0000000000000001e207

   1. Initial program 67.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. unswap-sqr N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 77.6%

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

   3. Applied rewrites 77.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. pow2 N/A

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

      6. unswap-sqr N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 97.0%

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

   5. Applied rewrites 97.0%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 97.0%

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. pow-prod-down N/A

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

      9. lift-pow.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-*r* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 74.9%

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

      18. lift-pow.f64 N/A

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

      19. pow2 N/A

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

      20. lift-*.f64 74.9%

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

   7. Applied rewrites 74.9%

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

   if 2.0000000000000001e207 < (pow.f64 s #s(literal 2 binary64))

   1. Initial program 67.3%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 59.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*l* N/A

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

      16. associate-*l* N/A

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

      17. *-commutative N/A

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

      18. associate-*r* N/A

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

      19. lower-*.f64 N/A

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

      20. lower-*.f64 N/A

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

      21. lift-pow.f64 N/A

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

      22. pow2 N/A

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

      23. lift-*.f64 N/A

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

      24. lift-*.f64 N/A

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

      25. *-commutative N/A

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

   6. Applied rewrites 60.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-pow.f64 N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. swap-sqr N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-pow.f64 N/A

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

      17. pow2 N/A

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

      18. swap-sqr N/A

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

      19. lift-*.f64 N/A

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

      20. lift-*.f64 N/A

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

   8. Applied rewrites 78.4%

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

Alternative 4: 83.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{max}\left(\left|c\right|, \left|s\right|\right)\\ t_1 := \mathsf{min}\left(\left|c\right|, \left|s\right|\right)\\ t_2 := t\_0 \cdot x\\ t_3 := t\_1 \cdot t\_2\\ \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{{t\_1}^{2} \cdot \left(\left(x \cdot {t\_0}^{2}\right) \cdot x\right)} \leq -5 \cdot 10^{-102}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2, x \cdot x, 1\right)}{\left(\left(t\_1 \cdot t\_1\right) \cdot x\right) \cdot \left(t\_2 \cdot t\_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{t\_3}}{t\_3}\\ \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (fmax (fabs c) (fabs s)))
        (t_1 (fmin (fabs c) (fabs s)))
        (t_2 (* t_0 x))
        (t_3 (* t_1 t_2)))
   (if (<=
        (/ (cos (* 2.0 x)) (* (pow t_1 2.0) (* (* x (pow t_0 2.0)) x)))
        -5e-102)
     (/ (fma -2.0 (* x x) 1.0) (* (* (* t_1 t_1) x) (* t_2 t_0)))
     (/ (/ 1.0 t_3) t_3))))

C

double code(double x, double c, double s) {
	double t_0 = fmax(fabs(c), fabs(s));
	double t_1 = fmin(fabs(c), fabs(s));
	double t_2 = t_0 * x;
	double t_3 = t_1 * t_2;
	double tmp;
	if ((cos((2.0 * x)) / (pow(t_1, 2.0) * ((x * pow(t_0, 2.0)) * x))) <= -5e-102) {
		tmp = fma(-2.0, (x * x), 1.0) / (((t_1 * t_1) * x) * (t_2 * t_0));
	} else {
		tmp = (1.0 / t_3) / t_3;
	}
	return tmp;
}

Julia

function code(x, c, s)
	t_0 = fmax(abs(c), abs(s))
	t_1 = fmin(abs(c), abs(s))
	t_2 = Float64(t_0 * x)
	t_3 = Float64(t_1 * t_2)
	tmp = 0.0
	if (Float64(cos(Float64(2.0 * x)) / Float64((t_1 ^ 2.0) * Float64(Float64(x * (t_0 ^ 2.0)) * x))) <= -5e-102)
		tmp = Float64(fma(-2.0, Float64(x * x), 1.0) / Float64(Float64(Float64(t_1 * t_1) * x) * Float64(t_2 * t_0)));
	else
		tmp = Float64(Float64(1.0 / t_3) / t_3);
	end
	return tmp
end

Wolfram

code[x_, c_, s_] := Block[{t$95$0 = N[Max[N[Abs[c], $MachinePrecision], N[Abs[s], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Min[N[Abs[c], $MachinePrecision], N[Abs[s], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 * x), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 * t$95$2), $MachinePrecision]}, If[LessEqual[N[(N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] / N[(N[Power[t$95$1, 2.0], $MachinePrecision] * N[(N[(x * N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -5e-102], N[(N[(-2.0 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] / N[(N[(N[(t$95$1 * t$95$1), $MachinePrecision] * x), $MachinePrecision] * N[(t$95$2 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / t$95$3), $MachinePrecision] / t$95$3), $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))) < -5.00000000000000026e-102

   1. Initial program 67.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. unswap-sqr N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 77.6%

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

   3. Applied rewrites 77.6%

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

   4. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 51.5%

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

   6. Applied rewrites 51.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-fma.f64 51.5%

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

      8. lower-fma.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-fma.f64 N/A

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

      18. lower-fma.f64 N/A

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

      19. lower-fma.f64 N/A

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

   8. Applied rewrites 50.1%

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

   if -5.00000000000000026e-102 < (/.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 67.3%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 59.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*l* N/A

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

      16. associate-*l* N/A

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

      17. *-commutative N/A

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

      18. associate-*r* N/A

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

      19. lower-*.f64 N/A

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

      20. lower-*.f64 N/A

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

      21. lift-pow.f64 N/A

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

      22. pow2 N/A

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

      23. lift-*.f64 N/A

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

      24. lift-*.f64 N/A

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

      25. *-commutative N/A

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

   6. Applied rewrites 60.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-pow.f64 N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. swap-sqr N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-pow.f64 N/A

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

      17. pow2 N/A

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

      18. swap-sqr N/A

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

      19. lift-*.f64 N/A

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

      20. lift-*.f64 N/A

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

   8. Applied rewrites 78.4%

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

Alternative 5: 83.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{max}\left(\left|c\right|, \left|s\right|\right)\\ t_1 := t\_0 \cdot \left|x\right|\\ t_2 := \mathsf{min}\left(\left|c\right|, \left|s\right|\right)\\ t_3 := t\_2 \cdot t\_1\\ \mathbf{if}\;\frac{\cos \left(2 \cdot \left|x\right|\right)}{{t\_2}^{2} \cdot \left(\left(\left|x\right| \cdot {t\_0}^{2}\right) \cdot \left|x\right|\right)} \leq -5 \cdot 10^{-102}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(-2, \left|x\right| \cdot \left|x\right|, 1\right)}{t\_2}}{e^{\log t\_1 \cdot 2} \cdot t\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{t\_3}}{t\_3}\\ \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (fmax (fabs c) (fabs s)))
        (t_1 (* t_0 (fabs x)))
        (t_2 (fmin (fabs c) (fabs s)))
        (t_3 (* t_2 t_1)))
   (if (<=
        (/
         (cos (* 2.0 (fabs x)))
         (* (pow t_2 2.0) (* (* (fabs x) (pow t_0 2.0)) (fabs x))))
        -5e-102)
     (/
      (/ (fma -2.0 (* (fabs x) (fabs x)) 1.0) t_2)
      (* (exp (* (log t_1) 2.0)) t_2))
     (/ (/ 1.0 t_3) t_3))))

C

double code(double x, double c, double s) {
	double t_0 = fmax(fabs(c), fabs(s));
	double t_1 = t_0 * fabs(x);
	double t_2 = fmin(fabs(c), fabs(s));
	double t_3 = t_2 * t_1;
	double tmp;
	if ((cos((2.0 * fabs(x))) / (pow(t_2, 2.0) * ((fabs(x) * pow(t_0, 2.0)) * fabs(x)))) <= -5e-102) {
		tmp = (fma(-2.0, (fabs(x) * fabs(x)), 1.0) / t_2) / (exp((log(t_1) * 2.0)) * t_2);
	} else {
		tmp = (1.0 / t_3) / t_3;
	}
	return tmp;
}

Julia

function code(x, c, s)
	t_0 = fmax(abs(c), abs(s))
	t_1 = Float64(t_0 * abs(x))
	t_2 = fmin(abs(c), abs(s))
	t_3 = Float64(t_2 * t_1)
	tmp = 0.0
	if (Float64(cos(Float64(2.0 * abs(x))) / Float64((t_2 ^ 2.0) * Float64(Float64(abs(x) * (t_0 ^ 2.0)) * abs(x)))) <= -5e-102)
		tmp = Float64(Float64(fma(-2.0, Float64(abs(x) * abs(x)), 1.0) / t_2) / Float64(exp(Float64(log(t_1) * 2.0)) * t_2));
	else
		tmp = Float64(Float64(1.0 / t_3) / t_3);
	end
	return tmp
end

Wolfram

code[x_, c_, s_] := Block[{t$95$0 = N[Max[N[Abs[c], $MachinePrecision], N[Abs[s], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Min[N[Abs[c], $MachinePrecision], N[Abs[s], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * t$95$1), $MachinePrecision]}, If[LessEqual[N[(N[Cos[N[(2.0 * N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[Power[t$95$2, 2.0], $MachinePrecision] * N[(N[(N[Abs[x], $MachinePrecision] * N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -5e-102], N[(N[(N[(-2.0 * N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / t$95$2), $MachinePrecision] / N[(N[Exp[N[(N[Log[t$95$1], $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / t$95$3), $MachinePrecision] / t$95$3), $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))) < -5.00000000000000026e-102

   1. Initial program 67.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. unswap-sqr N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 77.6%

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

   3. Applied rewrites 77.6%

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

   4. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 51.5%

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

   6. Applied rewrites 51.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow2 N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. swap-sqr N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*l* N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lift-*.f64 N/A

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

      17. associate-/r* N/A

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

   8. Applied rewrites 52.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. pow-to-exp N/A

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

      7. lower-unsound-exp.f64 N/A

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

      8. lower-unsound-*.f64 N/A

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

      9. lower-unsound-log.f64 32.3%

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

   10. Applied rewrites 32.3%

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

   if -5.00000000000000026e-102 < (/.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 67.3%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 59.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*l* N/A

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

      16. associate-*l* N/A

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

      17. *-commutative N/A

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

      18. associate-*r* N/A

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

      19. lower-*.f64 N/A

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

      20. lower-*.f64 N/A

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

      21. lift-pow.f64 N/A

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

      22. pow2 N/A

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

      23. lift-*.f64 N/A

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

      24. lift-*.f64 N/A

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

      25. *-commutative N/A

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

   6. Applied rewrites 60.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-pow.f64 N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. swap-sqr N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-pow.f64 N/A

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

      17. pow2 N/A

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

      18. swap-sqr N/A

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

      19. lift-*.f64 N/A

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

      20. lift-*.f64 N/A

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

   8. Applied rewrites 78.4%

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

Alternative 6: 79.5% accurate, 1.9× speedup

Math

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

FPCore

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

C

double code(double x, double c, double s) {
	double t_0 = fmin(fabs(c), fabs(s)) * (fmax(fabs(c), fabs(s)) * x);
	return (1.0 / t_0) / t_0;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, c, s)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: t_0
    t_0 = fmin(abs(c), abs(s)) * (fmax(abs(c), abs(s)) * x)
    code = (1.0d0 / t_0) / t_0
end function

Java

public static double code(double x, double c, double s) {
	double t_0 = fmin(Math.abs(c), Math.abs(s)) * (fmax(Math.abs(c), Math.abs(s)) * x);
	return (1.0 / t_0) / t_0;
}

Python

def code(x, c, s):
	t_0 = fmin(math.fabs(c), math.fabs(s)) * (fmax(math.fabs(c), math.fabs(s)) * x)
	return (1.0 / t_0) / t_0

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.3%

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

2. Taylor expanded in x around 0

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

4. Applied rewrites 59.2%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. lift-pow.f64 N/A

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

   7. unpow2 N/A

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

   8. associate-*l* N/A

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

   9. *-commutative N/A

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. associate-*r* N/A

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

   13. lift-*.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. associate-*l* N/A

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

   16. associate-*l* N/A

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

   17. *-commutative N/A

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

   18. associate-*r* N/A

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

   19. lower-*.f64 N/A

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

   20. lower-*.f64 N/A

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

   21. lift-pow.f64 N/A

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

   22. pow2 N/A

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

   23. lift-*.f64 N/A

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

   24. lift-*.f64 N/A

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

   25. *-commutative N/A

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

6. Applied rewrites 60.6%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. pow2 N/A

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

   7. lift-pow.f64 N/A

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

   8. associate-*l* N/A

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

   9. lift-*.f64 N/A

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

   10. associate-*l* N/A

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

   11. *-commutative N/A

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

   12. lift-*.f64 N/A

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

   13. swap-sqr N/A

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

   14. lift-*.f64 N/A

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

   15. lift-*.f64 N/A

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

   16. lift-pow.f64 N/A

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

   17. pow2 N/A

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

   18. swap-sqr N/A

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

   19. lift-*.f64 N/A

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

   20. lift-*.f64 N/A

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

8. Applied rewrites 78.4%

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

Alternative 7: 77.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{max}\left(\left|c\right|, \left|s\right|\right) \cdot x\\ t_1 := \mathsf{min}\left(\left|c\right|, \left|s\right|\right)\\ \frac{1}{\left(\left(t\_0 \cdot t\_1\right) \cdot t\_1\right) \cdot t\_0} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (* (fmax (fabs c) (fabs s)) x)) (t_1 (fmin (fabs c) (fabs s))))
   (/ 1.0 (* (* (* t_0 t_1) t_1) t_0))))

C

double code(double x, double c, double s) {
	double t_0 = fmax(fabs(c), fabs(s)) * x;
	double t_1 = fmin(fabs(c), fabs(s));
	return 1.0 / (((t_0 * t_1) * t_1) * t_0);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, c, s)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: t_0
    real(8) :: t_1
    t_0 = fmax(abs(c), abs(s)) * x
    t_1 = fmin(abs(c), abs(s))
    code = 1.0d0 / (((t_0 * t_1) * t_1) * t_0)
end function

Java

public static double code(double x, double c, double s) {
	double t_0 = fmax(Math.abs(c), Math.abs(s)) * x;
	double t_1 = fmin(Math.abs(c), Math.abs(s));
	return 1.0 / (((t_0 * t_1) * t_1) * t_0);
}

Python

def code(x, c, s):
	t_0 = fmax(math.fabs(c), math.fabs(s)) * x
	t_1 = fmin(math.fabs(c), math.fabs(s))
	return 1.0 / (((t_0 * t_1) * t_1) * t_0)

Julia

function code(x, c, s)
	t_0 = Float64(fmax(abs(c), abs(s)) * x)
	t_1 = fmin(abs(c), abs(s))
	return Float64(1.0 / Float64(Float64(Float64(t_0 * t_1) * t_1) * t_0))
end

MATLAB

function tmp = code(x, c, s)
	t_0 = max(abs(c), abs(s)) * x;
	t_1 = min(abs(c), abs(s));
	tmp = 1.0 / (((t_0 * t_1) * t_1) * t_0);
end

Wolfram

code[x_, c_, s_] := Block[{t$95$0 = N[(N[Max[N[Abs[c], $MachinePrecision], N[Abs[s], $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision]}, Block[{t$95$1 = N[Min[N[Abs[c], $MachinePrecision], N[Abs[s], $MachinePrecision]], $MachinePrecision]}, N[(1.0 / N[(N[(N[(t$95$0 * t$95$1), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 67.3%

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

2. Taylor expanded in x around 0

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

4. Applied rewrites 59.2%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. lift-pow.f64 N/A

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

   7. unpow2 N/A

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

   8. associate-*l* N/A

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

   9. *-commutative N/A

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. associate-*r* N/A

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

   13. lift-*.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. associate-*l* N/A

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

   16. associate-*l* N/A

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

   17. *-commutative N/A

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

   18. associate-*r* N/A

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

   19. lower-*.f64 N/A

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

   20. lower-*.f64 N/A

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

   21. lift-pow.f64 N/A

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

   22. pow2 N/A

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

   23. lift-*.f64 N/A

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

   24. lift-*.f64 N/A

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

   25. *-commutative N/A

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

6. Applied rewrites 60.6%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

   5. lift-*.f64 N/A

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

   6. lower-*.f64 64.6%

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

8. Applied rewrites 64.6%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*r* N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. associate-*r* N/A

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

   10. *-commutative N/A

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

   11. lift-*.f64 N/A

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

   12. associate-*r* N/A

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

   13. lift-*.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. pow2 N/A

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

   16. pow2 N/A

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

   17. pow-prod-down N/A

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

   18. lift-*.f64 N/A

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

   19. lift-*.f64 N/A

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

   20. lift-*.f64 N/A

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

   21. associate-*l* N/A

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

   22. lift-*.f64 N/A

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

   23. lift-*.f64 N/A

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

10. Applied rewrites 76.3%

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

Alternative 8: 73.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{max}\left(\left|c\right|, \left|s\right|\right)\\ t_1 := \mathsf{min}\left(\left|c\right|, \left|s\right|\right)\\ \frac{1}{\left(\left(t\_0 \cdot t\_1\right) \cdot t\_1\right) \cdot \left(\left(t\_0 \cdot x\right) \cdot x\right)} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (fmax (fabs c) (fabs s))) (t_1 (fmin (fabs c) (fabs s))))
   (/ 1.0 (* (* (* t_0 t_1) t_1) (* (* t_0 x) x)))))

C

double code(double x, double c, double s) {
	double t_0 = fmax(fabs(c), fabs(s));
	double t_1 = fmin(fabs(c), fabs(s));
	return 1.0 / (((t_0 * t_1) * t_1) * ((t_0 * x) * 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
    real(8) :: t_0
    real(8) :: t_1
    t_0 = fmax(abs(c), abs(s))
    t_1 = fmin(abs(c), abs(s))
    code = 1.0d0 / (((t_0 * t_1) * t_1) * ((t_0 * x) * x))
end function

Java

public static double code(double x, double c, double s) {
	double t_0 = fmax(Math.abs(c), Math.abs(s));
	double t_1 = fmin(Math.abs(c), Math.abs(s));
	return 1.0 / (((t_0 * t_1) * t_1) * ((t_0 * x) * x));
}

Python

def code(x, c, s):
	t_0 = fmax(math.fabs(c), math.fabs(s))
	t_1 = fmin(math.fabs(c), math.fabs(s))
	return 1.0 / (((t_0 * t_1) * t_1) * ((t_0 * x) * x))

Julia

function code(x, c, s)
	t_0 = fmax(abs(c), abs(s))
	t_1 = fmin(abs(c), abs(s))
	return Float64(1.0 / Float64(Float64(Float64(t_0 * t_1) * t_1) * Float64(Float64(t_0 * x) * x)))
end

MATLAB

function tmp = code(x, c, s)
	t_0 = max(abs(c), abs(s));
	t_1 = min(abs(c), abs(s));
	tmp = 1.0 / (((t_0 * t_1) * t_1) * ((t_0 * x) * x));
end

Wolfram

code[x_, c_, s_] := Block[{t$95$0 = N[Max[N[Abs[c], $MachinePrecision], N[Abs[s], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Min[N[Abs[c], $MachinePrecision], N[Abs[s], $MachinePrecision]], $MachinePrecision]}, N[(1.0 / N[(N[(N[(t$95$0 * t$95$1), $MachinePrecision] * t$95$1), $MachinePrecision] * N[(N[(t$95$0 * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 67.3%

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

2. Taylor expanded in x around 0

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

4. Applied rewrites 59.2%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. lift-pow.f64 N/A

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

   7. unpow2 N/A

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

   8. associate-*l* N/A

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

   9. *-commutative N/A

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. associate-*r* N/A

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

   13. lift-*.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. associate-*l* N/A

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

   16. associate-*l* N/A

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

   17. *-commutative N/A

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

   18. associate-*r* N/A

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

   19. lower-*.f64 N/A

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

   20. lower-*.f64 N/A

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

   21. lift-pow.f64 N/A

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

   22. pow2 N/A

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

   23. lift-*.f64 N/A

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

   24. lift-*.f64 N/A

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

   25. *-commutative N/A

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

6. Applied rewrites 60.6%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

   5. lift-*.f64 N/A

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

   6. lower-*.f64 64.6%

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

8. Applied rewrites 64.6%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. lower-*.f64 69.6%

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 69.6%

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

10. Applied rewrites 69.6%

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

Alternative 9: 73.2% accurate, 0.6× speedup

Math

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

FPCore

(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (fmax (fabs c) (fabs s))) (t_1 (fmin (fabs c) (fabs s))))
   (if (<=
        (/ (cos (* 2.0 x)) (* (pow t_1 2.0) (* (* x (pow t_0 2.0)) x)))
        INFINITY)
     (/ 1.0 (* (* t_0 (* t_1 t_1)) (* (* t_0 x) x)))
     (/ 1.0 (* (* (* t_1 t_0) t_1) (* (* x x) t_0))))))

C

double code(double x, double c, double s) {
	double t_0 = fmax(fabs(c), fabs(s));
	double t_1 = fmin(fabs(c), fabs(s));
	double tmp;
	if ((cos((2.0 * x)) / (pow(t_1, 2.0) * ((x * pow(t_0, 2.0)) * x))) <= ((double) INFINITY)) {
		tmp = 1.0 / ((t_0 * (t_1 * t_1)) * ((t_0 * x) * x));
	} else {
		tmp = 1.0 / (((t_1 * t_0) * t_1) * ((x * x) * t_0));
	}
	return tmp;
}

Java

public static double code(double x, double c, double s) {
	double t_0 = fmax(Math.abs(c), Math.abs(s));
	double t_1 = fmin(Math.abs(c), Math.abs(s));
	double tmp;
	if ((Math.cos((2.0 * x)) / (Math.pow(t_1, 2.0) * ((x * Math.pow(t_0, 2.0)) * x))) <= Double.POSITIVE_INFINITY) {
		tmp = 1.0 / ((t_0 * (t_1 * t_1)) * ((t_0 * x) * x));
	} else {
		tmp = 1.0 / (((t_1 * t_0) * t_1) * ((x * x) * t_0));
	}
	return tmp;
}

Python

def code(x, c, s):
	t_0 = fmax(math.fabs(c), math.fabs(s))
	t_1 = fmin(math.fabs(c), math.fabs(s))
	tmp = 0
	if (math.cos((2.0 * x)) / (math.pow(t_1, 2.0) * ((x * math.pow(t_0, 2.0)) * x))) <= math.inf:
		tmp = 1.0 / ((t_0 * (t_1 * t_1)) * ((t_0 * x) * x))
	else:
		tmp = 1.0 / (((t_1 * t_0) * t_1) * ((x * x) * t_0))
	return tmp

Julia

function code(x, c, s)
	t_0 = fmax(abs(c), abs(s))
	t_1 = fmin(abs(c), abs(s))
	tmp = 0.0
	if (Float64(cos(Float64(2.0 * x)) / Float64((t_1 ^ 2.0) * Float64(Float64(x * (t_0 ^ 2.0)) * x))) <= Inf)
		tmp = Float64(1.0 / Float64(Float64(t_0 * Float64(t_1 * t_1)) * Float64(Float64(t_0 * x) * x)));
	else
		tmp = Float64(1.0 / Float64(Float64(Float64(t_1 * t_0) * t_1) * Float64(Float64(x * x) * t_0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	t_0 = max(abs(c), abs(s));
	t_1 = min(abs(c), abs(s));
	tmp = 0.0;
	if ((cos((2.0 * x)) / ((t_1 ^ 2.0) * ((x * (t_0 ^ 2.0)) * x))) <= Inf)
		tmp = 1.0 / ((t_0 * (t_1 * t_1)) * ((t_0 * x) * x));
	else
		tmp = 1.0 / (((t_1 * t_0) * t_1) * ((x * x) * t_0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := Block[{t$95$0 = N[Max[N[Abs[c], $MachinePrecision], N[Abs[s], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Min[N[Abs[c], $MachinePrecision], N[Abs[s], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] / N[(N[Power[t$95$1, 2.0], $MachinePrecision] * N[(N[(x * N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(1.0 / N[(N[(t$95$0 * N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$0 * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[(t$95$1 * t$95$0), $MachinePrecision] * t$95$1), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * t$95$0), $MachinePrecision]), $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 67.3%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 59.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*l* N/A

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

      16. associate-*l* N/A

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

      17. *-commutative N/A

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

      18. associate-*r* N/A

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

      19. lower-*.f64 N/A

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

      20. lower-*.f64 N/A

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

      21. lift-pow.f64 N/A

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

      22. pow2 N/A

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

      23. lift-*.f64 N/A

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

      24. lift-*.f64 N/A

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

      25. *-commutative N/A

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

   6. Applied rewrites 60.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. lower-*.f64 64.6%

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

   8. Applied rewrites 64.6%

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

   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 67.3%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 59.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*l* N/A

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

      16. associate-*l* N/A

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

      17. *-commutative N/A

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

      18. associate-*r* N/A

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

      19. lower-*.f64 N/A

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

      20. lower-*.f64 N/A

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

      21. lift-pow.f64 N/A

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

      22. pow2 N/A

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

      23. lift-*.f64 N/A

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

      24. lift-*.f64 N/A

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

      25. *-commutative N/A

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

   6. Applied rewrites 60.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 65.4%

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

   8. Applied rewrites 65.4%

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

Alternative 10: 68.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{max}\left(\left|c\right|, \left|s\right|\right)\\ t_1 := \mathsf{min}\left(\left|c\right|, \left|s\right|\right)\\ \frac{1}{\left(t\_0 \cdot \left(t\_1 \cdot t\_1\right)\right) \cdot \left(\left(t\_0 \cdot x\right) \cdot x\right)} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (fmax (fabs c) (fabs s))) (t_1 (fmin (fabs c) (fabs s))))
   (/ 1.0 (* (* t_0 (* t_1 t_1)) (* (* t_0 x) x)))))

C

double code(double x, double c, double s) {
	double t_0 = fmax(fabs(c), fabs(s));
	double t_1 = fmin(fabs(c), fabs(s));
	return 1.0 / ((t_0 * (t_1 * t_1)) * ((t_0 * x) * 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
    real(8) :: t_0
    real(8) :: t_1
    t_0 = fmax(abs(c), abs(s))
    t_1 = fmin(abs(c), abs(s))
    code = 1.0d0 / ((t_0 * (t_1 * t_1)) * ((t_0 * x) * x))
end function

Java

public static double code(double x, double c, double s) {
	double t_0 = fmax(Math.abs(c), Math.abs(s));
	double t_1 = fmin(Math.abs(c), Math.abs(s));
	return 1.0 / ((t_0 * (t_1 * t_1)) * ((t_0 * x) * x));
}

Python

def code(x, c, s):
	t_0 = fmax(math.fabs(c), math.fabs(s))
	t_1 = fmin(math.fabs(c), math.fabs(s))
	return 1.0 / ((t_0 * (t_1 * t_1)) * ((t_0 * x) * x))

Julia

function code(x, c, s)
	t_0 = fmax(abs(c), abs(s))
	t_1 = fmin(abs(c), abs(s))
	return Float64(1.0 / Float64(Float64(t_0 * Float64(t_1 * t_1)) * Float64(Float64(t_0 * x) * x)))
end

MATLAB

function tmp = code(x, c, s)
	t_0 = max(abs(c), abs(s));
	t_1 = min(abs(c), abs(s));
	tmp = 1.0 / ((t_0 * (t_1 * t_1)) * ((t_0 * x) * x));
end

Wolfram

code[x_, c_, s_] := Block[{t$95$0 = N[Max[N[Abs[c], $MachinePrecision], N[Abs[s], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Min[N[Abs[c], $MachinePrecision], N[Abs[s], $MachinePrecision]], $MachinePrecision]}, N[(1.0 / N[(N[(t$95$0 * N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$0 * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 67.3%

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

2. Taylor expanded in x around 0

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

4. Applied rewrites 59.2%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. lift-pow.f64 N/A

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

   7. unpow2 N/A

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

   8. associate-*l* N/A

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

   9. *-commutative N/A

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. associate-*r* N/A

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

   13. lift-*.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. associate-*l* N/A

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

   16. associate-*l* N/A

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

   17. *-commutative N/A

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

   18. associate-*r* N/A

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

   19. lower-*.f64 N/A

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

   20. lower-*.f64 N/A

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

   21. lift-pow.f64 N/A

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

   22. pow2 N/A

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

   23. lift-*.f64 N/A

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

   24. lift-*.f64 N/A

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

   25. *-commutative N/A

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

6. Applied rewrites 60.6%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

   5. lift-*.f64 N/A

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

   6. lower-*.f64 64.6%

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

8. Applied rewrites 64.6%

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

Reproduce

herbie shell --seed 2025183 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))