mixedcos

Average Error: 28.4 → 2.5

Time: 15.3s

Precision: binary64

Cost: 7748

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

Math

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

↓

\[\begin{array}{l} t_0 := \cos \left(x + x\right)\\ t_1 := x \cdot \left(s \cdot c\right)\\ \mathbf{if}\;s \leq 10^{-20}:\\ \;\;\;\;\frac{1}{\frac{x \cdot c}{\frac{1}{s}}} \cdot \frac{t_0}{s \cdot \left(x \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t_0}{t_1}}{t_1}\\ \end{array} \]

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))) (t_1 (* x (* s c))))
   (if (<= s 1e-20)
     (* (/ 1.0 (/ (* x c) (/ 1.0 s))) (/ t_0 (* s (* x c))))
     (/ (/ t_0 t_1) t_1))))

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 t_1 = x * (s * c);
	double tmp;
	if (s <= 1e-20) {
		tmp = (1.0 / ((x * c) / (1.0 / s))) * (t_0 / (s * (x * c)));
	} else {
		tmp = (t_0 / t_1) / t_1;
	}
	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) :: tmp
    t_0 = cos((x + x))
    t_1 = x * (s * c)
    if (s <= 1d-20) then
        tmp = (1.0d0 / ((x * c) / (1.0d0 / s))) * (t_0 / (s * (x * c)))
    else
        tmp = (t_0 / t_1) / t_1
    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 t_1 = x * (s * c);
	double tmp;
	if (s <= 1e-20) {
		tmp = (1.0 / ((x * c) / (1.0 / s))) * (t_0 / (s * (x * c)));
	} else {
		tmp = (t_0 / t_1) / t_1;
	}
	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))
	t_1 = x * (s * c)
	tmp = 0
	if s <= 1e-20:
		tmp = (1.0 / ((x * c) / (1.0 / s))) * (t_0 / (s * (x * c)))
	else:
		tmp = (t_0 / t_1) / t_1
	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))
	t_1 = Float64(x * Float64(s * c))
	tmp = 0.0
	if (s <= 1e-20)
		tmp = Float64(Float64(1.0 / Float64(Float64(x * c) / Float64(1.0 / s))) * Float64(t_0 / Float64(s * Float64(x * c))));
	else
		tmp = Float64(Float64(t_0 / t_1) / t_1);
	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));
	t_1 = x * (s * c);
	tmp = 0.0;
	if (s <= 1e-20)
		tmp = (1.0 / ((x * c) / (1.0 / s))) * (t_0 / (s * (x * c)));
	else
		tmp = (t_0 / t_1) / t_1;
	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]}, Block[{t$95$1 = N[(x * N[(s * c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[s, 1e-20], N[(N[(1.0 / N[(N[(x * c), $MachinePrecision] / N[(1.0 / s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 / N[(s * N[(x * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 / t$95$1), $MachinePrecision] / t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if s < 9.99999999999999945e-21

   1. Initial program 33.7

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

   2. Simplified 17.6

      \[\leadsto \color{blue}{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(x \cdot \left(\left(c \cdot s\right) \cdot \left(c \cdot s\right)\right)\right)}} \]
      Proof
      (/.f64 (cos.f64 (*.f64 2 x)) (*.f64 x (*.f64 x (*.f64 (*.f64 c s) (*.f64 c s))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (cos.f64 (*.f64 2 x)) (*.f64 x (*.f64 x (Rewrite<= unswap-sqr_binary64 (*.f64 (*.f64 c c) (*.f64 s s)))))): 70 points increase in error, 7 points decrease in error
      (/.f64 (cos.f64 (*.f64 2 x)) (*.f64 x (*.f64 x (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 c 2)) (*.f64 s s))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (cos.f64 (*.f64 2 x)) (*.f64 x (*.f64 x (*.f64 (pow.f64 c 2) (Rewrite<= unpow2_binary64 (pow.f64 s 2)))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (cos.f64 (*.f64 2 x)) (*.f64 x (*.f64 x (Rewrite=> *-commutative_binary64 (*.f64 (pow.f64 s 2) (pow.f64 c 2)))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (cos.f64 (*.f64 2 x)) (*.f64 x (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 x (pow.f64 s 2)) (pow.f64 c 2))))): 7 points increase in error, 14 points decrease in error
      (/.f64 (cos.f64 (*.f64 2 x)) (*.f64 x (Rewrite<= *-commutative_binary64 (*.f64 (pow.f64 c 2) (*.f64 x (pow.f64 s 2)))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (cos.f64 (*.f64 2 x)) (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 (pow.f64 c 2) (*.f64 x (pow.f64 s 2))) x))): 0 points increase in error, 0 points decrease in error
      (/.f64 (cos.f64 (*.f64 2 x)) (Rewrite<= associate-*r*_binary64 (*.f64 (pow.f64 c 2) (*.f64 (*.f64 x (pow.f64 s 2)) x)))): 22 points increase in error, 8 points decrease in error

   3. Applied egg-rr 3.1

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

   4. Applied egg-rr 3.0

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

   if 9.99999999999999945e-21 < s

   1. Initial program 24.4

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

   2. Taylor expanded in x around inf 27.9

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

   3. Simplified 2.8

      \[\leadsto \color{blue}{\frac{\cos \left(2 \cdot x\right)}{\left(c \cdot \left(s \cdot x\right)\right) \cdot \left(c \cdot \left(s \cdot x\right)\right)}} \]
      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))))): 94 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))))): 52 points increase in error, 12 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 2.1

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

   5. Applied egg-rr 2.0

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

4. Final simplification 2.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;s \leq 10^{-20}:\\ \;\;\;\;\frac{1}{\frac{x \cdot c}{\frac{1}{s}}} \cdot \frac{\cos \left(x + x\right)}{s \cdot \left(x \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(x + x\right)}{x \cdot \left(s \cdot c\right)}}{x \cdot \left(s \cdot c\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	2.5
Cost	7620

\[\begin{array}{l} t_0 := x \cdot \left(s \cdot c\right)\\ t_1 := \cos \left(x + x\right)\\ t_2 := s \cdot \left(x \cdot c\right)\\ \mathbf{if}\;s \leq 10^{-25}:\\ \;\;\;\;\frac{t_1}{t_2} \cdot \frac{1}{t_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t_1}{t_0}}{t_0}\\ \end{array} \]

Alternative 2
Error	3.0
Cost	7360

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

Alternative 3
Error	16.9
Cost	6912

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

Alternative 4
Error	21.1
Cost	1228

\[\begin{array}{l} t_0 := \frac{\frac{\frac{1}{s}}{x}}{s \cdot \left(c \cdot \left(x \cdot c\right)\right)}\\ \mathbf{if}\;s \leq -1 \cdot 10^{-250}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;s \leq 2 \cdot 10^{+18}:\\ \;\;\;\;\frac{1}{s \cdot \left(c \cdot \left(\left(s \cdot c\right) \cdot \left(x \cdot x\right)\right)\right)}\\ \mathbf{elif}\;s \leq 7.442514089317075 \cdot 10^{+92}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{c \cdot \left(\left(s \cdot c\right) \cdot \left(x \cdot \left(s \cdot x\right)\right)\right)}\\ \end{array} \]

Alternative 5
Error	21.1
Cost	1096

\[\begin{array}{l} t_0 := \frac{1}{s \cdot \left(c \cdot \left(\left(s \cdot c\right) \cdot \left(x \cdot x\right)\right)\right)}\\ \mathbf{if}\;x \leq -1 \cdot 10^{-115}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 10^{-100}:\\ \;\;\;\;\frac{\frac{\frac{1}{s}}{x}}{s \cdot \left(c \cdot \left(x \cdot c\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	21.9
Cost	1096

\[\begin{array}{l} t_0 := \frac{\frac{\frac{1}{s}}{x}}{s \cdot \left(c \cdot \left(x \cdot c\right)\right)}\\ \mathbf{if}\;c \leq -5 \cdot 10^{-148}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 1.4 \cdot 10^{-132}:\\ \;\;\;\;\frac{1}{c \cdot \left(c \cdot \left(s \cdot \left(s \cdot \left(x \cdot x\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	21.3
Cost	1096

\[\begin{array}{l} \mathbf{if}\;c \leq -5 \cdot 10^{-148}:\\ \;\;\;\;\frac{\frac{\frac{1}{s}}{x}}{s \cdot \left(c \cdot \left(x \cdot c\right)\right)}\\ \mathbf{elif}\;c \leq 1.1 \cdot 10^{-160}:\\ \;\;\;\;\frac{1}{c \cdot \left(c \cdot \left(s \cdot \left(s \cdot \left(x \cdot x\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s \cdot x}}{\left(s \cdot x\right) \cdot \left(c \cdot c\right)}\\ \end{array} \]

Alternative 8
Error	18.8
Cost	964

\[\begin{array}{l} \mathbf{if}\;s \leq 5.470111598275961 \cdot 10^{+127}:\\ \;\;\;\;\frac{1}{s \cdot \left(\left(x \cdot \left(s \cdot c\right)\right) \cdot \left(x \cdot c\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{c \cdot \left(\left(s \cdot c\right) \cdot \left(x \cdot \left(s \cdot x\right)\right)\right)}\\ \end{array} \]

Alternative 9
Error	16.7
Cost	960

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

Alternative 10
Error	22.8
Cost	832

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

Alternative 11
Error	18.1
Cost	832

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

Alternative 12
Error	16.8
Cost	832

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

Alternative 13
Error	41.1
Cost	576

\[\frac{\frac{\frac{-2}{c \cdot c}}{s}}{s} \]

Reproduce

herbie shell --seed 2022298 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))