mixedcos

Percentage Accurate: 66.1% → 99.1%

Time: 4.8s

Alternatives: 10

Speedup: 2.0×

• 68.3% 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: 66.1% 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: 99.1% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

function tmp_2 = code(x, c, s)
	t_0 = max(abs(c), s);
	t_1 = cos((x + x));
	t_2 = min(abs(c), s);
	t_3 = (t_0 * x) * t_2;
	tmp = 0.0;
	if ((cos((2.0 * x)) / ((t_2 ^ 2.0) * ((x * (t_0 ^ 2.0)) * x))) <= Inf)
		tmp = (t_1 / t_3) / t_3;
	else
		tmp = t_1 / (((t_0 * t_2) * x) ^ 2.0);
	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[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Min[N[Abs[c], $MachinePrecision], s], $MachinePrecision]}, Block[{t$95$3 = N[(N[(t$95$0 * x), $MachinePrecision] * t$95$2), $MachinePrecision]}, If[LessEqual[N[(N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] / N[(N[Power[t$95$2, 2.0], $MachinePrecision] * N[(N[(x * N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(t$95$1 / t$95$3), $MachinePrecision] / t$95$3), $MachinePrecision], N[(t$95$1 / N[Power[N[(N[(t$95$0 * t$95$2), $MachinePrecision] * x), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 66.1%

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

      1. lift-*.f64 N/A

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

      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 77.0%

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

   3. Applied rewrites 77.0%

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \color{blue}{\left(\left(s \cdot x\right) \cdot \left(s \cdot x\right)\right)}} \]
   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. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. swap-sqr N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. unswap-sqr N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 N/A

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

      17. pow2 N/A

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

      18. pow-prod-down N/A

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

      19. lower-pow.f64 N/A

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

      20. lower-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f64 97.1%

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

   5. Applied rewrites 97.1%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. frac-times N/A

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

      16. mult-flip N/A

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

      17. lower-/.f64 N/A

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

   7. Applied rewrites 97.1%

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

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

   1. Initial program 66.1%

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

      1. lift-*.f64 N/A

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

      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 77.0%

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

   3. Applied rewrites 77.0%

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \color{blue}{\left(\left(s \cdot x\right) \cdot \left(s \cdot x\right)\right)}} \]
   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. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. swap-sqr N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. unswap-sqr N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 N/A

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

      17. pow2 N/A

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

      18. pow-prod-down N/A

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

      19. lower-pow.f64 N/A

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

      20. lower-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f64 97.1%

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

   5. Applied rewrites 97.1%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 97.1%

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

   7. Applied rewrites 97.1%

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

Alternative 2: 97.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 66.1%

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

   1. lift-*.f64 N/A

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

   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 77.0%

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

3. Applied rewrites 77.0%

   \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \color{blue}{\left(\left(s \cdot x\right) \cdot \left(s \cdot x\right)\right)}} \]
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. lift-pow.f64 N/A

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

   3. pow2 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. swap-sqr N/A

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

   9. lift-*.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. associate-*l* N/A

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

   12. lift-*.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. unswap-sqr N/A

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

   15. pow2 N/A

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

   16. lift-*.f64 N/A

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

   17. pow2 N/A

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

   18. pow-prod-down N/A

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

   19. lower-pow.f64 N/A

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

   20. lower-*.f64 N/A

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

   21. *-commutative N/A

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

   22. lower-*.f64 97.1%

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

5. Applied rewrites 97.1%

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

   1. lift-/.f64 N/A

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

   2. mult-flip N/A

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

   3. associate-*r/ N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. associate-*l* N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lift-*.f64 N/A

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

   11. lower-pow.f64 N/A

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

   12. pow2 N/A

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

   13. lift-*.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. frac-times N/A

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

   16. mult-flip N/A

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

   17. lower-/.f64 N/A

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

7. Applied rewrites 97.1%

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

Alternative 3: 96.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 66.1%

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

   1. lift-*.f64 N/A

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

   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 77.0%

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

3. Applied rewrites 77.0%

   \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \color{blue}{\left(\left(s \cdot x\right) \cdot \left(s \cdot x\right)\right)}} \]
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. lift-pow.f64 N/A

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

   3. pow2 N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. unswap-sqr N/A

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

   9. lower-*.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-*.f64 96.8%

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

5. Applied rewrites 96.8%

   \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(\left(s \cdot x\right) \cdot c\right) \cdot \left(\left(s \cdot x\right) \cdot c\right)}} \]
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 96.8%

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

7. Applied rewrites 96.8%

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

Alternative 4: 96.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(\left|c\right|, \left|s\right|\right)\\ t_1 := \mathsf{max}\left(\left|c\right|, \left|s\right|\right)\\ t_2 := \left(t\_1 \cdot \left|x\right|\right) \cdot t\_0\\ t_3 := \frac{1}{t\_2}\\ \mathbf{if}\;\left|x\right| \leq 4.5 \cdot 10^{-31}:\\ \;\;\;\;t\_3 \cdot t\_3\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(\left|x\right| + \left|x\right|\right)}{t\_1 \cdot \left(\left(t\_0 \cdot \left|x\right|\right) \cdot t\_2\right)}\\ \end{array} \]

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 (fabs x)) t_0))
        (t_3 (/ 1.0 t_2)))
   (if (<= (fabs x) 4.5e-31)
     (* t_3 t_3)
     (/ (cos (+ (fabs x) (fabs x))) (* t_1 (* (* t_0 (fabs x)) t_2))))))

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 * fabs(x)) * t_0;
	double t_3 = 1.0 / t_2;
	double tmp;
	if (fabs(x) <= 4.5e-31) {
		tmp = t_3 * t_3;
	} else {
		tmp = cos((fabs(x) + fabs(x))) / (t_1 * ((t_0 * fabs(x)) * t_2));
	}
	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 * abs(x)) * t_0
    t_3 = 1.0d0 / t_2
    if (abs(x) <= 4.5d-31) then
        tmp = t_3 * t_3
    else
        tmp = cos((abs(x) + abs(x))) / (t_1 * ((t_0 * abs(x)) * t_2))
    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 * Math.abs(x)) * t_0;
	double t_3 = 1.0 / t_2;
	double tmp;
	if (Math.abs(x) <= 4.5e-31) {
		tmp = t_3 * t_3;
	} else {
		tmp = Math.cos((Math.abs(x) + Math.abs(x))) / (t_1 * ((t_0 * Math.abs(x)) * t_2));
	}
	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 * math.fabs(x)) * t_0
	t_3 = 1.0 / t_2
	tmp = 0
	if math.fabs(x) <= 4.5e-31:
		tmp = t_3 * t_3
	else:
		tmp = math.cos((math.fabs(x) + math.fabs(x))) / (t_1 * ((t_0 * math.fabs(x)) * t_2))
	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(Float64(t_1 * abs(x)) * t_0)
	t_3 = Float64(1.0 / t_2)
	tmp = 0.0
	if (abs(x) <= 4.5e-31)
		tmp = Float64(t_3 * t_3);
	else
		tmp = Float64(cos(Float64(abs(x) + abs(x))) / Float64(t_1 * Float64(Float64(t_0 * abs(x)) * t_2)));
	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 * abs(x)) * t_0;
	t_3 = 1.0 / t_2;
	tmp = 0.0;
	if (abs(x) <= 4.5e-31)
		tmp = t_3 * t_3;
	else
		tmp = cos((abs(x) + abs(x))) / (t_1 * ((t_0 * abs(x)) * t_2));
	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[(N[(t$95$1 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$3 = N[(1.0 / t$95$2), $MachinePrecision]}, If[LessEqual[N[Abs[x], $MachinePrecision], 4.5e-31], N[(t$95$3 * t$95$3), $MachinePrecision], N[(N[Cos[N[(N[Abs[x], $MachinePrecision] + N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$1 * N[(N[(t$95$0 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if x < 4.5000000000000004e-31

   1. Initial program 66.1%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 58.2%

      \[\leadsto \frac{\color{blue}{1}}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   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. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lift-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*l* N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 64.5%

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

   6. Applied rewrites 64.5%

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

   8. Applied rewrites 78.0%

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

   if 4.5000000000000004e-31 < x

   1. Initial program 66.1%

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

      1. lift-*.f64 N/A

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

      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 77.0%

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

   3. Applied rewrites 77.0%

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \color{blue}{\left(\left(s \cdot x\right) \cdot \left(s \cdot x\right)\right)}} \]
   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. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. swap-sqr N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. unswap-sqr N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 N/A

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

      17. pow2 N/A

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

      18. pow-prod-down N/A

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

      19. lower-pow.f64 N/A

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

      20. lower-*.f64 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f64 97.1%

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

   5. Applied rewrites 97.1%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 97.1%

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

   7. Applied rewrites 97.1%

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

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 90.9%

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

   9. Applied rewrites 90.9%

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

Alternative 5: 82.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{max}\left(\left|c\right|, \left|s\right|\right)\\ t_1 := \mathsf{min}\left(\left|c\right|, \left|s\right|\right)\\ t_2 := \left(t\_0 \cdot x\right) \cdot t\_1\\ t_3 := \frac{1}{t\_2}\\ \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{{t\_1}^{2} \cdot \left(\left(x \cdot {t\_0}^{2}\right) \cdot x\right)} \leq -5 \cdot 10^{-126}:\\ \;\;\;\;\frac{1 + -2 \cdot {x}^{2}}{t\_2 \cdot t\_2}\\ \mathbf{else}:\\ \;\;\;\;t\_3 \cdot t\_3\\ \end{array} \]

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_1))
        (t_3 (/ 1.0 t_2)))
   (if (<=
        (/ (cos (* 2.0 x)) (* (pow t_1 2.0) (* (* x (pow t_0 2.0)) x)))
        -5e-126)
     (/ (+ 1.0 (* -2.0 (pow x 2.0))) (* t_2 t_2))
     (* 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) * t_1;
	double t_3 = 1.0 / t_2;
	double tmp;
	if ((cos((2.0 * x)) / (pow(t_1, 2.0) * ((x * pow(t_0, 2.0)) * x))) <= -5e-126) {
		tmp = (1.0 + (-2.0 * pow(x, 2.0))) / (t_2 * t_2);
	} 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), abs(s))
    t_1 = fmin(abs(c), abs(s))
    t_2 = (t_0 * x) * t_1
    t_3 = 1.0d0 / t_2
    if ((cos((2.0d0 * x)) / ((t_1 ** 2.0d0) * ((x * (t_0 ** 2.0d0)) * x))) <= (-5d-126)) then
        tmp = (1.0d0 + ((-2.0d0) * (x ** 2.0d0))) / (t_2 * t_2)
    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), Math.abs(s));
	double t_1 = fmin(Math.abs(c), Math.abs(s));
	double t_2 = (t_0 * x) * t_1;
	double t_3 = 1.0 / t_2;
	double tmp;
	if ((Math.cos((2.0 * x)) / (Math.pow(t_1, 2.0) * ((x * Math.pow(t_0, 2.0)) * x))) <= -5e-126) {
		tmp = (1.0 + (-2.0 * Math.pow(x, 2.0))) / (t_2 * t_2);
	} else {
		tmp = t_3 * 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_0 * x) * t_1
	t_3 = 1.0 / t_2
	tmp = 0
	if (math.cos((2.0 * x)) / (math.pow(t_1, 2.0) * ((x * math.pow(t_0, 2.0)) * x))) <= -5e-126:
		tmp = (1.0 + (-2.0 * math.pow(x, 2.0))) / (t_2 * t_2)
	else:
		tmp = 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(Float64(t_0 * x) * t_1)
	t_3 = Float64(1.0 / 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-126)
		tmp = Float64(Float64(1.0 + Float64(-2.0 * (x ^ 2.0))) / Float64(t_2 * t_2));
	else
		tmp = Float64(t_3 * 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_0 * x) * t_1;
	t_3 = 1.0 / t_2;
	tmp = 0.0;
	if ((cos((2.0 * x)) / ((t_1 ^ 2.0) * ((x * (t_0 ^ 2.0)) * x))) <= -5e-126)
		tmp = (1.0 + (-2.0 * (x ^ 2.0))) / (t_2 * t_2);
	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], 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$0 * x), $MachinePrecision] * t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(1.0 / 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-126], N[(N[(1.0 + N[(-2.0 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$2 * t$95$2), $MachinePrecision]), $MachinePrecision], N[(t$95$3 * 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))) < -5.0000000000000001e-126

   1. Initial program 66.1%

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

      1. lift-*.f64 N/A

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

      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 77.0%

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

   3. Applied rewrites 77.0%

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \color{blue}{\left(\left(s \cdot x\right) \cdot \left(s \cdot x\right)\right)}} \]
   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. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. unswap-sqr N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 96.8%

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

   5. Applied rewrites 96.8%

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

   6. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 61.3%

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

   8. Applied rewrites 61.3%

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

   if -5.0000000000000001e-126 < (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x)))

   1. Initial program 66.1%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 58.2%

      \[\leadsto \frac{\color{blue}{1}}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   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. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lift-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*l* N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 64.5%

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

   6. Applied rewrites 64.5%

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

   8. Applied rewrites 78.0%

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

Alternative 6: 79.1% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 66.1%

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

2. Taylor expanded in x around 0

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

4. Applied rewrites 58.2%

   \[\leadsto \frac{\color{blue}{1}}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
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. lift-pow.f64 N/A

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

   3. pow2 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*r* N/A

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

   7. lift-*.f64 N/A

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

   8. lift-pow.f64 N/A

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

   9. pow2 N/A

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

   10. lift-*.f64 N/A

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

   11. associate-*l* N/A

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

   12. lift-*.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. associate-*l* N/A

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

   15. lift-*.f64 N/A

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

   16. lift-*.f64 N/A

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

   17. lift-*.f64 64.5%

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

6. Applied rewrites 64.5%

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

8. Applied rewrites 78.0%

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

Alternative 7: 79.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

double code(double x, double c, double s) {
	double t_0 = (fmax(fabs(c), fabs(s)) * x) * fmin(fabs(c), fabs(s));
	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 = (fmax(abs(c), abs(s)) * x) * fmin(abs(c), abs(s))
    code = 1.0d0 / (t_0 * 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) * fmin(Math.abs(c), Math.abs(s));
	return 1.0 / (t_0 * t_0);
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 66.1%

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

   1. lift-*.f64 N/A

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

   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 77.0%

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

3. Applied rewrites 77.0%

   \[\leadsto \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \color{blue}{\left(\left(s \cdot x\right) \cdot \left(s \cdot x\right)\right)}} \]
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. lift-pow.f64 N/A

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

   3. pow2 N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. unswap-sqr N/A

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

   9. lower-*.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-*.f64 96.8%

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

5. Applied rewrites 96.8%

   \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(\left(s \cdot x\right) \cdot c\right) \cdot \left(\left(s \cdot x\right) \cdot c\right)}} \]
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 96.8%

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

7. Applied rewrites 96.8%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 78.0%

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

Alternative 8: 76.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{max}\left(\left|c\right|, \left|s\right|\right)\\ t_1 := \mathsf{min}\left(\left|c\right|, \left|s\right|\right)\\ \mathbf{if}\;t\_1 \leq 2.3 \cdot 10^{-26}:\\ \;\;\;\;\frac{1}{\left(t\_1 \cdot \left(t\_0 \cdot \left(\left(t\_0 \cdot x\right) \cdot t\_1\right)\right)\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(\left(\left(t\_1 \cdot t\_1\right) \cdot x\right) \cdot t\_0\right) \cdot x\right) \cdot t\_0}\\ \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (fmax (fabs c) (fabs s))) (t_1 (fmin (fabs c) (fabs s))))
   (if (<= t_1 2.3e-26)
     (/ 1.0 (* (* t_1 (* t_0 (* (* t_0 x) t_1))) x))
     (/ 1.0 (* (* (* (* (* t_1 t_1) x) t_0) 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 (t_1 <= 2.3e-26) {
		tmp = 1.0 / ((t_1 * (t_0 * ((t_0 * x) * t_1))) * x);
	} else {
		tmp = 1.0 / (((((t_1 * t_1) * x) * t_0) * x) * t_0);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, c, s)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = fmax(abs(c), abs(s))
    t_1 = fmin(abs(c), abs(s))
    if (t_1 <= 2.3d-26) then
        tmp = 1.0d0 / ((t_1 * (t_0 * ((t_0 * x) * t_1))) * x)
    else
        tmp = 1.0d0 / (((((t_1 * t_1) * x) * t_0) * x) * t_0)
    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 tmp;
	if (t_1 <= 2.3e-26) {
		tmp = 1.0 / ((t_1 * (t_0 * ((t_0 * x) * t_1))) * x);
	} else {
		tmp = 1.0 / (((((t_1 * t_1) * x) * t_0) * 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 t_1 <= 2.3e-26:
		tmp = 1.0 / ((t_1 * (t_0 * ((t_0 * x) * t_1))) * x)
	else:
		tmp = 1.0 / (((((t_1 * t_1) * x) * t_0) * 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 (t_1 <= 2.3e-26)
		tmp = Float64(1.0 / Float64(Float64(t_1 * Float64(t_0 * Float64(Float64(t_0 * x) * t_1))) * x));
	else
		tmp = Float64(1.0 / Float64(Float64(Float64(Float64(Float64(t_1 * t_1) * x) * t_0) * 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 (t_1 <= 2.3e-26)
		tmp = 1.0 / ((t_1 * (t_0 * ((t_0 * x) * t_1))) * x);
	else
		tmp = 1.0 / (((((t_1 * t_1) * x) * t_0) * 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[t$95$1, 2.3e-26], N[(1.0 / N[(N[(t$95$1 * N[(t$95$0 * N[(N[(t$95$0 * x), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[(N[(N[(t$95$1 * t$95$1), $MachinePrecision] * x), $MachinePrecision] * t$95$0), $MachinePrecision] * x), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if c < 2.3000000000000001e-26

   1. Initial program 66.1%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 58.2%

      \[\leadsto \frac{\color{blue}{1}}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   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. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lift-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*l* N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 64.5%

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

   6. Applied rewrites 64.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. sqr-neg-rev N/A

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

      5. lift-neg.f64 N/A

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

      6. lift-neg.f64 N/A

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

      7. unpow2 N/A

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

      8. exp-to-pow N/A

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

      9. lift-log.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-exp.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*l* N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*l* N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 32.1%

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-*.f64 32.1%

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

   8. Applied rewrites 65.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. lift-*.f64 N/A

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

      9. lower-*.f64 74.3%

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

   10. Applied rewrites 74.3%

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

   if 2.3000000000000001e-26 < c

   1. Initial program 66.1%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 58.2%

      \[\leadsto \frac{\color{blue}{1}}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   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. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lift-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*l* N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 64.5%

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

   6. Applied rewrites 64.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. sqr-neg-rev N/A

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

      5. lift-neg.f64 N/A

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

      6. lift-neg.f64 N/A

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

      7. unpow2 N/A

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

      8. exp-to-pow N/A

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

      9. lift-log.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-exp.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*l* N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*l* N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 32.1%

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-*.f64 32.1%

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

   8. Applied rewrites 65.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-*l* N/A

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

      16. lift-*.f64 N/A

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

      17. associate-*l* N/A

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

      18. lift-*.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. lower-*.f64 65.7%

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

   10. Applied rewrites 65.7%

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

Alternative 9: 75.1% accurate, 2.0× speedup

Math

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

FPCore

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

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));
	return 1.0 / ((t_0 * (t_1 * ((t_1 * x) * t_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
    real(8) :: t_0
    real(8) :: t_1
    t_0 = fmin(abs(c), abs(s))
    t_1 = fmax(abs(c), abs(s))
    code = 1.0d0 / ((t_0 * (t_1 * ((t_1 * x) * t_0))) * x)
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));
	return 1.0 / ((t_0 * (t_1 * ((t_1 * x) * t_0))) * x);
}

Python

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

Julia

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

MATLAB

function tmp = code(x, c, s)
	t_0 = min(abs(c), abs(s));
	t_1 = max(abs(c), abs(s));
	tmp = 1.0 / ((t_0 * (t_1 * ((t_1 * x) * t_0))) * x);
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]}, N[(1.0 / N[(N[(t$95$0 * N[(t$95$1 * N[(N[(t$95$1 * x), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 66.1%

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

2. Taylor expanded in x around 0

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

4. Applied rewrites 58.2%

   \[\leadsto \frac{\color{blue}{1}}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
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. lift-pow.f64 N/A

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

   3. pow2 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*r* N/A

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

   7. lift-*.f64 N/A

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

   8. lift-pow.f64 N/A

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

   9. pow2 N/A

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

   10. lift-*.f64 N/A

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

   11. associate-*l* N/A

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

   12. lift-*.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. associate-*l* N/A

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

   15. lift-*.f64 N/A

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

   16. lift-*.f64 N/A

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

   17. lift-*.f64 64.5%

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

6. Applied rewrites 64.5%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. sqr-neg-rev N/A

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

   5. lift-neg.f64 N/A

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

   6. lift-neg.f64 N/A

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

   7. unpow2 N/A

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

   8. exp-to-pow N/A

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

   9. lift-log.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. lift-exp.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. associate-*l* N/A

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

   14. lift-*.f64 N/A

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

   15. lift-*.f64 N/A

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

   16. associate-*l* N/A

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

   17. lower-*.f64 N/A

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

   18. lower-*.f64 32.1%

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

   19. lift-*.f64 N/A

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

   20. *-commutative N/A

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

   21. lower-*.f64 32.1%

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

8. Applied rewrites 65.0%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*l* N/A

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

   6. lift-*.f64 N/A

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

   7. associate-*l* N/A

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

   8. lift-*.f64 N/A

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

   9. lower-*.f64 74.3%

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

10. Applied rewrites 74.3%

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

Alternative 10: 65.0% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 66.1%

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

2. Taylor expanded in x around 0

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

4. Applied rewrites 58.2%

   \[\leadsto \frac{\color{blue}{1}}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
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. lift-pow.f64 N/A

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

   3. pow2 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*r* N/A

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

   7. lift-*.f64 N/A

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

   8. lift-pow.f64 N/A

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

   9. pow2 N/A

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

   10. lift-*.f64 N/A

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

   11. associate-*l* N/A

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

   12. lift-*.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. associate-*l* N/A

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

   15. lift-*.f64 N/A

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

   16. lift-*.f64 N/A

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

   17. lift-*.f64 64.5%

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

6. Applied rewrites 64.5%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. sqr-neg-rev N/A

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

   5. lift-neg.f64 N/A

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

   6. lift-neg.f64 N/A

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

   7. unpow2 N/A

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

   8. exp-to-pow N/A

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

   9. lift-log.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. lift-exp.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. associate-*l* N/A

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

   14. lift-*.f64 N/A

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

   15. lift-*.f64 N/A

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

   16. associate-*l* N/A

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

   17. lower-*.f64 N/A

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

   18. lower-*.f64 32.1%

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

   19. lift-*.f64 N/A

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

   20. *-commutative N/A

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

   21. lower-*.f64 32.1%

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

8. Applied rewrites 65.0%

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

Reproduce

herbie shell --seed 2025191 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))