mixedcos

Average Error: 28.3 → 0.9

Time: 16.2s

Precision: binary64

Cost: 33540

Math

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

↓

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

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 (cos (+ x x))) (t_1 (* c (* x s))))
   (if (<=
        (/ (cos (* 2.0 x)) (* (pow c 2.0) (* x (* x (pow s 2.0)))))
        INFINITY)
     (/ (/ t_0 t_1) t_1)
     (* t_0 (pow (* x (* c s)) -2.0)))))

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 = cos((x + x));
	double t_1 = c * (x * s);
	double tmp;
	if ((cos((2.0 * x)) / (pow(c, 2.0) * (x * (x * pow(s, 2.0))))) <= ((double) INFINITY)) {
		tmp = (t_0 / t_1) / t_1;
	} else {
		tmp = t_0 * pow((x * (c * s)), -2.0);
	}
	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 = Math.cos((x + x));
	double t_1 = c * (x * s);
	double tmp;
	if ((Math.cos((2.0 * x)) / (Math.pow(c, 2.0) * (x * (x * Math.pow(s, 2.0))))) <= Double.POSITIVE_INFINITY) {
		tmp = (t_0 / t_1) / t_1;
	} else {
		tmp = t_0 * Math.pow((x * (c * s)), -2.0);
	}
	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 = math.cos((x + x))
	t_1 = c * (x * s)
	tmp = 0
	if (math.cos((2.0 * x)) / (math.pow(c, 2.0) * (x * (x * math.pow(s, 2.0))))) <= math.inf:
		tmp = (t_0 / t_1) / t_1
	else:
		tmp = t_0 * math.pow((x * (c * s)), -2.0)
	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 = cos(Float64(x + x))
	t_1 = Float64(c * Float64(x * s))
	tmp = 0.0
	if (Float64(cos(Float64(2.0 * x)) / Float64((c ^ 2.0) * Float64(x * Float64(x * (s ^ 2.0))))) <= Inf)
		tmp = Float64(Float64(t_0 / t_1) / t_1);
	else
		tmp = Float64(t_0 * (Float64(x * Float64(c * s)) ^ -2.0));
	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 = cos((x + x));
	t_1 = c * (x * s);
	tmp = 0.0;
	if ((cos((2.0 * x)) / ((c ^ 2.0) * (x * (x * (s ^ 2.0))))) <= Inf)
		tmp = (t_0 / t_1) / t_1;
	else
		tmp = t_0 * ((x * (c * s)) ^ -2.0);
	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[Cos[N[(x + x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] / N[(N[Power[c, 2.0], $MachinePrecision] * N[(x * N[(x * N[Power[s, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(t$95$0 / t$95$1), $MachinePrecision] / t$95$1), $MachinePrecision], N[(t$95$0 * N[Power[N[(x * N[(c * s), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $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 18.3

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

   2. Simplified 12.9

      \[\leadsto \color{blue}{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(x \cdot \left(\left(c \cdot s\right) \cdot \left(c \cdot s\right)\right)\right)}} \]
      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)))))): 67 points increase in error, 1 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))))): 9 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)))): 14 points increase in error, 8 points decrease in error

   3. Applied egg-rr 1.3

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

   4. Applied egg-rr 0.3

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

   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. Applied egg-rr 61.2

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

   3. Applied egg-rr 2.8

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

4. Final simplification 0.9

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

Alternatives

Alternative 1
Error	7.1
Cost	8020

\[\begin{array}{l} t_0 := c \cdot \left(x \cdot s\right)\\ t_1 := \frac{1}{t_0}\\ t_2 := \frac{\cos \left(2 \cdot x\right)}{s \cdot \left(\left(x \cdot c\right) \cdot \left(x \cdot \left(c \cdot s\right)\right)\right)}\\ t_3 := \frac{\cos \left(x + x\right)}{t_0}\\ \mathbf{if}\;s \leq 10^{-250}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;s \leq 3.2435672357147274 \cdot 10^{+132}:\\ \;\;\;\;\frac{\frac{t_3}{s}}{x \cdot c}\\ \mathbf{elif}\;s \leq 2.1022107265610716 \cdot 10^{+176}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;s \leq 4.640382977886413 \cdot 10^{+196}:\\ \;\;\;\;\frac{1}{\frac{t_0}{t_1}}\\ \mathbf{elif}\;s \leq 6.291346953052861 \cdot 10^{+216}:\\ \;\;\;\;\frac{\frac{t_3}{x}}{c \cdot s}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot t_1\\ \end{array} \]

Alternative 2
Error	6.2
Cost	7624

\[\begin{array}{l} t_0 := \frac{1}{c \cdot \left(x \cdot s\right)}\\ t_1 := \frac{\cos \left(2 \cdot x\right)}{s \cdot \left(\left(x \cdot c\right) \cdot \left(x \cdot \left(c \cdot s\right)\right)\right)}\\ \mathbf{if}\;x \leq -1 \cdot 10^{-54}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 10^{-60}:\\ \;\;\;\;t_0 \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	5.7
Cost	7624

\[\begin{array}{l} t_0 := c \cdot \left(x \cdot s\right)\\ \mathbf{if}\;x \leq -1 \cdot 10^{-25}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{s \cdot \left(\left(x \cdot c\right) \cdot \left(x \cdot \left(c \cdot s\right)\right)\right)}\\ \mathbf{elif}\;x \leq 10^{-150}:\\ \;\;\;\;{t_0}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\cos \left(x + x\right)}{t_0}}{x}}{c \cdot s}\\ \end{array} \]

Alternative 4
Error	3.1
Cost	7624

\[\begin{array}{l} t_0 := c \cdot \left(x \cdot s\right)\\ t_1 := \frac{\frac{\cos \left(x + x\right)}{t_0}}{t_0}\\ \mathbf{if}\;c \leq -1 \cdot 10^{-200}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq 10^{-224}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{s \cdot \left(\left(x \cdot c\right) \cdot \left(x \cdot \left(c \cdot s\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	3.2
Cost	7624

\[\begin{array}{l} t_0 := \cos \left(2 \cdot x\right)\\ t_1 := c \cdot \left(x \cdot s\right)\\ \mathbf{if}\;c \leq -1 \cdot 10^{-200}:\\ \;\;\;\;\frac{\frac{\cos \left(x + x\right)}{t_1}}{t_1}\\ \mathbf{elif}\;c \leq 10^{-224}:\\ \;\;\;\;\frac{t_0}{s \cdot \left(\left(x \cdot c\right) \cdot \left(x \cdot \left(c \cdot s\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{t_1 \cdot t_1}\\ \end{array} \]

Alternative 6
Error	2.5
Cost	7620

\[\begin{array}{l} t_0 := \cos \left(x + x\right)\\ t_1 := c \cdot \left(x \cdot s\right)\\ \mathbf{if}\;c \leq -1 \cdot 10^{-200}:\\ \;\;\;\;\frac{\frac{t_0}{t_1}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot \left(x \cdot c\right)} \cdot \frac{\frac{t_0}{x \cdot c}}{s}\\ \end{array} \]

Alternative 7
Error	2.4
Cost	7620

\[\begin{array}{l} t_0 := s \cdot \left(x \cdot c\right)\\ t_1 := \cos \left(x + x\right)\\ t_2 := c \cdot \left(x \cdot s\right)\\ \mathbf{if}\;c \leq -1 \cdot 10^{-200}:\\ \;\;\;\;\frac{\frac{t_1}{t_2}}{t_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t_0} \cdot \frac{t_1}{t_0}\\ \end{array} \]

Alternative 8
Error	17.7
Cost	6784

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

Alternative 9
Error	19.0
Cost	1096

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

Alternative 10
Error	19.1
Cost	964

\[\begin{array}{l} \mathbf{if}\;s \leq 1.7159308517144218 \cdot 10^{+108}:\\ \;\;\;\;\frac{1}{s \cdot \left(\left(x \cdot c\right) \cdot \left(x \cdot \left(c \cdot s\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{c \cdot \left(\left(x \cdot s\right) \cdot \left(c \cdot \left(x \cdot s\right)\right)\right)}\\ \end{array} \]

Alternative 11
Error	17.8
Cost	960

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

Alternative 12
Error	17.7
Cost	960

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

Alternative 13
Error	25.1
Cost	832

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

Alternative 14
Error	20.4
Cost	832

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

Alternative 15
Error	19.6
Cost	832

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

Alternative 16
Error	17.8
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 2022308 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))