mixedcos

Average Error: 28.8 → 1.1

Time: 16.1s

Precision: binary64

Cost: 27460

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

Math

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

↓

\[\begin{array}{l} t_0 := c \cdot \left(x \cdot s\right)\\ t_1 := \cos \left(2 \cdot x\right)\\ t_2 := x \cdot \left(c \cdot s\right)\\ \mathbf{if}\;\frac{t_1}{{c}^{2} \cdot \left(x \cdot \left(x \cdot {s}^{2}\right)\right)} \leq \infty:\\ \;\;\;\;\frac{t_1}{t_0 \cdot t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(x + x\right)}{t_2}}{t_2}\\ \end{array} \]

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 (* x s))) (t_1 (cos (* 2.0 x))) (t_2 (* x (* c s))))
   (if (<= (/ t_1 (* (pow c 2.0) (* x (* x (pow s 2.0))))) INFINITY)
     (/ t_1 (* t_0 t_0))
     (/ (/ (cos (+ x x)) t_2) 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 * (x * s);
	double t_1 = cos((2.0 * x));
	double t_2 = x * (c * s);
	double tmp;
	if ((t_1 / (pow(c, 2.0) * (x * (x * pow(s, 2.0))))) <= ((double) INFINITY)) {
		tmp = t_1 / (t_0 * t_0);
	} else {
		tmp = (cos((x + x)) / t_2) / t_2;
	}
	return tmp;
}

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 = Math.cos((2.0 * x));
	double t_2 = x * (c * s);
	double tmp;
	if ((t_1 / (Math.pow(c, 2.0) * (x * (x * Math.pow(s, 2.0))))) <= Double.POSITIVE_INFINITY) {
		tmp = t_1 / (t_0 * t_0);
	} else {
		tmp = (Math.cos((x + x)) / t_2) / 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 * (x * s)
	t_1 = math.cos((2.0 * x))
	t_2 = x * (c * s)
	tmp = 0
	if (t_1 / (math.pow(c, 2.0) * (x * (x * math.pow(s, 2.0))))) <= math.inf:
		tmp = t_1 / (t_0 * t_0)
	else:
		tmp = (math.cos((x + x)) / t_2) / 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 * Float64(x * s))
	t_1 = cos(Float64(2.0 * x))
	t_2 = Float64(x * Float64(c * s))
	tmp = 0.0
	if (Float64(t_1 / Float64((c ^ 2.0) * Float64(x * Float64(x * (s ^ 2.0))))) <= Inf)
		tmp = Float64(t_1 / Float64(t_0 * t_0));
	else
		tmp = Float64(Float64(cos(Float64(x + x)) / t_2) / 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 * (x * s);
	t_1 = cos((2.0 * x));
	t_2 = x * (c * s);
	tmp = 0.0;
	if ((t_1 / ((c ^ 2.0) * (x * (x * (s ^ 2.0))))) <= Inf)
		tmp = t_1 / (t_0 * t_0);
	else
		tmp = (cos((x + x)) / t_2) / 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[(x * s), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(x * N[(c * s), $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], Infinity], N[(t$95$1 / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision] / t$95$2), $MachinePrecision] / t$95$2), $MachinePrecision]]]]]

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))) < +inf.0

   1. Initial program 18.9

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

   2. Taylor expanded in c around 0 23.9

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

   3. Simplified 0.6

      \[\leadsto \frac{\cos \left(2 \cdot x\right)}{\color{blue}{\left(c \cdot \left(s \cdot x\right)\right) \cdot \left(c \cdot \left(s \cdot x\right)\right)}} \]
      Proof
      (*.f64 (*.f64 c (*.f64 s x)) (*.f64 c (*.f64 s x))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= unswap-sqr_binary64 (*.f64 (*.f64 c c) (*.f64 (*.f64 s x) (*.f64 s x)))): 91 points increase in error, 18 points decrease in error
      (*.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 (pow.f64 c 2) (Rewrite<= unswap-sqr_binary64 (*.f64 (*.f64 s s) (*.f64 x x)))): 44 points increase in error, 10 points decrease in error
      (*.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 (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
      (Rewrite=> associate-*r*_binary64 (*.f64 (*.f64 (pow.f64 c 2) (pow.f64 s 2)) (pow.f64 x 2))): 6 points increase in error, 15 points decrease in error
      (*.f64 (Rewrite=> *-commutative_binary64 (*.f64 (pow.f64 s 2) (pow.f64 c 2))) (pow.f64 x 2)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*r*_binary64 (*.f64 (pow.f64 s 2) (*.f64 (pow.f64 c 2) (pow.f64 x 2)))): 9 points increase in error, 7 points decrease in error

   if +inf.0 < (/.f64 (cos.f64 (*.f64 2 x)) (*.f64 (pow.f64 c 2) (*.f64 (*.f64 x (pow.f64 s 2)) x)))

   1. Initial program 64.0

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

   2. Simplified 14.9

      \[\leadsto \color{blue}{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(x \cdot \left(\left(c \cdot s\right) \cdot \left(c \cdot s\right)\right)\right)}} \]
      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)))))): 75 points increase in error, 10 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))))): 13 points increase in error, 8 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)))): 18 points increase in error, 8 points decrease in error

   3. Applied egg-rr 29.5

      \[\leadsto \color{blue}{\sqrt[3]{\frac{{\cos \left(x + x\right)}^{3}}{{\left({\left(\left(x \cdot c\right) \cdot s\right)}^{2}\right)}^{3}}}} \]

   4. Applied egg-rr 2.6

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

4. Final simplification 1.1

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

Alternatives

Alternative 1
Error	3.7
Cost	7624

\[\begin{array}{l} t_0 := \frac{\frac{\cos \left(x + x\right)}{s \cdot \left(x \cdot c\right)}}{x \cdot \left(c \cdot s\right)}\\ \mathbf{if}\;x \leq -1 \cdot 10^{-23}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 5 \cdot 10^{-182}:\\ \;\;\;\;{\left(c \cdot \left(x \cdot s\right)\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	2.6
Cost	7360

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

Alternative 3
Error	16.2
Cost	6916

\[\begin{array}{l} \mathbf{if}\;s \leq 6.292089059420481 \cdot 10^{+139}:\\ \;\;\;\;\frac{\frac{\frac{1}{s}}{x \cdot c}}{s \cdot \left(x \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(c \cdot \left(x \cdot s\right)\right)}^{-2}\\ \end{array} \]

Alternative 4
Error	20.4
Cost	1228

\[\begin{array}{l} t_0 := \frac{1}{c \cdot \left(c \cdot \left(\left(x \cdot s\right) \cdot \left(x \cdot s\right)\right)\right)}\\ t_1 := \frac{\frac{1}{\left(x \cdot c\right) \cdot \left(x \cdot c\right)}}{s \cdot s}\\ \mathbf{if}\;s \leq -1 \cdot 10^{-130}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;s \leq 10^{-115}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;s \leq 10^{+15}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	28.3
Cost	1096

\[\begin{array}{l} t_0 := \frac{1}{s \cdot \left(s \cdot \left(c \cdot \left(c \cdot \left(x \cdot x\right)\right)\right)\right)}\\ \mathbf{if}\;x \leq -8 \cdot 10^{-145}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 2.9 \cdot 10^{-132}:\\ \;\;\;\;\frac{-2}{\left(s \cdot s\right) \cdot \left(c \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	26.9
Cost	1096

\[\begin{array}{l} t_0 := \frac{1}{s \cdot \left(s \cdot \left(c \cdot \left(c \cdot \left(x \cdot x\right)\right)\right)\right)}\\ \mathbf{if}\;x \leq -1 \cdot 10^{-145}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 10^{-165}:\\ \;\;\;\;\frac{\frac{1}{x \cdot \left(x \cdot \left(s \cdot s\right)\right)}}{c \cdot c}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	26.2
Cost	1096

\[\begin{array}{l} t_0 := \frac{1}{s \cdot \left(s \cdot \left(c \cdot \left(c \cdot \left(x \cdot x\right)\right)\right)\right)}\\ \mathbf{if}\;x \leq -1 \cdot 10^{-145}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 10^{-150}:\\ \;\;\;\;\frac{\frac{1}{s}}{\left(c \cdot c\right) \cdot \left(x \cdot \left(x \cdot s\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	21.8
Cost	1096

\[\begin{array}{l} t_0 := \frac{1}{c \cdot \left(c \cdot \left(\left(x \cdot s\right) \cdot \left(x \cdot s\right)\right)\right)}\\ \mathbf{if}\;s \leq -1 \cdot 10^{-28}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;s \leq 10^{+15}:\\ \;\;\;\;\frac{1}{s \cdot \left(s \cdot \left(c \cdot \left(c \cdot \left(x \cdot x\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 9
Error	17.3
Cost	1096

\[\begin{array}{l} t_0 := s \cdot \left(x \cdot c\right)\\ t_1 := \frac{\frac{1}{t_0}}{t_0}\\ \mathbf{if}\;s \leq 5.804671208592295 \cdot 10^{+217}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;s \leq 4.553357778883606 \cdot 10^{+294}:\\ \;\;\;\;\frac{1}{c \cdot \left(c \cdot \left(\left(x \cdot s\right) \cdot \left(x \cdot s\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Error	17.2
Cost	1096

\[\begin{array}{l} t_0 := s \cdot \left(x \cdot c\right)\\ \mathbf{if}\;s \leq 5.804671208592295 \cdot 10^{+217}:\\ \;\;\;\;\frac{\frac{1}{t_0}}{t_0}\\ \mathbf{elif}\;s \leq 4.553357778883606 \cdot 10^{+294}:\\ \;\;\;\;\frac{1}{c \cdot \left(c \cdot \left(\left(x \cdot s\right) \cdot \left(x \cdot s\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{1}{c}}{s}}{x}}{x \cdot \left(c \cdot s\right)}\\ \end{array} \]

Alternative 11
Error	17.2
Cost	1096

\[\begin{array}{l} \mathbf{if}\;s \leq 5.804671208592295 \cdot 10^{+217}:\\ \;\;\;\;\frac{\frac{\frac{1}{s}}{x \cdot c}}{s \cdot \left(x \cdot c\right)}\\ \mathbf{elif}\;s \leq 4.553357778883606 \cdot 10^{+294}:\\ \;\;\;\;\frac{1}{c \cdot \left(c \cdot \left(\left(x \cdot s\right) \cdot \left(x \cdot s\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{1}{c}}{s}}{x}}{x \cdot \left(c \cdot s\right)}\\ \end{array} \]

Alternative 12
Error	17.2
Cost	1096

\[\begin{array}{l} \mathbf{if}\;s \leq 5.804671208592295 \cdot 10^{+217}:\\ \;\;\;\;\frac{\frac{\frac{1}{s}}{x \cdot c}}{s \cdot \left(x \cdot c\right)}\\ \mathbf{elif}\;s \leq 4.553357778883606 \cdot 10^{+294}:\\ \;\;\;\;\frac{1}{c \cdot \left(c \cdot \left(\left(x \cdot s\right) \cdot \left(x \cdot s\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{c \cdot \left(\left(x \cdot s\right) \cdot \left(x \cdot \left(c \cdot s\right)\right)\right)}\\ \end{array} \]

Alternative 13
Error	16.2
Cost	1092

\[\begin{array}{l} t_0 := \frac{1}{c \cdot \left(x \cdot s\right)}\\ \mathbf{if}\;s \leq 6.292089059420481 \cdot 10^{+139}:\\ \;\;\;\;\frac{\frac{\frac{1}{s}}{x \cdot c}}{s \cdot \left(x \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot t_0\\ \end{array} \]

Alternative 14
Error	16.2
Cost	964

\[\begin{array}{l} t_0 := c \cdot \left(x \cdot s\right)\\ \mathbf{if}\;s \leq 2.551422346977415 \cdot 10^{+190}:\\ \;\;\;\;\frac{\frac{\frac{1}{s}}{x \cdot c}}{s \cdot \left(x \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t_0 \cdot t_0}\\ \end{array} \]

Alternative 15
Error	41.6
Cost	576

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

Alternative 16
Error	41.4
Cost	576

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

Reproduce

herbie shell --seed 2022295 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))