mixedcos

Average Error: 28.6 → 1.3

Time: 9.3s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := c \cdot \left|x \cdot s\right|\\ t_1 := \cos \left(2 \cdot x\right)\\ t_2 := \left(c \cdot \left|s\right|\right) \cdot \left|x\right|\\ \mathbf{if}\;\frac{t_1}{{c}^{2} \cdot \left(x \cdot \left(x \cdot {s}^{2}\right)\right)} \leq 1.27068303241079 \cdot 10^{-254}:\\ \;\;\;\;\frac{\frac{t_1}{t_0}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t_2} \cdot \frac{t_1}{t_2}\\ \end{array} \]

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 (* c (fabs (* x s))))
        (t_1 (cos (* 2.0 x)))
        (t_2 (* (* c (fabs s)) (fabs x))))
   (if (<=
        (/ t_1 (* (pow c 2.0) (* x (* x (pow s 2.0)))))
        1.27068303241079e-254)
     (/ (/ t_1 t_0) t_0)
     (* (/ 1.0 t_2) (/ t_1 t_2)))))

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 = c * fabs((x * s));
	double t_1 = cos((2.0 * x));
	double t_2 = (c * fabs(s)) * fabs(x);
	double tmp;
	if ((t_1 / (pow(c, 2.0) * (x * (x * pow(s, 2.0))))) <= 1.27068303241079e-254) {
		tmp = (t_1 / t_0) / t_0;
	} else {
		tmp = (1.0 / t_2) * (t_1 / t_2);
	}
	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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = c * abs((x * s))
    t_1 = cos((2.0d0 * x))
    t_2 = (c * abs(s)) * abs(x)
    if ((t_1 / ((c ** 2.0d0) * (x * (x * (s ** 2.0d0))))) <= 1.27068303241079d-254) then
        tmp = (t_1 / t_0) / t_0
    else
        tmp = (1.0d0 / t_2) * (t_1 / t_2)
    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 = c * Math.abs((x * s));
	double t_1 = Math.cos((2.0 * x));
	double t_2 = (c * Math.abs(s)) * Math.abs(x);
	double tmp;
	if ((t_1 / (Math.pow(c, 2.0) * (x * (x * Math.pow(s, 2.0))))) <= 1.27068303241079e-254) {
		tmp = (t_1 / t_0) / t_0;
	} else {
		tmp = (1.0 / t_2) * (t_1 / t_2);
	}
	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 = c * math.fabs((x * s))
	t_1 = math.cos((2.0 * x))
	t_2 = (c * math.fabs(s)) * math.fabs(x)
	tmp = 0
	if (t_1 / (math.pow(c, 2.0) * (x * (x * math.pow(s, 2.0))))) <= 1.27068303241079e-254:
		tmp = (t_1 / t_0) / t_0
	else:
		tmp = (1.0 / t_2) * (t_1 / t_2)
	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 = Float64(c * abs(Float64(x * s)))
	t_1 = cos(Float64(2.0 * x))
	t_2 = Float64(Float64(c * abs(s)) * abs(x))
	tmp = 0.0
	if (Float64(t_1 / Float64((c ^ 2.0) * Float64(x * Float64(x * (s ^ 2.0))))) <= 1.27068303241079e-254)
		tmp = Float64(Float64(t_1 / t_0) / t_0);
	else
		tmp = Float64(Float64(1.0 / t_2) * Float64(t_1 / t_2));
	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 = c * abs((x * s));
	t_1 = cos((2.0 * x));
	t_2 = (c * abs(s)) * abs(x);
	tmp = 0.0;
	if ((t_1 / ((c ^ 2.0) * (x * (x * (s ^ 2.0))))) <= 1.27068303241079e-254)
		tmp = (t_1 / t_0) / t_0;
	else
		tmp = (1.0 / t_2) * (t_1 / t_2);
	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[(c * N[Abs[N[(x * s), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(c * N[Abs[s], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$1 / N[(N[Power[c, 2.0], $MachinePrecision] * N[(x * N[(x * N[Power[s, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.27068303241079e-254], N[(N[(t$95$1 / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(1.0 / t$95$2), $MachinePrecision] * N[(t$95$1 / t$95$2), $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))) < 1.2706830324107899e-254

   1. Initial program 17.1

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

   2. Applied add-sqr-sqrt_binary64 17.1

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

   3. Simplified 17.1

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

   4. Simplified 8.6

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

   5. Applied pow2_binary64 8.6

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

   6. Applied pow-prod-down_binary64 0.6

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

   7. Applied unpow2_binary64 0.6

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

   8. Applied associate-/r*_binary64 0.2

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

   if 1.2706830324107899e-254 < (/.f64 (cos.f64 (*.f64 2 x)) (*.f64 (pow.f64 c 2) (*.f64 (*.f64 x (pow.f64 s 2)) x)))

   1. Initial program 47.1

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

   2. Applied add-sqr-sqrt_binary64 47.2

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

   3. Simplified 47.1

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

   4. Simplified 39.8

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

   5. Applied pow2_binary64 39.8

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

   6. Applied pow-prod-down_binary64 7.0

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

   7. Applied fabs-mul_binary64 7.0

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

   8. Applied associate-*r*_binary64 3.2

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

   9. Applied unpow2_binary64 3.2

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

   10. Applied *-un-lft-identity_binary64 3.2

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

   11. Applied times-frac_binary64 3.1

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

4. Final simplification 1.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(x \cdot \left(x \cdot {s}^{2}\right)\right)} \leq 1.27068303241079 \cdot 10^{-254}:\\ \;\;\;\;\frac{\frac{\cos \left(2 \cdot x\right)}{c \cdot \left|x \cdot s\right|}}{c \cdot \left|x \cdot s\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(c \cdot \left|s\right|\right) \cdot \left|x\right|} \cdot \frac{\cos \left(2 \cdot x\right)}{\left(c \cdot \left|s\right|\right) \cdot \left|x\right|}\\ \end{array} \]

Reproduce

herbie shell --seed 2022129 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))