mixedcos

Average Error: 28.3 → 0.8

Time: 9.9s

Precision: binary64

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

Math

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

↓

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

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 (* 2.0 x))) (t_1 (fabs (* x s))))
   (if (<= (/ t_0 (* (pow c 2.0) (* x (* x (pow s 2.0))))) INFINITY)
     (* (/ (/ 1.0 c) t_1) (/ t_0 (* c t_1)))
     (/ t_0 (pow (* (* c (fabs s)) (fabs x)) 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((2.0 * x));
	double t_1 = fabs((x * s));
	double tmp;
	if ((t_0 / (pow(c, 2.0) * (x * (x * pow(s, 2.0))))) <= ((double) INFINITY)) {
		tmp = ((1.0 / c) / t_1) * (t_0 / (c * t_1));
	} else {
		tmp = t_0 / pow(((c * fabs(s)) * fabs(x)), 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((2.0 * x));
	double t_1 = Math.abs((x * s));
	double tmp;
	if ((t_0 / (Math.pow(c, 2.0) * (x * (x * Math.pow(s, 2.0))))) <= Double.POSITIVE_INFINITY) {
		tmp = ((1.0 / c) / t_1) * (t_0 / (c * t_1));
	} else {
		tmp = t_0 / Math.pow(((c * Math.abs(s)) * Math.abs(x)), 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((2.0 * x))
	t_1 = math.fabs((x * s))
	tmp = 0
	if (t_0 / (math.pow(c, 2.0) * (x * (x * math.pow(s, 2.0))))) <= math.inf:
		tmp = ((1.0 / c) / t_1) * (t_0 / (c * t_1))
	else:
		tmp = t_0 / math.pow(((c * math.fabs(s)) * math.fabs(x)), 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(2.0 * x))
	t_1 = abs(Float64(x * s))
	tmp = 0.0
	if (Float64(t_0 / Float64((c ^ 2.0) * Float64(x * Float64(x * (s ^ 2.0))))) <= Inf)
		tmp = Float64(Float64(Float64(1.0 / c) / t_1) * Float64(t_0 / Float64(c * t_1)));
	else
		tmp = Float64(t_0 / (Float64(Float64(c * abs(s)) * abs(x)) ^ 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((2.0 * x));
	t_1 = abs((x * s));
	tmp = 0.0;
	if ((t_0 / ((c ^ 2.0) * (x * (x * (s ^ 2.0))))) <= Inf)
		tmp = ((1.0 / c) / t_1) * (t_0 / (c * t_1));
	else
		tmp = t_0 / (((c * abs(s)) * abs(x)) ^ 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[(2.0 * x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Abs[N[(x * s), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(t$95$0 / N[(N[Power[c, 2.0], $MachinePrecision] * N[(x * N[(x * N[Power[s, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(N[(1.0 / c), $MachinePrecision] / t$95$1), $MachinePrecision] * N[(t$95$0 / N[(c * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[Power[N[(N[(c * N[Abs[s], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]]

Error

Bits error versus x

Bits error versus c

Bits error versus s

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 18.7

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

   2. Applied add-sqr-sqrt_binary64 18.8

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

   3. Simplified 18.7

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

   4. Simplified 9.9

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

   5. Applied pow2_binary64 9.9

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

   6. Applied pow-prod-down_binary64 0.7

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

   7. Applied unpow2_binary64 0.7

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

   8. Applied *-un-lft-identity_binary64 0.7

      \[\leadsto \frac{\color{blue}{1 \cdot \cos \left(2 \cdot x\right)}}{\left(c \cdot \left|s \cdot x\right|\right) \cdot \left(c \cdot \left|s \cdot x\right|\right)} \]

   9. Applied times-frac_binary64 0.4

      \[\leadsto \color{blue}{\frac{1}{c \cdot \left|s \cdot x\right|} \cdot \frac{\cos \left(2 \cdot x\right)}{c \cdot \left|s \cdot x\right|}} \]

   10. Applied associate-/r*_binary64 0.3

       \[\leadsto \color{blue}{\frac{\frac{1}{c}}{\left|s \cdot x\right|}} \cdot \frac{\cos \left(2 \cdot x\right)}{c \cdot \left|s \cdot x\right|} \]

   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 add-sqr-sqrt_binary64 64.0

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

   3. Simplified 64.0

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

   4. Simplified 57.7

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

   5. Applied pow2_binary64 57.7

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

   6. Applied pow-prod-down_binary64 10.1

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

   7. Applied fabs-mul_binary64 10.1

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

   8. Applied associate-*r*_binary64 2.7

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

4. Final simplification 0.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(x \cdot \left(x \cdot {s}^{2}\right)\right)} \leq \infty:\\ \;\;\;\;\frac{\frac{1}{c}}{\left|x \cdot s\right|} \cdot \frac{\cos \left(2 \cdot x\right)}{c \cdot \left|x \cdot s\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(2 \cdot x\right)}{{\left(\left(c \cdot \left|s\right|\right) \cdot \left|x\right|\right)}^{2}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022137 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))