mixedcos

Percentage Accurate: 67.4% → 97.1%

Time: 3.3s

Alternatives: 8

Speedup: 1.5×

• 31.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 67.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.1% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.4%

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

   1. lift-*.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-pow.f64 N/A

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

   8. associate-*r* N/A

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

   9. associate-*r* N/A

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

   10. unpow2 N/A

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

   11. pow-prod-down N/A

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

   12. pow-prod-down N/A

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

   13. lower-pow.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 97.1

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

3. Applied rewrites 97.1%

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

   1. lift-pow.f64 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f64 97.1

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 97.1

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 97.1

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

5. Applied rewrites 97.1%

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

   1. lift-*.f64 N/A

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

   2. count-2-rev N/A

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

   3. lower-+.f64 97.1

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

7. Applied rewrites 97.1%

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

Alternative 2: 81.9% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (* (* c s) x)) (t_1 (* t_0 t_0)))
   (if (<= (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))) -1e-102)
     (/ (fma -2.0 (* x x) 1.0) t_1)
     (/ 1.0 t_1))))

C

double code(double x, double c, double s) {
	double t_0 = (c * s) * x;
	double t_1 = t_0 * t_0;
	double tmp;
	if ((cos((2.0 * x)) / (pow(c, 2.0) * ((x * pow(s, 2.0)) * x))) <= -1e-102) {
		tmp = fma(-2.0, (x * x), 1.0) / t_1;
	} else {
		tmp = 1.0 / t_1;
	}
	return tmp;
}

Julia

function code(x, c, s)
	t_0 = Float64(Float64(c * s) * x)
	t_1 = Float64(t_0 * t_0)
	tmp = 0.0
	if (Float64(cos(Float64(2.0 * x)) / Float64((c ^ 2.0) * Float64(Float64(x * (s ^ 2.0)) * x))) <= -1e-102)
		tmp = Float64(fma(-2.0, Float64(x * x), 1.0) / t_1);
	else
		tmp = Float64(1.0 / t_1);
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x))) < -9.99999999999999933e-103

   1. Initial program 67.4%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. associate-*r* N/A

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

      10. unpow2 N/A

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

      11. pow-prod-down N/A

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

      12. pow-prod-down N/A

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

      13. lower-pow.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 97.1

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

   3. Applied rewrites 97.1%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 97.1

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 97.1

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 97.1

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

   5. Applied rewrites 97.1%

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

   6. Taylor expanded in x around 0

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

      1. count-2-rev N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 60.5

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

   8. Applied rewrites 60.5%

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

   if -9.99999999999999933e-103 < (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x)))

   1. Initial program 67.4%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. associate-*r* N/A

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

      10. unpow2 N/A

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

      11. pow-prod-down N/A

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

      12. pow-prod-down N/A

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

      13. lower-pow.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 97.1

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

   3. Applied rewrites 97.1%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 97.1

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 97.1

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 97.1

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

   5. Applied rewrites 97.1%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 97.1

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

   7. Applied rewrites 97.1%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 78.3%

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

Alternative 3: 81.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(c \cdot s\right) \cdot x\\ \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \leq -1 \cdot 10^{-102}:\\ \;\;\;\;\frac{-2}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t\_0 \cdot t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (* (* c s) x)))
   (if (<= (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))) -1e-102)
     (/ -2.0 (* (* (* s s) c) c))
     (/ 1.0 (* t_0 t_0)))))

C

double code(double x, double c, double s) {
	double t_0 = (c * s) * x;
	double tmp;
	if ((cos((2.0 * x)) / (pow(c, 2.0) * ((x * pow(s, 2.0)) * x))) <= -1e-102) {
		tmp = -2.0 / (((s * s) * c) * c);
	} else {
		tmp = 1.0 / (t_0 * t_0);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, c, s)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (c * s) * x
    if ((cos((2.0d0 * x)) / ((c ** 2.0d0) * ((x * (s ** 2.0d0)) * x))) <= (-1d-102)) then
        tmp = (-2.0d0) / (((s * s) * c) * c)
    else
        tmp = 1.0d0 / (t_0 * t_0)
    end if
    code = tmp
end function

Java

public static double code(double x, double c, double s) {
	double t_0 = (c * s) * x;
	double tmp;
	if ((Math.cos((2.0 * x)) / (Math.pow(c, 2.0) * ((x * Math.pow(s, 2.0)) * x))) <= -1e-102) {
		tmp = -2.0 / (((s * s) * c) * c);
	} else {
		tmp = 1.0 / (t_0 * t_0);
	}
	return tmp;
}

Python

def code(x, c, s):
	t_0 = (c * s) * x
	tmp = 0
	if (math.cos((2.0 * x)) / (math.pow(c, 2.0) * ((x * math.pow(s, 2.0)) * x))) <= -1e-102:
		tmp = -2.0 / (((s * s) * c) * c)
	else:
		tmp = 1.0 / (t_0 * t_0)
	return tmp

Julia

function code(x, c, s)
	t_0 = Float64(Float64(c * s) * x)
	tmp = 0.0
	if (Float64(cos(Float64(2.0 * x)) / Float64((c ^ 2.0) * Float64(Float64(x * (s ^ 2.0)) * x))) <= -1e-102)
		tmp = Float64(-2.0 / Float64(Float64(Float64(s * s) * c) * c));
	else
		tmp = Float64(1.0 / Float64(t_0 * t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	t_0 = (c * s) * x;
	tmp = 0.0;
	if ((cos((2.0 * x)) / ((c ^ 2.0) * ((x * (s ^ 2.0)) * x))) <= -1e-102)
		tmp = -2.0 / (((s * s) * c) * c);
	else
		tmp = 1.0 / (t_0 * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := Block[{t$95$0 = N[(N[(c * s), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[N[(N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] / N[(N[Power[c, 2.0], $MachinePrecision] * N[(N[(x * N[Power[s, 2.0], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1e-102], N[(-2.0 / N[(N[(N[(s * s), $MachinePrecision] * c), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(t$95$0 * t$95$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))) < -9.99999999999999933e-103

   1. Initial program 67.4%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. associate-*r* N/A

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

      10. unpow2 N/A

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

      11. pow-prod-down N/A

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

      12. pow-prod-down N/A

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

      13. lower-pow.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 97.1

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

   3. Applied rewrites 97.1%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 97.1

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 97.1

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 97.1

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

   5. Applied rewrites 97.1%

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

   6. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

   8. Applied rewrites 21.9%

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

   9. Taylor expanded in x around inf

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

       1. pow2 N/A

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

       2. lift-*.f64 N/A

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

       3. *-commutative N/A

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

       4. lower-*.f64 N/A

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

       5. pow2 N/A

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

       6. lift-*.f64 N/A

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

       7. lower-/.f64 28.5

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

       8. lift-*.f64 N/A

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

       9. lift-*.f64 N/A

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

       10. associate-*r* N/A

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

       11. lift-*.f64 N/A

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

       12. lower-*.f64 28.8

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

   11. Applied rewrites 28.8%

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

   if -9.99999999999999933e-103 < (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x)))

   1. Initial program 67.4%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. associate-*r* N/A

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

      10. unpow2 N/A

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

      11. pow-prod-down N/A

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

      12. pow-prod-down N/A

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

      13. lower-pow.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 97.1

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

   3. Applied rewrites 97.1%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 97.1

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 97.1

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 97.1

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

   5. Applied rewrites 97.1%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 97.1

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

   7. Applied rewrites 97.1%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 78.3%

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

Alternative 4: 74.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \leq -1 \cdot 10^{-102}:\\ \;\;\;\;\frac{-2}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(\left(s \cdot \left(s \cdot x\right)\right) \cdot x\right) \cdot c\right) \cdot c}\\ \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (if (<= (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))) -1e-102)
   (/ -2.0 (* (* (* s s) c) c))
   (/ 1.0 (* (* (* (* s (* s x)) x) c) c))))

C

double code(double x, double c, double s) {
	double tmp;
	if ((cos((2.0 * x)) / (pow(c, 2.0) * ((x * pow(s, 2.0)) * x))) <= -1e-102) {
		tmp = -2.0 / (((s * s) * c) * c);
	} else {
		tmp = 1.0 / ((((s * (s * x)) * x) * c) * c);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, c, s)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: tmp
    if ((cos((2.0d0 * x)) / ((c ** 2.0d0) * ((x * (s ** 2.0d0)) * x))) <= (-1d-102)) then
        tmp = (-2.0d0) / (((s * s) * c) * c)
    else
        tmp = 1.0d0 / ((((s * (s * x)) * x) * c) * c)
    end if
    code = tmp
end function

Java

public static double code(double x, double c, double s) {
	double tmp;
	if ((Math.cos((2.0 * x)) / (Math.pow(c, 2.0) * ((x * Math.pow(s, 2.0)) * x))) <= -1e-102) {
		tmp = -2.0 / (((s * s) * c) * c);
	} else {
		tmp = 1.0 / ((((s * (s * x)) * x) * c) * c);
	}
	return tmp;
}

Python

def code(x, c, s):
	tmp = 0
	if (math.cos((2.0 * x)) / (math.pow(c, 2.0) * ((x * math.pow(s, 2.0)) * x))) <= -1e-102:
		tmp = -2.0 / (((s * s) * c) * c)
	else:
		tmp = 1.0 / ((((s * (s * x)) * x) * c) * c)
	return tmp

Julia

function code(x, c, s)
	tmp = 0.0
	if (Float64(cos(Float64(2.0 * x)) / Float64((c ^ 2.0) * Float64(Float64(x * (s ^ 2.0)) * x))) <= -1e-102)
		tmp = Float64(-2.0 / Float64(Float64(Float64(s * s) * c) * c));
	else
		tmp = Float64(1.0 / Float64(Float64(Float64(Float64(s * Float64(s * x)) * x) * c) * c));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	tmp = 0.0;
	if ((cos((2.0 * x)) / ((c ^ 2.0) * ((x * (s ^ 2.0)) * x))) <= -1e-102)
		tmp = -2.0 / (((s * s) * c) * c);
	else
		tmp = 1.0 / ((((s * (s * x)) * x) * c) * c);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := If[LessEqual[N[(N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] / N[(N[Power[c, 2.0], $MachinePrecision] * N[(N[(x * N[Power[s, 2.0], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1e-102], N[(-2.0 / N[(N[(N[(s * s), $MachinePrecision] * c), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[(N[(s * N[(s * x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] * c), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x))) < -9.99999999999999933e-103

   1. Initial program 67.4%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. associate-*r* N/A

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

      10. unpow2 N/A

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

      11. pow-prod-down N/A

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

      12. pow-prod-down N/A

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

      13. lower-pow.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 97.1

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

   3. Applied rewrites 97.1%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 97.1

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 97.1

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 97.1

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

   5. Applied rewrites 97.1%

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

   6. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

   8. Applied rewrites 21.9%

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

   9. Taylor expanded in x around inf

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

       1. pow2 N/A

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

       2. lift-*.f64 N/A

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

       3. *-commutative N/A

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

       4. lower-*.f64 N/A

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

       5. pow2 N/A

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

       6. lift-*.f64 N/A

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

       7. lower-/.f64 28.5

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

       8. lift-*.f64 N/A

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

       9. lift-*.f64 N/A

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

       10. associate-*r* N/A

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

       11. lift-*.f64 N/A

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

       12. lower-*.f64 28.8

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

   11. Applied rewrites 28.8%

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

   if -9.99999999999999933e-103 < (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x)))

   1. Initial program 67.4%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 65.0

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

   4. Applied rewrites 65.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 70.7

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

   6. Applied rewrites 70.7%

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

Alternative 5: 73.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-102}:\\ \;\;\;\;\frac{-2}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c}\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\frac{1}{\left(\left(\left(s \cdot s\right) \cdot x\right) \cdot \left(x \cdot c\right)\right) \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(s \cdot \left(s \cdot c\right)\right) \cdot \left(x \cdot x\right)\right) \cdot c}\\ \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x)))))
   (if (<= t_0 -1e-102)
     (/ -2.0 (* (* (* s s) c) c))
     (if (<= t_0 INFINITY)
       (/ 1.0 (* (* (* (* s s) x) (* x c)) c))
       (/ 1.0 (* (* (* s (* s c)) (* x x)) c))))))

C

double code(double x, double c, double s) {
	double t_0 = cos((2.0 * x)) / (pow(c, 2.0) * ((x * pow(s, 2.0)) * x));
	double tmp;
	if (t_0 <= -1e-102) {
		tmp = -2.0 / (((s * s) * c) * c);
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = 1.0 / ((((s * s) * x) * (x * c)) * c);
	} else {
		tmp = 1.0 / (((s * (s * c)) * (x * x)) * c);
	}
	return tmp;
}

Java

public static double code(double x, double c, double s) {
	double t_0 = Math.cos((2.0 * x)) / (Math.pow(c, 2.0) * ((x * Math.pow(s, 2.0)) * x));
	double tmp;
	if (t_0 <= -1e-102) {
		tmp = -2.0 / (((s * s) * c) * c);
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = 1.0 / ((((s * s) * x) * (x * c)) * c);
	} else {
		tmp = 1.0 / (((s * (s * c)) * (x * x)) * c);
	}
	return tmp;
}

Python

def code(x, c, s):
	t_0 = math.cos((2.0 * x)) / (math.pow(c, 2.0) * ((x * math.pow(s, 2.0)) * x))
	tmp = 0
	if t_0 <= -1e-102:
		tmp = -2.0 / (((s * s) * c) * c)
	elif t_0 <= math.inf:
		tmp = 1.0 / ((((s * s) * x) * (x * c)) * c)
	else:
		tmp = 1.0 / (((s * (s * c)) * (x * x)) * c)
	return tmp

Julia

function code(x, c, s)
	t_0 = Float64(cos(Float64(2.0 * x)) / Float64((c ^ 2.0) * Float64(Float64(x * (s ^ 2.0)) * x)))
	tmp = 0.0
	if (t_0 <= -1e-102)
		tmp = Float64(-2.0 / Float64(Float64(Float64(s * s) * c) * c));
	elseif (t_0 <= Inf)
		tmp = Float64(1.0 / Float64(Float64(Float64(Float64(s * s) * x) * Float64(x * c)) * c));
	else
		tmp = Float64(1.0 / Float64(Float64(Float64(s * Float64(s * c)) * Float64(x * x)) * c));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	t_0 = cos((2.0 * x)) / ((c ^ 2.0) * ((x * (s ^ 2.0)) * x));
	tmp = 0.0;
	if (t_0 <= -1e-102)
		tmp = -2.0 / (((s * s) * c) * c);
	elseif (t_0 <= Inf)
		tmp = 1.0 / ((((s * s) * x) * (x * c)) * c);
	else
		tmp = 1.0 / (((s * (s * c)) * (x * x)) * c);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := Block[{t$95$0 = 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]}, If[LessEqual[t$95$0, -1e-102], N[(-2.0 / N[(N[(N[(s * s), $MachinePrecision] * c), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(1.0 / N[(N[(N[(N[(s * s), $MachinePrecision] * x), $MachinePrecision] * N[(x * c), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[(s * N[(s * c), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 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))) < -9.99999999999999933e-103

   1. Initial program 67.4%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. associate-*r* N/A

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

      10. unpow2 N/A

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

      11. pow-prod-down N/A

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

      12. pow-prod-down N/A

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

      13. lower-pow.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 97.1

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

   3. Applied rewrites 97.1%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 97.1

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 97.1

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 97.1

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

   5. Applied rewrites 97.1%

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

   6. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

   8. Applied rewrites 21.9%

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

   9. Taylor expanded in x around inf

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

       1. pow2 N/A

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

       2. lift-*.f64 N/A

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

       3. *-commutative N/A

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

       4. lower-*.f64 N/A

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

       5. pow2 N/A

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

       6. lift-*.f64 N/A

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

       7. lower-/.f64 28.5

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

       8. lift-*.f64 N/A

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

       9. lift-*.f64 N/A

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

       10. associate-*r* N/A

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

       11. lift-*.f64 N/A

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

       12. lower-*.f64 28.8

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

   11. Applied rewrites 28.8%

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

   if -9.99999999999999933e-103 < (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x))) < +inf.0

   1. Initial program 67.4%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 65.0

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

   4. Applied rewrites 65.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 67.0

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

   6. Applied rewrites 67.0%

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

   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 67.4%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 65.0

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

   4. Applied rewrites 65.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. pow2 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 61.5

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

   6. Applied rewrites 61.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 66.2

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

   8. Applied rewrites 66.2%

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

Alternative 6: 69.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \leq -1 \cdot 10^{-102}:\\ \;\;\;\;\frac{-2}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(s \cdot \left(s \cdot c\right)\right) \cdot \left(x \cdot x\right)\right) \cdot c}\\ \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (if (<= (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))) -1e-102)
   (/ -2.0 (* (* (* s s) c) c))
   (/ 1.0 (* (* (* s (* s c)) (* x x)) c))))

C

double code(double x, double c, double s) {
	double tmp;
	if ((cos((2.0 * x)) / (pow(c, 2.0) * ((x * pow(s, 2.0)) * x))) <= -1e-102) {
		tmp = -2.0 / (((s * s) * c) * c);
	} else {
		tmp = 1.0 / (((s * (s * c)) * (x * x)) * c);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, c, s)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: tmp
    if ((cos((2.0d0 * x)) / ((c ** 2.0d0) * ((x * (s ** 2.0d0)) * x))) <= (-1d-102)) then
        tmp = (-2.0d0) / (((s * s) * c) * c)
    else
        tmp = 1.0d0 / (((s * (s * c)) * (x * x)) * c)
    end if
    code = tmp
end function

Java

public static double code(double x, double c, double s) {
	double tmp;
	if ((Math.cos((2.0 * x)) / (Math.pow(c, 2.0) * ((x * Math.pow(s, 2.0)) * x))) <= -1e-102) {
		tmp = -2.0 / (((s * s) * c) * c);
	} else {
		tmp = 1.0 / (((s * (s * c)) * (x * x)) * c);
	}
	return tmp;
}

Python

def code(x, c, s):
	tmp = 0
	if (math.cos((2.0 * x)) / (math.pow(c, 2.0) * ((x * math.pow(s, 2.0)) * x))) <= -1e-102:
		tmp = -2.0 / (((s * s) * c) * c)
	else:
		tmp = 1.0 / (((s * (s * c)) * (x * x)) * c)
	return tmp

Julia

function code(x, c, s)
	tmp = 0.0
	if (Float64(cos(Float64(2.0 * x)) / Float64((c ^ 2.0) * Float64(Float64(x * (s ^ 2.0)) * x))) <= -1e-102)
		tmp = Float64(-2.0 / Float64(Float64(Float64(s * s) * c) * c));
	else
		tmp = Float64(1.0 / Float64(Float64(Float64(s * Float64(s * c)) * Float64(x * x)) * c));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	tmp = 0.0;
	if ((cos((2.0 * x)) / ((c ^ 2.0) * ((x * (s ^ 2.0)) * x))) <= -1e-102)
		tmp = -2.0 / (((s * s) * c) * c);
	else
		tmp = 1.0 / (((s * (s * c)) * (x * x)) * c);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := If[LessEqual[N[(N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] / N[(N[Power[c, 2.0], $MachinePrecision] * N[(N[(x * N[Power[s, 2.0], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1e-102], N[(-2.0 / N[(N[(N[(s * s), $MachinePrecision] * c), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[(s * N[(s * c), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x))) < -9.99999999999999933e-103

   1. Initial program 67.4%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. associate-*r* N/A

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

      10. unpow2 N/A

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

      11. pow-prod-down N/A

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

      12. pow-prod-down N/A

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

      13. lower-pow.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 97.1

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

   3. Applied rewrites 97.1%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 97.1

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 97.1

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 97.1

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

   5. Applied rewrites 97.1%

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

   6. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

   8. Applied rewrites 21.9%

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

   9. Taylor expanded in x around inf

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

       1. pow2 N/A

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

       2. lift-*.f64 N/A

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

       3. *-commutative N/A

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

       4. lower-*.f64 N/A

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

       5. pow2 N/A

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

       6. lift-*.f64 N/A

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

       7. lower-/.f64 28.5

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

       8. lift-*.f64 N/A

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

       9. lift-*.f64 N/A

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

       10. associate-*r* N/A

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

       11. lift-*.f64 N/A

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

       12. lower-*.f64 28.8

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

   11. Applied rewrites 28.8%

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

   if -9.99999999999999933e-103 < (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x)))

   1. Initial program 67.4%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 65.0

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

   4. Applied rewrites 65.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. pow2 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 61.5

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

   6. Applied rewrites 61.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 66.2

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

   8. Applied rewrites 66.2%

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

Alternative 7: 69.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \leq -1 \cdot 10^{-102}:\\ \;\;\;\;\frac{-2}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(s \cdot \left(s \cdot c\right)\right) \cdot \left(\left(x \cdot x\right) \cdot c\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (if (<= (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))) -1e-102)
   (/ -2.0 (* (* (* s s) c) c))
   (/ 1.0 (* (* s (* s c)) (* (* x x) c)))))

C

double code(double x, double c, double s) {
	double tmp;
	if ((cos((2.0 * x)) / (pow(c, 2.0) * ((x * pow(s, 2.0)) * x))) <= -1e-102) {
		tmp = -2.0 / (((s * s) * c) * c);
	} else {
		tmp = 1.0 / ((s * (s * c)) * ((x * x) * c));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, c, s)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: tmp
    if ((cos((2.0d0 * x)) / ((c ** 2.0d0) * ((x * (s ** 2.0d0)) * x))) <= (-1d-102)) then
        tmp = (-2.0d0) / (((s * s) * c) * c)
    else
        tmp = 1.0d0 / ((s * (s * c)) * ((x * x) * c))
    end if
    code = tmp
end function

Java

public static double code(double x, double c, double s) {
	double tmp;
	if ((Math.cos((2.0 * x)) / (Math.pow(c, 2.0) * ((x * Math.pow(s, 2.0)) * x))) <= -1e-102) {
		tmp = -2.0 / (((s * s) * c) * c);
	} else {
		tmp = 1.0 / ((s * (s * c)) * ((x * x) * c));
	}
	return tmp;
}

Python

def code(x, c, s):
	tmp = 0
	if (math.cos((2.0 * x)) / (math.pow(c, 2.0) * ((x * math.pow(s, 2.0)) * x))) <= -1e-102:
		tmp = -2.0 / (((s * s) * c) * c)
	else:
		tmp = 1.0 / ((s * (s * c)) * ((x * x) * c))
	return tmp

Julia

function code(x, c, s)
	tmp = 0.0
	if (Float64(cos(Float64(2.0 * x)) / Float64((c ^ 2.0) * Float64(Float64(x * (s ^ 2.0)) * x))) <= -1e-102)
		tmp = Float64(-2.0 / Float64(Float64(Float64(s * s) * c) * c));
	else
		tmp = Float64(1.0 / Float64(Float64(s * Float64(s * c)) * Float64(Float64(x * x) * c)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	tmp = 0.0;
	if ((cos((2.0 * x)) / ((c ^ 2.0) * ((x * (s ^ 2.0)) * x))) <= -1e-102)
		tmp = -2.0 / (((s * s) * c) * c);
	else
		tmp = 1.0 / ((s * (s * c)) * ((x * x) * c));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := If[LessEqual[N[(N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] / N[(N[Power[c, 2.0], $MachinePrecision] * N[(N[(x * N[Power[s, 2.0], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1e-102], N[(-2.0 / N[(N[(N[(s * s), $MachinePrecision] * c), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(s * N[(s * c), $MachinePrecision]), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x))) < -9.99999999999999933e-103

   1. Initial program 67.4%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. associate-*r* N/A

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

      10. unpow2 N/A

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

      11. pow-prod-down N/A

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

      12. pow-prod-down N/A

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

      13. lower-pow.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 97.1

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

   3. Applied rewrites 97.1%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 97.1

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 97.1

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 97.1

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

   5. Applied rewrites 97.1%

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

   6. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

   8. Applied rewrites 21.9%

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

   9. Taylor expanded in x around inf

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

       1. pow2 N/A

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

       2. lift-*.f64 N/A

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

       3. *-commutative N/A

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

       4. lower-*.f64 N/A

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

       5. pow2 N/A

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

       6. lift-*.f64 N/A

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

       7. lower-/.f64 28.5

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

       8. lift-*.f64 N/A

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

       9. lift-*.f64 N/A

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

       10. associate-*r* N/A

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

       11. lift-*.f64 N/A

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

       12. lower-*.f64 28.8

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

   11. Applied rewrites 28.8%

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

   if -9.99999999999999933e-103 < (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x)))

   1. Initial program 67.4%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 65.0

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

   4. Applied rewrites 65.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. pow2 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 61.5

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

   6. Applied rewrites 61.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. pow2 N/A

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

      4. lower-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 61.4

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

   8. Applied rewrites 61.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 66.3

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

   10. Applied rewrites 66.3%

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

Alternative 8: 28.8% accurate, 6.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

public static double code(double x, double c, double s) {
	return -2.0 / (((s * s) * c) * c);
}

Python

def code(x, c, s):
	return -2.0 / (((s * s) * c) * c)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.4%

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

   1. lift-*.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-pow.f64 N/A

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

   8. associate-*r* N/A

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

   9. associate-*r* N/A

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

   10. unpow2 N/A

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

   11. pow-prod-down N/A

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

   12. pow-prod-down N/A

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

   13. lower-pow.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 97.1

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

3. Applied rewrites 97.1%

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

   1. lift-pow.f64 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f64 97.1

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 97.1

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 97.1

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

5. Applied rewrites 97.1%

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

6. Taylor expanded in x around 0

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

   1. lower-/.f64 N/A

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

8. Applied rewrites 21.9%

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

9. Taylor expanded in x around inf

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

    1. pow2 N/A

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

    2. lift-*.f64 N/A

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

    3. *-commutative N/A

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

    4. lower-*.f64 N/A

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

    5. pow2 N/A

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

    6. lift-*.f64 N/A

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

    7. lower-/.f64 28.5

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

    8. lift-*.f64 N/A

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

    9. lift-*.f64 N/A

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

    10. associate-*r* N/A

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

    11. lift-*.f64 N/A

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

    12. lower-*.f64 28.8

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

11. Applied rewrites 28.8%

    \[\leadsto \frac{-2}{\color{blue}{\left(\left(s \cdot s\right) \cdot c\right) \cdot c}} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2025139 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))