mixedcos

Average Error: 28.2 → 1.9

Time: 7.1s

Precision: binary64

\[ \begin{array}{c}[c, s] = \mathsf{sort}([c, s])\\ \end{array} \]

Math

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

↓

\[\begin{array}{l} t_0 := \cos \left(x + x\right)\\ \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(x \cdot \left(x \cdot {s}^{2}\right)\right)} \leq 0:\\ \;\;\;\;\frac{\frac{-t_0}{c \cdot \left(x \cdot s\right)}}{s \cdot \left(x \cdot \left(-c\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot {\left(x \cdot \left(c \cdot s\right)\right)}^{-2}\\ \end{array} \]

FPCore

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

↓

(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (cos (+ x x))))
   (if (<= (/ (cos (* 2.0 x)) (* (pow c 2.0) (* x (* x (pow s 2.0))))) 0.0)
     (/ (/ (- t_0) (* c (* x s))) (* s (* x (- c))))
     (* t_0 (pow (* x (* c s)) -2.0)))))

C

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

↓

double code(double x, double c, double s) {
	double t_0 = cos((x + x));
	double tmp;
	if ((cos((2.0 * x)) / (pow(c, 2.0) * (x * (x * pow(s, 2.0))))) <= 0.0) {
		tmp = (-t_0 / (c * (x * s))) / (s * (x * -c));
	} else {
		tmp = t_0 * pow((x * (c * s)), -2.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, c, s)
    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

↓

real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos((x + x))
    if ((cos((2.0d0 * x)) / ((c ** 2.0d0) * (x * (x * (s ** 2.0d0))))) <= 0.0d0) then
        tmp = (-t_0 / (c * (x * s))) / (s * (x * -c))
    else
        tmp = t_0 * ((x * (c * s)) ** (-2.0d0))
    end if
    code = tmp
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));
}

↓

public static double code(double x, double c, double s) {
	double t_0 = Math.cos((x + x));
	double tmp;
	if ((Math.cos((2.0 * x)) / (Math.pow(c, 2.0) * (x * (x * Math.pow(s, 2.0))))) <= 0.0) {
		tmp = (-t_0 / (c * (x * s))) / (s * (x * -c));
	} else {
		tmp = t_0 * Math.pow((x * (c * s)), -2.0);
	}
	return tmp;
}

Python

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

↓

def code(x, c, s):
	t_0 = math.cos((x + x))
	tmp = 0
	if (math.cos((2.0 * x)) / (math.pow(c, 2.0) * (x * (x * math.pow(s, 2.0))))) <= 0.0:
		tmp = (-t_0 / (c * (x * s))) / (s * (x * -c))
	else:
		tmp = t_0 * math.pow((x * (c * s)), -2.0)
	return tmp

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

↓

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

MATLAB

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

↓

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

↓

code[x_, c_, s_] := Block[{t$95$0 = N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] / N[(N[Power[c, 2.0], $MachinePrecision] * N[(x * N[(x * N[Power[s, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(N[((-t$95$0) / N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(s * N[(x * (-c)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[Power[N[(x * N[(c * s), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]]]

Error

Bits error versus x

Bits error versus c

Bits error versus s

Derivation

1. Split input into 2 regimes

2. if (/.f64 (cos.f64 (*.f64 2 x)) (*.f64 (pow.f64 c 2) (*.f64 (*.f64 x (pow.f64 s 2)) x))) < -0.0

   1. Initial program 16.8

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

   2. Simplified 2.8

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

   3. Applied egg-rr 2.5

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

   4. Applied egg-rr 1.3

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

   5. Applied egg-rr 1.2

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

   if -0.0 < (/.f64 (cos.f64 (*.f64 2 x)) (*.f64 (pow.f64 c 2) (*.f64 (*.f64 x (pow.f64 s 2)) x)))

   1. Initial program 46.6

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

   2. Simplified 3.3

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

   3. Applied egg-rr 3.0

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

4. Final simplification 1.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(x \cdot \left(x \cdot {s}^{2}\right)\right)} \leq 0:\\ \;\;\;\;\frac{\frac{-\cos \left(x + x\right)}{c \cdot \left(x \cdot s\right)}}{s \cdot \left(x \cdot \left(-c\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(x + x\right) \cdot {\left(x \cdot \left(c \cdot s\right)\right)}^{-2}\\ \end{array} \]

Reproduce

herbie shell --seed 2022151 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))