mixedcos

Average Error: 28.0 → 2.1

Time: 17.0s

Precision: binary64

Cost: 7620

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

Math

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

↓

\[\begin{array}{l} t_0 := c \cdot \left(x \cdot s\right)\\ t_1 := \left(c \cdot x\right) \cdot s\\ t_2 := \cos \left(x + x\right)\\ \mathbf{if}\;c \leq -7 \cdot 10^{+208}:\\ \;\;\;\;\frac{1}{t_1} \cdot \frac{t_2}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t_2}{t_0}}{t_0}\\ \end{array} \]

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 (* x s))) (t_1 (* (* c x) s)) (t_2 (cos (+ x x))))
   (if (<= c -7e+208) (* (/ 1.0 t_1) (/ t_2 t_1)) (/ (/ t_2 t_0) t_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 = c * (x * s);
	double t_1 = (c * x) * s;
	double t_2 = cos((x + x));
	double tmp;
	if (c <= -7e+208) {
		tmp = (1.0 / t_1) * (t_2 / t_1);
	} else {
		tmp = (t_2 / t_0) / t_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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = c * (x * s)
    t_1 = (c * x) * s
    t_2 = cos((x + x))
    if (c <= (-7d+208)) then
        tmp = (1.0d0 / t_1) * (t_2 / t_1)
    else
        tmp = (t_2 / t_0) / t_0
    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 * (x * s);
	double t_1 = (c * x) * s;
	double t_2 = Math.cos((x + x));
	double tmp;
	if (c <= -7e+208) {
		tmp = (1.0 / t_1) * (t_2 / t_1);
	} else {
		tmp = (t_2 / t_0) / t_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 = c * (x * s)
	t_1 = (c * x) * s
	t_2 = math.cos((x + x))
	tmp = 0
	if c <= -7e+208:
		tmp = (1.0 / t_1) * (t_2 / t_1)
	else:
		tmp = (t_2 / t_0) / t_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 = Float64(c * Float64(x * s))
	t_1 = Float64(Float64(c * x) * s)
	t_2 = cos(Float64(x + x))
	tmp = 0.0
	if (c <= -7e+208)
		tmp = Float64(Float64(1.0 / t_1) * Float64(t_2 / t_1));
	else
		tmp = Float64(Float64(t_2 / t_0) / t_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 = c * (x * s);
	t_1 = (c * x) * s;
	t_2 = cos((x + x));
	tmp = 0.0;
	if (c <= -7e+208)
		tmp = (1.0 / t_1) * (t_2 / t_1);
	else
		tmp = (t_2 / t_0) / t_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[(c * N[(x * s), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(c * x), $MachinePrecision] * s), $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[c, -7e+208], N[(N[(1.0 / t$95$1), $MachinePrecision] * N[(t$95$2 / t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$2 / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if c < -7.00000000000000033e208

   1. Initial program 24.2

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

   2. Simplified 8.3

      \[\leadsto \color{blue}{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(x \cdot \left(\left(c \cdot s\right) \cdot \left(c \cdot s\right)\right)\right)}} \]
      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)))))): 65 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, 11 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)))): 20 points increase in error, 7 points decrease in error

   3. Applied egg-rr 2.4

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

   if -7.00000000000000033e208 < c

   1. Initial program 29.1

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

   2. Simplified 27.2

      \[\leadsto \color{blue}{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(\left(c \cdot c\right) \cdot \left(x \cdot \left(s \cdot s\right)\right)\right)}} \]
      Proof
      (/.f64 (cos.f64 (*.f64 2 x)) (*.f64 x (*.f64 (*.f64 c c) (*.f64 x (*.f64 s s))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (cos.f64 (*.f64 2 x)) (*.f64 x (*.f64 (*.f64 c c) (*.f64 x (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 (Rewrite<= unpow2_binary64 (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)))): 20 points increase in error, 7 points decrease in error

   3. Applied egg-rr 32.5

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

   4. Simplified 2.0

      \[\leadsto \color{blue}{\frac{\frac{\cos \left(x + x\right)}{c \cdot \left(s \cdot x\right)}}{c \cdot \left(s \cdot x\right)}} \]
      Proof
      (/.f64 (/.f64 (cos.f64 (+.f64 x x)) (*.f64 c (*.f64 s x))) (*.f64 c (*.f64 s x))): 0 points increase in error, 0 points decrease in error
      (/.f64 (/.f64 (cos.f64 (+.f64 x x)) (*.f64 c (*.f64 s (Rewrite<= rem-square-sqrt_binary64 (*.f64 (sqrt.f64 x) (sqrt.f64 x)))))) (*.f64 c (*.f64 s x))): 131 points increase in error, 9 points decrease in error
      (/.f64 (/.f64 (cos.f64 (+.f64 x x)) (*.f64 c (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 s (sqrt.f64 x)) (sqrt.f64 x))))) (*.f64 c (*.f64 s x))): 7 points increase in error, 9 points decrease in error
      (/.f64 (/.f64 (cos.f64 (+.f64 x x)) (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 c (*.f64 s (sqrt.f64 x))) (sqrt.f64 x)))) (*.f64 c (*.f64 s x))): 4 points increase in error, 13 points decrease in error
      (/.f64 (Rewrite<= *-lft-identity_binary64 (*.f64 1 (/.f64 (cos.f64 (+.f64 x x)) (*.f64 (*.f64 c (*.f64 s (sqrt.f64 x))) (sqrt.f64 x))))) (*.f64 c (*.f64 s x))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 1 (/.f64 (cos.f64 (+.f64 x x)) (*.f64 (*.f64 c (*.f64 s (sqrt.f64 x))) (sqrt.f64 x)))) (*.f64 c (*.f64 s (Rewrite<= rem-square-sqrt_binary64 (*.f64 (sqrt.f64 x) (sqrt.f64 x)))))): 24 points increase in error, 1 points decrease in error
      (/.f64 (*.f64 1 (/.f64 (cos.f64 (+.f64 x x)) (*.f64 (*.f64 c (*.f64 s (sqrt.f64 x))) (sqrt.f64 x)))) (*.f64 c (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 s (sqrt.f64 x)) (sqrt.f64 x))))): 10 points increase in error, 8 points decrease in error
      (/.f64 (*.f64 1 (/.f64 (cos.f64 (+.f64 x x)) (*.f64 (*.f64 c (*.f64 s (sqrt.f64 x))) (sqrt.f64 x)))) (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 c (*.f64 s (sqrt.f64 x))) (sqrt.f64 x)))): 9 points increase in error, 10 points decrease in error
      (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 1 (*.f64 (*.f64 c (*.f64 s (sqrt.f64 x))) (sqrt.f64 x))) (/.f64 (cos.f64 (+.f64 x x)) (*.f64 (*.f64 c (*.f64 s (sqrt.f64 x))) (sqrt.f64 x))))): 10 points increase in error, 5 points decrease in error
3. Recombined 2 regimes into one program.

4. Final simplification 2.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -7 \cdot 10^{+208}:\\ \;\;\;\;\frac{1}{\left(c \cdot x\right) \cdot s} \cdot \frac{\cos \left(x + x\right)}{\left(c \cdot x\right) \cdot s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(x + x\right)}{c \cdot \left(x \cdot s\right)}}{c \cdot \left(x \cdot s\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	4.9
Cost	7624

\[\begin{array}{l} t_0 := \frac{1}{x \cdot \left(c \cdot s\right)}\\ t_1 := \frac{\cos \left(x \cdot 2\right)}{s \cdot \left(\left(c \cdot x\right) \cdot \left(\left(c \cdot x\right) \cdot s\right)\right)}\\ \mathbf{if}\;x \leq -2.2 \cdot 10^{-72}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{-30}:\\ \;\;\;\;t_0 \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	4.0
Cost	7624

\[\begin{array}{l} t_0 := \frac{1}{x \cdot \left(c \cdot s\right)}\\ t_1 := \frac{\frac{\cos \left(x + x\right)}{c \cdot \left(x \cdot s\right)}}{\left(c \cdot x\right) \cdot s}\\ \mathbf{if}\;x \leq -1.65 \cdot 10^{-72}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-30}:\\ \;\;\;\;t_0 \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	4.0
Cost	7624

\[\begin{array}{l} t_0 := \frac{1}{x \cdot \left(c \cdot s\right)}\\ t_1 := c \cdot \left(x \cdot s\right)\\ t_2 := \left(c \cdot x\right) \cdot s\\ \mathbf{if}\;x \leq -7.2 \cdot 10^{-72}:\\ \;\;\;\;\frac{\frac{\cos \left(x + x\right)}{t_1}}{t_2}\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{-30}:\\ \;\;\;\;t_0 \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(x \cdot -2\right)}{t_2}}{t_1}\\ \end{array} \]

Alternative 4
Error	2.4
Cost	7360

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

Alternative 5
Error	16.1
Cost	1604

\[\begin{array}{l} t_0 := \frac{\frac{1}{x \cdot s}}{c}\\ t_1 := \frac{1}{\left(c \cdot x\right) \cdot s}\\ \mathbf{if}\;c \leq -6.2 \cdot 10^{+208}:\\ \;\;\;\;t_1 \cdot \left(t_1 + -2 \cdot \frac{x}{c \cdot s}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot t_0\\ \end{array} \]

Alternative 6
Error	16.2
Cost	964

\[\begin{array}{l} t_0 := \left(c \cdot x\right) \cdot s\\ \mathbf{if}\;s \leq 5 \cdot 10^{+32}:\\ \;\;\;\;\frac{1}{t_0 \cdot t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{c \cdot \left(x \cdot s\right)}}{x \cdot s}}{c}\\ \end{array} \]

Alternative 7
Error	15.9
Cost	964

\[\begin{array}{l} t_0 := c \cdot \left(x \cdot s\right)\\ t_1 := \left(c \cdot x\right) \cdot s\\ \mathbf{if}\;c \leq -1 \cdot 10^{+209}:\\ \;\;\;\;\frac{1}{t_1 \cdot t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{t_0}}{t_0}\\ \end{array} \]

Alternative 8
Error	16.9
Cost	960

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

Alternative 9
Error	19.5
Cost	832

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

Alternative 10
Error	16.8
Cost	832

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

Reproduce

herbie shell --seed 2022318 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))