mixedcos

Average Error: 27.9 → 2.4

Time: 15.2s

Precision: binary64

Cost: 7752

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

Math

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

↓

\[\begin{array}{l} t_0 := x \cdot \left(s \cdot c\right)\\ t_1 := \cos \left(x + x\right)\\ t_2 := s \cdot \left(x \cdot c\right)\\ t_3 := \frac{\frac{t_1}{t_2}}{t_2}\\ \mathbf{if}\;x \leq 10^{-258}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x \leq 7.927357837510631 \cdot 10^{+224}:\\ \;\;\;\;\frac{t_1}{t_0} \cdot \frac{1}{t_0}\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

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 (* x (* s c)))
        (t_1 (cos (+ x x)))
        (t_2 (* s (* x c)))
        (t_3 (/ (/ t_1 t_2) t_2)))
   (if (<= x 1e-258)
     t_3
     (if (<= x 7.927357837510631e+224) (* (/ t_1 t_0) (/ 1.0 t_0)) t_3))))

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 = x * (s * c);
	double t_1 = cos((x + x));
	double t_2 = s * (x * c);
	double t_3 = (t_1 / t_2) / t_2;
	double tmp;
	if (x <= 1e-258) {
		tmp = t_3;
	} else if (x <= 7.927357837510631e+224) {
		tmp = (t_1 / t_0) * (1.0 / t_0);
	} else {
		tmp = t_3;
	}
	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) :: t_3
    real(8) :: tmp
    t_0 = x * (s * c)
    t_1 = cos((x + x))
    t_2 = s * (x * c)
    t_3 = (t_1 / t_2) / t_2
    if (x <= 1d-258) then
        tmp = t_3
    else if (x <= 7.927357837510631d+224) then
        tmp = (t_1 / t_0) * (1.0d0 / t_0)
    else
        tmp = t_3
    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 = x * (s * c);
	double t_1 = Math.cos((x + x));
	double t_2 = s * (x * c);
	double t_3 = (t_1 / t_2) / t_2;
	double tmp;
	if (x <= 1e-258) {
		tmp = t_3;
	} else if (x <= 7.927357837510631e+224) {
		tmp = (t_1 / t_0) * (1.0 / t_0);
	} else {
		tmp = t_3;
	}
	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 = x * (s * c)
	t_1 = math.cos((x + x))
	t_2 = s * (x * c)
	t_3 = (t_1 / t_2) / t_2
	tmp = 0
	if x <= 1e-258:
		tmp = t_3
	elif x <= 7.927357837510631e+224:
		tmp = (t_1 / t_0) * (1.0 / t_0)
	else:
		tmp = t_3
	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(x * Float64(s * c))
	t_1 = cos(Float64(x + x))
	t_2 = Float64(s * Float64(x * c))
	t_3 = Float64(Float64(t_1 / t_2) / t_2)
	tmp = 0.0
	if (x <= 1e-258)
		tmp = t_3;
	elseif (x <= 7.927357837510631e+224)
		tmp = Float64(Float64(t_1 / t_0) * Float64(1.0 / t_0));
	else
		tmp = t_3;
	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 = x * (s * c);
	t_1 = cos((x + x));
	t_2 = s * (x * c);
	t_3 = (t_1 / t_2) / t_2;
	tmp = 0.0;
	if (x <= 1e-258)
		tmp = t_3;
	elseif (x <= 7.927357837510631e+224)
		tmp = (t_1 / t_0) * (1.0 / t_0);
	else
		tmp = t_3;
	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[(x * N[(s * c), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(s * N[(x * c), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(t$95$1 / t$95$2), $MachinePrecision] / t$95$2), $MachinePrecision]}, If[LessEqual[x, 1e-258], t$95$3, If[LessEqual[x, 7.927357837510631e+224], N[(N[(t$95$1 / t$95$0), $MachinePrecision] * N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]

Derivation

1. Split input into 2 regimes

2. if x < 9.99999999999999954e-259 or 7.92735783751063116e224 < x

   1. Initial program 28.6

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

   2. Taylor expanded in x around inf 31.8

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

   3. Simplified 3.0

      \[\leadsto \color{blue}{\frac{\cos \left(2 \cdot x\right)}{\left(c \cdot \left(s \cdot x\right)\right) \cdot \left(c \cdot \left(s \cdot x\right)\right)}} \]
      Proof
      (/.f64 (cos.f64 (*.f64 2 x)) (*.f64 (*.f64 c (*.f64 s x)) (*.f64 c (*.f64 s x)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (cos.f64 (*.f64 2 x)) (Rewrite<= unswap-sqr_binary64 (*.f64 (*.f64 c c) (*.f64 (*.f64 s x) (*.f64 s x))))): 81 points increase in error, 16 points decrease in error
      (/.f64 (cos.f64 (*.f64 2 x)) (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 c 2)) (*.f64 (*.f64 s x) (*.f64 s x)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (cos.f64 (*.f64 2 x)) (*.f64 (pow.f64 c 2) (Rewrite<= unswap-sqr_binary64 (*.f64 (*.f64 s s) (*.f64 x x))))): 67 points increase in error, 5 points decrease in error
      (/.f64 (cos.f64 (*.f64 2 x)) (*.f64 (pow.f64 c 2) (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 s 2)) (*.f64 x x)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (cos.f64 (*.f64 2 x)) (*.f64 (pow.f64 c 2) (*.f64 (pow.f64 s 2) (Rewrite<= unpow2_binary64 (pow.f64 x 2))))): 0 points increase in error, 0 points decrease in error

   4. Applied egg-rr 3.7

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

   5. Applied egg-rr 2.9

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

   if 9.99999999999999954e-259 < x < 7.92735783751063116e224

   1. Initial program 27.0

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

   2. Taylor expanded in x around inf 30.7

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

   3. Simplified 2.5

      \[\leadsto \color{blue}{\frac{\cos \left(2 \cdot x\right)}{\left(c \cdot \left(s \cdot x\right)\right) \cdot \left(c \cdot \left(s \cdot x\right)\right)}} \]
      Proof
      (/.f64 (cos.f64 (*.f64 2 x)) (*.f64 (*.f64 c (*.f64 s x)) (*.f64 c (*.f64 s x)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (cos.f64 (*.f64 2 x)) (Rewrite<= unswap-sqr_binary64 (*.f64 (*.f64 c c) (*.f64 (*.f64 s x) (*.f64 s x))))): 81 points increase in error, 16 points decrease in error
      (/.f64 (cos.f64 (*.f64 2 x)) (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 c 2)) (*.f64 (*.f64 s x) (*.f64 s x)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (cos.f64 (*.f64 2 x)) (*.f64 (pow.f64 c 2) (Rewrite<= unswap-sqr_binary64 (*.f64 (*.f64 s s) (*.f64 x x))))): 67 points increase in error, 5 points decrease in error
      (/.f64 (cos.f64 (*.f64 2 x)) (*.f64 (pow.f64 c 2) (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 s 2)) (*.f64 x x)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (cos.f64 (*.f64 2 x)) (*.f64 (pow.f64 c 2) (*.f64 (pow.f64 s 2) (Rewrite<= unpow2_binary64 (pow.f64 x 2))))): 0 points increase in error, 0 points decrease in error

   4. Applied egg-rr 1.5

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

4. Final simplification 2.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 10^{-258}:\\ \;\;\;\;\frac{\frac{\cos \left(x + x\right)}{s \cdot \left(x \cdot c\right)}}{s \cdot \left(x \cdot c\right)}\\ \mathbf{elif}\;x \leq 7.927357837510631 \cdot 10^{+224}:\\ \;\;\;\;\frac{\cos \left(x + x\right)}{x \cdot \left(s \cdot c\right)} \cdot \frac{1}{x \cdot \left(s \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(x + x\right)}{s \cdot \left(x \cdot c\right)}}{s \cdot \left(x \cdot c\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	4.0
Cost	7624

\[\begin{array}{l} t_0 := \frac{\frac{\cos \left(x + x\right)}{x \cdot \left(s \cdot c\right)}}{s \cdot \left(x \cdot c\right)}\\ t_1 := c \cdot \left(x \cdot s\right)\\ \mathbf{if}\;x \leq -1 \cdot 10^{-114}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 10^{-105}:\\ \;\;\;\;\frac{1}{t_1 \cdot t_1} + \frac{\frac{\frac{-2}{c}}{s \cdot c}}{s}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	4.6
Cost	7624

\[\begin{array}{l} t_0 := \frac{\frac{\frac{\cos \left(x + x\right)}{s \cdot \left(x \cdot c\right)}}{x \cdot c}}{s}\\ \mathbf{if}\;x \leq -10000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 10^{-45}:\\ \;\;\;\;{\left(c \cdot \left(x \cdot s\right)\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	2.7
Cost	7360

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

Alternative 4
Error	2.8
Cost	7360

\[\begin{array}{l} t_0 := c \cdot \left(x \cdot s\right)\\ \frac{\cos \left(x \cdot 2\right)}{t_0 \cdot t_0} \end{array} \]

Alternative 5
Error	16.8
Cost	6784

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

Alternative 6
Error	19.1
Cost	1228

\[\begin{array}{l} t_0 := \frac{\frac{1}{x \cdot s}}{s \cdot \left(c \cdot \left(x \cdot c\right)\right)}\\ \mathbf{if}\;c \leq -4.0385053962015554 \cdot 10^{+214}:\\ \;\;\;\;\frac{1}{c \cdot \left(x \cdot \left(s \cdot \left(s \cdot \left(x \cdot c\right)\right)\right)\right)}\\ \mathbf{elif}\;c \leq -1 \cdot 10^{-75}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 10^{-75}:\\ \;\;\;\;\frac{1}{x \cdot \left(s \cdot \left(c \cdot \left(x \cdot \left(s \cdot c\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	20.0
Cost	1096

\[\begin{array}{l} t_0 := \frac{1}{c \cdot \left(x \cdot \left(s \cdot \left(s \cdot \left(x \cdot c\right)\right)\right)\right)}\\ \mathbf{if}\;c \leq -4.0385053962015554 \cdot 10^{+214}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq -1 \cdot 10^{-80}:\\ \;\;\;\;\frac{\frac{1}{x \cdot s}}{s \cdot \left(c \cdot \left(x \cdot c\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	16.8
Cost	960

\[\begin{array}{l} t_0 := \frac{1}{c \cdot \left(x \cdot s\right)}\\ t_0 \cdot t_0 \end{array} \]

Alternative 9
Error	22.8
Cost	832

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

Alternative 10
Error	17.9
Cost	832

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

Alternative 11
Error	16.8
Cost	832

\[\begin{array}{l} t_0 := s \cdot \left(x \cdot c\right)\\ \frac{\frac{1}{t_0}}{t_0} \end{array} \]

Alternative 12
Error	16.9
Cost	832

\[\begin{array}{l} t_0 := c \cdot \left(x \cdot s\right)\\ \frac{1}{t_0 \cdot t_0} \end{array} \]

Reproduce

herbie shell --seed 2022297 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))