mixedcos

Percentage Accurate: 66.5% → 99.2%

Time: 18.3s

Alternatives: 13

Speedup: 24.1×

• 32.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

(FPCore (x c s)
 :precision binary64
 (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))

C

double code(double x, double c, double s) {
	return cos((2.0 * x)) / (pow(c, 2.0) * ((x * pow(s, 2.0)) * x));
}

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

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));
}

Python

def code(x, c, s):
	return math.cos((2.0 * x)) / (math.pow(c, 2.0) * ((x * math.pow(s, 2.0)) * x))

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

MATLAB

function tmp = code(x, c, s)
	tmp = cos((2.0 * x)) / ((c ^ 2.0) * ((x * (s ^ 2.0)) * x));
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]

Initial Program: 66.5% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x c s)
 :precision binary64
 (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))

C

double code(double x, double c, double s) {
	return cos((2.0 * x)) / (pow(c, 2.0) * ((x * pow(s, 2.0)) * x));
}

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

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));
}

Python

def code(x, c, s):
	return math.cos((2.0 * x)) / (math.pow(c, 2.0) * ((x * math.pow(s, 2.0)) * x))

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

MATLAB

function tmp = code(x, c, s)
	tmp = cos((2.0 * x)) / ((c ^ 2.0) * ((x * (s ^ 2.0)) * x));
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]

Alternative 1: 99.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := c \cdot \left(x\_m \cdot s\right)\\ t_1 := \cos \left(2 \cdot x\_m\right)\\ t_2 := \left(c \cdot s\right) \cdot \sqrt{x\_m}\\ \mathbf{if}\;\frac{t\_1}{{c}^{2} \cdot \left(x\_m \cdot \left(x\_m \cdot {s}^{2}\right)\right)} \leq \infty:\\ \;\;\;\;\frac{1}{t\_0} \cdot \frac{t\_1}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{t\_1}{x\_m}}{t\_2}}{t\_2}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m c s)
 :precision binary64
 (let* ((t_0 (* c (* x_m s)))
        (t_1 (cos (* 2.0 x_m)))
        (t_2 (* (* c s) (sqrt x_m))))
   (if (<= (/ t_1 (* (pow c 2.0) (* x_m (* x_m (pow s 2.0))))) INFINITY)
     (* (/ 1.0 t_0) (/ t_1 t_0))
     (/ (/ (/ t_1 x_m) t_2) t_2))))

C

x_m = fabs(x);
double code(double x_m, double c, double s) {
	double t_0 = c * (x_m * s);
	double t_1 = cos((2.0 * x_m));
	double t_2 = (c * s) * sqrt(x_m);
	double tmp;
	if ((t_1 / (pow(c, 2.0) * (x_m * (x_m * pow(s, 2.0))))) <= ((double) INFINITY)) {
		tmp = (1.0 / t_0) * (t_1 / t_0);
	} else {
		tmp = ((t_1 / x_m) / t_2) / t_2;
	}
	return tmp;
}

Java

x_m = Math.abs(x);
public static double code(double x_m, double c, double s) {
	double t_0 = c * (x_m * s);
	double t_1 = Math.cos((2.0 * x_m));
	double t_2 = (c * s) * Math.sqrt(x_m);
	double tmp;
	if ((t_1 / (Math.pow(c, 2.0) * (x_m * (x_m * Math.pow(s, 2.0))))) <= Double.POSITIVE_INFINITY) {
		tmp = (1.0 / t_0) * (t_1 / t_0);
	} else {
		tmp = ((t_1 / x_m) / t_2) / t_2;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m, c, s):
	t_0 = c * (x_m * s)
	t_1 = math.cos((2.0 * x_m))
	t_2 = (c * s) * math.sqrt(x_m)
	tmp = 0
	if (t_1 / (math.pow(c, 2.0) * (x_m * (x_m * math.pow(s, 2.0))))) <= math.inf:
		tmp = (1.0 / t_0) * (t_1 / t_0)
	else:
		tmp = ((t_1 / x_m) / t_2) / t_2
	return tmp

Julia

x_m = abs(x)
function code(x_m, c, s)
	t_0 = Float64(c * Float64(x_m * s))
	t_1 = cos(Float64(2.0 * x_m))
	t_2 = Float64(Float64(c * s) * sqrt(x_m))
	tmp = 0.0
	if (Float64(t_1 / Float64((c ^ 2.0) * Float64(x_m * Float64(x_m * (s ^ 2.0))))) <= Inf)
		tmp = Float64(Float64(1.0 / t_0) * Float64(t_1 / t_0));
	else
		tmp = Float64(Float64(Float64(t_1 / x_m) / t_2) / t_2);
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m, c, s)
	t_0 = c * (x_m * s);
	t_1 = cos((2.0 * x_m));
	t_2 = (c * s) * sqrt(x_m);
	tmp = 0.0;
	if ((t_1 / ((c ^ 2.0) * (x_m * (x_m * (s ^ 2.0))))) <= Inf)
		tmp = (1.0 / t_0) * (t_1 / t_0);
	else
		tmp = ((t_1 / x_m) / t_2) / t_2;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, c_, s_] := Block[{t$95$0 = N[(c * N[(x$95$m * s), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(2.0 * x$95$m), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(c * s), $MachinePrecision] * N[Sqrt[x$95$m], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$1 / N[(N[Power[c, 2.0], $MachinePrecision] * N[(x$95$m * N[(x$95$m * N[Power[s, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(1.0 / t$95$0), $MachinePrecision] * N[(t$95$1 / t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t$95$1 / x$95$m), $MachinePrecision] / t$95$2), $MachinePrecision] / t$95$2), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x))) < +inf.0

   1. Initial program 80.6%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-un-lft-identity 80.6%

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

      2. add-sqr-sqrt 80.6%

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

      3. times-frac 80.6%

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

      4. sqrt-prod 80.6%

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

      5. sqrt-pow1 60.0%

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

      6. metadata-eval 60.0%

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

      7. pow1 60.0%

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

      8. *-commutative 60.0%

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

      9. associate-*r* 54.5%

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

      10. unpow2 54.5%

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

      11. pow-prod-down 60.0%

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

      12. sqrt-pow1 57.0%

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

      13. metadata-eval 57.0%

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

      14. pow1 57.0%

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

      15. *-commutative 57.0%

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

   4. Applied egg-rr 99.6%

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

   if +inf.0 < (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x)))

   1. Initial program 0.0%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-un-lft-identity 0.0%

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

      2. add-sqr-sqrt 0.0%

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

      3. times-frac 0.0%

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

      4. sqrt-prod 0.0%

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

      5. sqrt-pow1 0.0%

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

      6. metadata-eval 0.0%

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

      7. pow1 0.0%

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

      8. *-commutative 0.0%

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

      9. associate-*r* 0.0%

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

      10. unpow2 0.0%

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

      11. pow-prod-down 0.0%

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

      12. sqrt-pow1 0.0%

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

      13. metadata-eval 0.0%

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

      14. pow1 0.0%

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

      15. *-commutative 0.0%

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

   4. Applied egg-rr 76.5%

      \[\leadsto \color{blue}{\frac{1}{c \cdot \left(x \cdot s\right)} \cdot \frac{\cos \left(2 \cdot x\right)}{c \cdot \left(x \cdot s\right)}} \]
   5. Step-by-step derivation

      1. frac-times 76.6%

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

      2. *-un-lft-identity 76.6%

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

      3. unpow2 76.6%

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

      4. *-commutative 76.6%

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

      5. *-commutative 76.6%

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

      6. *-commutative 76.6%

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

      7. associate-*r* 97.1%

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

      8. unpow-prod-down 67.2%

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

      9. pow2 67.2%

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

      10. associate-*l* 79.9%

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

      11. associate-/l/ 79.7%

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

      12. add-sqr-sqrt 39.6%

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

      13. associate-/r* 39.6%

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

   6. Applied egg-rr 44.2%

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

4. Final simplification 89.9%

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

Alternative 2: 98.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := c \cdot \left(x\_m \cdot s\right)\\ t_1 := \cos \left(2 \cdot x\_m\right)\\ \mathbf{if}\;\frac{t\_1}{{c}^{2} \cdot \left(x\_m \cdot \left(x\_m \cdot {s}^{2}\right)\right)} \leq \infty:\\ \;\;\;\;\frac{1}{t\_0} \cdot \frac{t\_1}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{1}{\left(c \cdot s\right) \cdot \sqrt{x\_m}}\right)}^{2} \cdot \frac{t\_1}{x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m c s)
 :precision binary64
 (let* ((t_0 (* c (* x_m s))) (t_1 (cos (* 2.0 x_m))))
   (if (<= (/ t_1 (* (pow c 2.0) (* x_m (* x_m (pow s 2.0))))) INFINITY)
     (* (/ 1.0 t_0) (/ t_1 t_0))
     (* (pow (/ 1.0 (* (* c s) (sqrt x_m))) 2.0) (/ t_1 x_m)))))

C

x_m = fabs(x);
double code(double x_m, double c, double s) {
	double t_0 = c * (x_m * s);
	double t_1 = cos((2.0 * x_m));
	double tmp;
	if ((t_1 / (pow(c, 2.0) * (x_m * (x_m * pow(s, 2.0))))) <= ((double) INFINITY)) {
		tmp = (1.0 / t_0) * (t_1 / t_0);
	} else {
		tmp = pow((1.0 / ((c * s) * sqrt(x_m))), 2.0) * (t_1 / x_m);
	}
	return tmp;
}

Java

x_m = Math.abs(x);
public static double code(double x_m, double c, double s) {
	double t_0 = c * (x_m * s);
	double t_1 = Math.cos((2.0 * x_m));
	double tmp;
	if ((t_1 / (Math.pow(c, 2.0) * (x_m * (x_m * Math.pow(s, 2.0))))) <= Double.POSITIVE_INFINITY) {
		tmp = (1.0 / t_0) * (t_1 / t_0);
	} else {
		tmp = Math.pow((1.0 / ((c * s) * Math.sqrt(x_m))), 2.0) * (t_1 / x_m);
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m, c, s):
	t_0 = c * (x_m * s)
	t_1 = math.cos((2.0 * x_m))
	tmp = 0
	if (t_1 / (math.pow(c, 2.0) * (x_m * (x_m * math.pow(s, 2.0))))) <= math.inf:
		tmp = (1.0 / t_0) * (t_1 / t_0)
	else:
		tmp = math.pow((1.0 / ((c * s) * math.sqrt(x_m))), 2.0) * (t_1 / x_m)
	return tmp

Julia

x_m = abs(x)
function code(x_m, c, s)
	t_0 = Float64(c * Float64(x_m * s))
	t_1 = cos(Float64(2.0 * x_m))
	tmp = 0.0
	if (Float64(t_1 / Float64((c ^ 2.0) * Float64(x_m * Float64(x_m * (s ^ 2.0))))) <= Inf)
		tmp = Float64(Float64(1.0 / t_0) * Float64(t_1 / t_0));
	else
		tmp = Float64((Float64(1.0 / Float64(Float64(c * s) * sqrt(x_m))) ^ 2.0) * Float64(t_1 / x_m));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m, c, s)
	t_0 = c * (x_m * s);
	t_1 = cos((2.0 * x_m));
	tmp = 0.0;
	if ((t_1 / ((c ^ 2.0) * (x_m * (x_m * (s ^ 2.0))))) <= Inf)
		tmp = (1.0 / t_0) * (t_1 / t_0);
	else
		tmp = ((1.0 / ((c * s) * sqrt(x_m))) ^ 2.0) * (t_1 / x_m);
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, c_, s_] := Block[{t$95$0 = N[(c * N[(x$95$m * s), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(2.0 * x$95$m), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(t$95$1 / N[(N[Power[c, 2.0], $MachinePrecision] * N[(x$95$m * N[(x$95$m * N[Power[s, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(1.0 / t$95$0), $MachinePrecision] * N[(t$95$1 / t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(1.0 / N[(N[(c * s), $MachinePrecision] * N[Sqrt[x$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(t$95$1 / x$95$m), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x))) < +inf.0

   1. Initial program 80.6%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-un-lft-identity 80.6%

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

      2. add-sqr-sqrt 80.6%

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

      3. times-frac 80.6%

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

      4. sqrt-prod 80.6%

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

      5. sqrt-pow1 60.0%

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

      6. metadata-eval 60.0%

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

      7. pow1 60.0%

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

      8. *-commutative 60.0%

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

      9. associate-*r* 54.5%

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

      10. unpow2 54.5%

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

      11. pow-prod-down 60.0%

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

      12. sqrt-pow1 57.0%

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

      13. metadata-eval 57.0%

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

      14. pow1 57.0%

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

      15. *-commutative 57.0%

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

   4. Applied egg-rr 99.6%

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

   if +inf.0 < (/.f64 (cos.f64 (*.f64 #s(literal 2 binary64) x)) (*.f64 (pow.f64 c #s(literal 2 binary64)) (*.f64 (*.f64 x (pow.f64 s #s(literal 2 binary64))) x)))

   1. Initial program 0.0%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-un-lft-identity 0.0%

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

      2. associate-*r* 0.5%

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

      3. times-frac 0.5%

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

      4. *-commutative 0.5%

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

      5. associate-*r* 0.9%

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

      6. pow-prod-down 79.8%

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

   4. Applied egg-rr 79.8%

      \[\leadsto \color{blue}{\frac{1}{{\left(c \cdot s\right)}^{2} \cdot x} \cdot \frac{\cos \left(2 \cdot x\right)}{x}} \]
   5. Step-by-step derivation

      1. add-sqr-sqrt 44.3%

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

      2. pow2 44.3%

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

      3. sqrt-div 39.6%

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

      4. metadata-eval 39.6%

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

      5. sqrt-prod 39.5%

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

      6. sqrt-pow1 44.1%

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

      7. metadata-eval 44.1%

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

      8. pow1 44.1%

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

   6. Applied egg-rr 44.1%

      \[\leadsto \color{blue}{{\left(\frac{1}{\left(c \cdot s\right) \cdot \sqrt{x}}\right)}^{2}} \cdot \frac{\cos \left(2 \cdot x\right)}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.9%

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

Alternative 3: 93.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \frac{\frac{1}{c}}{x\_m \cdot s}\\ \mathbf{if}\;x\_m \leq 4.4 \cdot 10^{-45}:\\ \;\;\;\;t\_0 \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(2 \cdot x\_m\right)}{c \cdot \left(\left(x\_m \cdot s\right) \cdot \left(x\_m \cdot c\right)\right)}}{s}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m c s)
 :precision binary64
 (let* ((t_0 (/ (/ 1.0 c) (* x_m s))))
   (if (<= x_m 4.4e-45)
     (* t_0 t_0)
     (/ (/ (cos (* 2.0 x_m)) (* c (* (* x_m s) (* x_m c)))) s))))

C

x_m = fabs(x);
double code(double x_m, double c, double s) {
	double t_0 = (1.0 / c) / (x_m * s);
	double tmp;
	if (x_m <= 4.4e-45) {
		tmp = t_0 * t_0;
	} else {
		tmp = (cos((2.0 * x_m)) / (c * ((x_m * s) * (x_m * c)))) / s;
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m, c, s)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 / c) / (x_m * s)
    if (x_m <= 4.4d-45) then
        tmp = t_0 * t_0
    else
        tmp = (cos((2.0d0 * x_m)) / (c * ((x_m * s) * (x_m * c)))) / s
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m, double c, double s) {
	double t_0 = (1.0 / c) / (x_m * s);
	double tmp;
	if (x_m <= 4.4e-45) {
		tmp = t_0 * t_0;
	} else {
		tmp = (Math.cos((2.0 * x_m)) / (c * ((x_m * s) * (x_m * c)))) / s;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m, c, s):
	t_0 = (1.0 / c) / (x_m * s)
	tmp = 0
	if x_m <= 4.4e-45:
		tmp = t_0 * t_0
	else:
		tmp = (math.cos((2.0 * x_m)) / (c * ((x_m * s) * (x_m * c)))) / s
	return tmp

Julia

x_m = abs(x)
function code(x_m, c, s)
	t_0 = Float64(Float64(1.0 / c) / Float64(x_m * s))
	tmp = 0.0
	if (x_m <= 4.4e-45)
		tmp = Float64(t_0 * t_0);
	else
		tmp = Float64(Float64(cos(Float64(2.0 * x_m)) / Float64(c * Float64(Float64(x_m * s) * Float64(x_m * c)))) / s);
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m, c, s)
	t_0 = (1.0 / c) / (x_m * s);
	tmp = 0.0;
	if (x_m <= 4.4e-45)
		tmp = t_0 * t_0;
	else
		tmp = (cos((2.0 * x_m)) / (c * ((x_m * s) * (x_m * c)))) / s;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, c_, s_] := Block[{t$95$0 = N[(N[(1.0 / c), $MachinePrecision] / N[(x$95$m * s), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$95$m, 4.4e-45], N[(t$95$0 * t$95$0), $MachinePrecision], N[(N[(N[Cos[N[(2.0 * x$95$m), $MachinePrecision]], $MachinePrecision] / N[(c * N[(N[(x$95$m * s), $MachinePrecision] * N[(x$95$m * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / s), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 4.39999999999999987e-45

   1. Initial program 64.6%

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

   3. Taylor expanded in x around 0 52.7%

      \[\leadsto \color{blue}{\frac{1}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)}} \]
   4. Step-by-step derivation

      1. associate-/r* 52.7%

         \[\leadsto \color{blue}{\frac{\frac{1}{{c}^{2}}}{{s}^{2} \cdot {x}^{2}}} \]

      2. *-commutative 52.7%

         \[\leadsto \frac{\frac{1}{{c}^{2}}}{\color{blue}{{x}^{2} \cdot {s}^{2}}} \]

      3. unpow2 52.7%

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

      4. unpow2 52.7%

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

      5. swap-sqr 67.1%

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

      6. unpow2 67.1%

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

      7. associate-/r* 67.0%

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

      8. unpow2 67.0%

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

      9. unpow2 67.0%

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

      10. swap-sqr 79.1%

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

      11. unpow2 79.1%

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

      12. *-commutative 79.1%

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

   5. Simplified 79.1%

      \[\leadsto \color{blue}{\frac{1}{{\left(c \cdot \left(s \cdot x\right)\right)}^{2}}} \]
   6. Step-by-step derivation

      1. pow-flip 79.4%

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

      2. *-commutative 79.4%

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

      3. pow-flip 79.1%

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

      4. unpow2 79.1%

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

      5. metadata-eval 79.1%

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

      6. frac-times 79.4%

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

      7. associate-/r* 79.4%

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

      8. associate-/r* 79.4%

         \[\leadsto \frac{\frac{1}{c}}{x \cdot s} \cdot \color{blue}{\frac{\frac{1}{c}}{x \cdot s}} \]

   7. Applied egg-rr 79.4%

      \[\leadsto \color{blue}{\frac{\frac{1}{c}}{x \cdot s} \cdot \frac{\frac{1}{c}}{x \cdot s}} \]

   if 4.39999999999999987e-45 < x

   1. Initial program 70.5%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-un-lft-identity 70.5%

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

      2. add-sqr-sqrt 70.5%

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

      3. times-frac 70.5%

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

      4. sqrt-prod 70.5%

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

      5. sqrt-pow1 57.8%

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

      6. metadata-eval 57.8%

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

      7. pow1 57.8%

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

      8. *-commutative 57.8%

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

      9. associate-*r* 52.7%

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

      10. unpow2 52.7%

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

      11. pow-prod-down 57.9%

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

      12. sqrt-pow1 58.4%

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

      13. metadata-eval 58.4%

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

      14. pow1 58.4%

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

      15. *-commutative 58.4%

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

   4. Applied egg-rr 97.4%

      \[\leadsto \color{blue}{\frac{1}{c \cdot \left(x \cdot s\right)} \cdot \frac{\cos \left(2 \cdot x\right)}{c \cdot \left(x \cdot s\right)}} \]
   5. Step-by-step derivation

      1. associate-*l/ 97.4%

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

      2. *-un-lft-identity 97.4%

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

      3. associate-*r* 96.4%

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

      4. associate-/r* 95.2%

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

      5. *-commutative 95.2%

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

   6. Applied egg-rr 95.2%

      \[\leadsto \color{blue}{\frac{\frac{\frac{\cos \left(x \cdot 2\right)}{c \cdot \left(x \cdot s\right)}}{c \cdot x}}{s}} \]
   7. Step-by-step derivation

      1. *-un-lft-identity 95.2%

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

      2. associate-/l/ 95.1%

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

      3. *-commutative 95.1%

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

   8. Applied egg-rr 95.1%

      \[\leadsto \frac{\color{blue}{1 \cdot \frac{\cos \left(x \cdot 2\right)}{\left(c \cdot x\right) \cdot \left(c \cdot \left(s \cdot x\right)\right)}}}{s} \]
   9. Step-by-step derivation

      1. *-lft-identity 95.1%

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

      2. *-commutative 95.1%

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

      3. associate-*l* 92.9%

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

      4. associate-*r* 92.9%

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

      5. *-commutative 92.9%

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

   10. Simplified 92.9%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.4 \cdot 10^{-45}:\\ \;\;\;\;\frac{\frac{1}{c}}{x \cdot s} \cdot \frac{\frac{1}{c}}{x \cdot s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos \left(2 \cdot x\right)}{c \cdot \left(\left(x \cdot s\right) \cdot \left(x \cdot c\right)\right)}}{s}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 93.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \frac{\frac{1}{c}}{x\_m \cdot s}\\ \mathbf{if}\;x\_m \leq 6.8 \cdot 10^{-46}:\\ \;\;\;\;t\_0 \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\cos \left(2 \cdot x\_m\right)}{c \cdot \left(x\_m \cdot \left(c \cdot s\right)\right)}}{x\_m}}{s}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m c s)
 :precision binary64
 (let* ((t_0 (/ (/ 1.0 c) (* x_m s))))
   (if (<= x_m 6.8e-46)
     (* t_0 t_0)
     (/ (/ (/ (cos (* 2.0 x_m)) (* c (* x_m (* c s)))) x_m) s))))

C

x_m = fabs(x);
double code(double x_m, double c, double s) {
	double t_0 = (1.0 / c) / (x_m * s);
	double tmp;
	if (x_m <= 6.8e-46) {
		tmp = t_0 * t_0;
	} else {
		tmp = ((cos((2.0 * x_m)) / (c * (x_m * (c * s)))) / x_m) / s;
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m, c, s)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 / c) / (x_m * s)
    if (x_m <= 6.8d-46) then
        tmp = t_0 * t_0
    else
        tmp = ((cos((2.0d0 * x_m)) / (c * (x_m * (c * s)))) / x_m) / s
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m, double c, double s) {
	double t_0 = (1.0 / c) / (x_m * s);
	double tmp;
	if (x_m <= 6.8e-46) {
		tmp = t_0 * t_0;
	} else {
		tmp = ((Math.cos((2.0 * x_m)) / (c * (x_m * (c * s)))) / x_m) / s;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m, c, s):
	t_0 = (1.0 / c) / (x_m * s)
	tmp = 0
	if x_m <= 6.8e-46:
		tmp = t_0 * t_0
	else:
		tmp = ((math.cos((2.0 * x_m)) / (c * (x_m * (c * s)))) / x_m) / s
	return tmp

Julia

x_m = abs(x)
function code(x_m, c, s)
	t_0 = Float64(Float64(1.0 / c) / Float64(x_m * s))
	tmp = 0.0
	if (x_m <= 6.8e-46)
		tmp = Float64(t_0 * t_0);
	else
		tmp = Float64(Float64(Float64(cos(Float64(2.0 * x_m)) / Float64(c * Float64(x_m * Float64(c * s)))) / x_m) / s);
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m, c, s)
	t_0 = (1.0 / c) / (x_m * s);
	tmp = 0.0;
	if (x_m <= 6.8e-46)
		tmp = t_0 * t_0;
	else
		tmp = ((cos((2.0 * x_m)) / (c * (x_m * (c * s)))) / x_m) / s;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, c_, s_] := Block[{t$95$0 = N[(N[(1.0 / c), $MachinePrecision] / N[(x$95$m * s), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$95$m, 6.8e-46], N[(t$95$0 * t$95$0), $MachinePrecision], N[(N[(N[(N[Cos[N[(2.0 * x$95$m), $MachinePrecision]], $MachinePrecision] / N[(c * N[(x$95$m * N[(c * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision] / s), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 6.79999999999999992e-46

   1. Initial program 64.6%

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

   3. Taylor expanded in x around 0 52.7%

      \[\leadsto \color{blue}{\frac{1}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)}} \]
   4. Step-by-step derivation

      1. associate-/r* 52.7%

         \[\leadsto \color{blue}{\frac{\frac{1}{{c}^{2}}}{{s}^{2} \cdot {x}^{2}}} \]

      2. *-commutative 52.7%

         \[\leadsto \frac{\frac{1}{{c}^{2}}}{\color{blue}{{x}^{2} \cdot {s}^{2}}} \]

      3. unpow2 52.7%

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

      4. unpow2 52.7%

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

      5. swap-sqr 67.1%

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

      6. unpow2 67.1%

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

      7. associate-/r* 67.0%

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

      8. unpow2 67.0%

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

      9. unpow2 67.0%

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

      10. swap-sqr 79.1%

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

      11. unpow2 79.1%

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

      12. *-commutative 79.1%

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

   5. Simplified 79.1%

      \[\leadsto \color{blue}{\frac{1}{{\left(c \cdot \left(s \cdot x\right)\right)}^{2}}} \]
   6. Step-by-step derivation

      1. pow-flip 79.4%

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

      2. *-commutative 79.4%

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

      3. pow-flip 79.1%

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

      4. unpow2 79.1%

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

      5. metadata-eval 79.1%

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

      6. frac-times 79.4%

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

      7. associate-/r* 79.4%

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

      8. associate-/r* 79.4%

         \[\leadsto \frac{\frac{1}{c}}{x \cdot s} \cdot \color{blue}{\frac{\frac{1}{c}}{x \cdot s}} \]

   7. Applied egg-rr 79.4%

      \[\leadsto \color{blue}{\frac{\frac{1}{c}}{x \cdot s} \cdot \frac{\frac{1}{c}}{x \cdot s}} \]

   if 6.79999999999999992e-46 < x

   1. Initial program 70.5%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-un-lft-identity 70.5%

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

      2. add-sqr-sqrt 70.5%

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

      3. times-frac 70.5%

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

      4. sqrt-prod 70.5%

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

      5. sqrt-pow1 57.8%

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

      6. metadata-eval 57.8%

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

      7. pow1 57.8%

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

      8. *-commutative 57.8%

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

      9. associate-*r* 52.7%

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

      10. unpow2 52.7%

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

      11. pow-prod-down 57.9%

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

      12. sqrt-pow1 58.4%

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

      13. metadata-eval 58.4%

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

      14. pow1 58.4%

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

      15. *-commutative 58.4%

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

   4. Applied egg-rr 97.4%

      \[\leadsto \color{blue}{\frac{1}{c \cdot \left(x \cdot s\right)} \cdot \frac{\cos \left(2 \cdot x\right)}{c \cdot \left(x \cdot s\right)}} \]
   5. Step-by-step derivation

      1. associate-*l/ 97.4%

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

      2. *-un-lft-identity 97.4%

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

      3. associate-*r* 96.4%

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

      4. associate-/r* 95.2%

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

      5. *-commutative 95.2%

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

   6. Applied egg-rr 95.2%

      \[\leadsto \color{blue}{\frac{\frac{\frac{\cos \left(x \cdot 2\right)}{c \cdot \left(x \cdot s\right)}}{c \cdot x}}{s}} \]
   7. Step-by-step derivation

      1. *-un-lft-identity 95.2%

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

      2. *-commutative 95.2%

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

      3. times-frac 91.9%

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

      4. *-commutative 91.9%

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

   8. Applied egg-rr 91.9%

      \[\leadsto \frac{\color{blue}{\frac{1}{x} \cdot \frac{\frac{\cos \left(x \cdot 2\right)}{c \cdot \left(s \cdot x\right)}}{c}}}{s} \]
   9. Step-by-step derivation

      1. associate-*l/ 91.9%

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

      2. *-rgt-identity 91.9%

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

      3. associate-*r/ 91.9%

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

      4. associate-*l/ 91.8%

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

      5. *-lft-identity 91.8%

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

      6. associate-*l/ 91.9%

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

      7. associate-*r/ 91.9%

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

      8. *-rgt-identity 91.9%

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

      9. associate-/l/ 91.9%

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

      10. *-commutative 91.9%

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

      11. associate-*r* 93.0%

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

      12. *-commutative 93.0%

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

   10. Simplified 93.0%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6.8 \cdot 10^{-46}:\\ \;\;\;\;\frac{\frac{1}{c}}{x \cdot s} \cdot \frac{\frac{1}{c}}{x \cdot s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\cos \left(2 \cdot x\right)}{c \cdot \left(x \cdot \left(c \cdot s\right)\right)}}{x}}{s}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 94.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \frac{\frac{1}{c}}{x\_m \cdot s}\\ \mathbf{if}\;x\_m \leq 1.02 \cdot 10^{-20}:\\ \;\;\;\;t\_0 \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\cos \left(2 \cdot x\_m\right)}{x\_m \cdot \left(c \cdot s\right)}}{x\_m \cdot c}}{s}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m c s)
 :precision binary64
 (let* ((t_0 (/ (/ 1.0 c) (* x_m s))))
   (if (<= x_m 1.02e-20)
     (* t_0 t_0)
     (/ (/ (/ (cos (* 2.0 x_m)) (* x_m (* c s))) (* x_m c)) s))))

C

x_m = fabs(x);
double code(double x_m, double c, double s) {
	double t_0 = (1.0 / c) / (x_m * s);
	double tmp;
	if (x_m <= 1.02e-20) {
		tmp = t_0 * t_0;
	} else {
		tmp = ((cos((2.0 * x_m)) / (x_m * (c * s))) / (x_m * c)) / s;
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m, c, s)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 / c) / (x_m * s)
    if (x_m <= 1.02d-20) then
        tmp = t_0 * t_0
    else
        tmp = ((cos((2.0d0 * x_m)) / (x_m * (c * s))) / (x_m * c)) / s
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m, double c, double s) {
	double t_0 = (1.0 / c) / (x_m * s);
	double tmp;
	if (x_m <= 1.02e-20) {
		tmp = t_0 * t_0;
	} else {
		tmp = ((Math.cos((2.0 * x_m)) / (x_m * (c * s))) / (x_m * c)) / s;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m, c, s):
	t_0 = (1.0 / c) / (x_m * s)
	tmp = 0
	if x_m <= 1.02e-20:
		tmp = t_0 * t_0
	else:
		tmp = ((math.cos((2.0 * x_m)) / (x_m * (c * s))) / (x_m * c)) / s
	return tmp

Julia

x_m = abs(x)
function code(x_m, c, s)
	t_0 = Float64(Float64(1.0 / c) / Float64(x_m * s))
	tmp = 0.0
	if (x_m <= 1.02e-20)
		tmp = Float64(t_0 * t_0);
	else
		tmp = Float64(Float64(Float64(cos(Float64(2.0 * x_m)) / Float64(x_m * Float64(c * s))) / Float64(x_m * c)) / s);
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m, c, s)
	t_0 = (1.0 / c) / (x_m * s);
	tmp = 0.0;
	if (x_m <= 1.02e-20)
		tmp = t_0 * t_0;
	else
		tmp = ((cos((2.0 * x_m)) / (x_m * (c * s))) / (x_m * c)) / s;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, c_, s_] := Block[{t$95$0 = N[(N[(1.0 / c), $MachinePrecision] / N[(x$95$m * s), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$95$m, 1.02e-20], N[(t$95$0 * t$95$0), $MachinePrecision], N[(N[(N[(N[Cos[N[(2.0 * x$95$m), $MachinePrecision]], $MachinePrecision] / N[(x$95$m * N[(c * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$95$m * c), $MachinePrecision]), $MachinePrecision] / s), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.02000000000000001e-20

   1. Initial program 64.8%

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

   3. Taylor expanded in x around 0 53.3%

      \[\leadsto \color{blue}{\frac{1}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)}} \]
   4. Step-by-step derivation

      1. associate-/r* 53.3%

         \[\leadsto \color{blue}{\frac{\frac{1}{{c}^{2}}}{{s}^{2} \cdot {x}^{2}}} \]

      2. *-commutative 53.3%

         \[\leadsto \frac{\frac{1}{{c}^{2}}}{\color{blue}{{x}^{2} \cdot {s}^{2}}} \]

      3. unpow2 53.3%

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

      4. unpow2 53.3%

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

      5. swap-sqr 67.3%

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

      6. unpow2 67.3%

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

      7. associate-/r* 67.2%

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

      8. unpow2 67.2%

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

      9. unpow2 67.2%

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

      10. swap-sqr 79.5%

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

      11. unpow2 79.5%

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

      12. *-commutative 79.5%

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

   5. Simplified 79.5%

      \[\leadsto \color{blue}{\frac{1}{{\left(c \cdot \left(s \cdot x\right)\right)}^{2}}} \]
   6. Step-by-step derivation

      1. pow-flip 79.9%

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

      2. *-commutative 79.9%

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

      3. pow-flip 79.5%

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

      4. unpow2 79.5%

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

      5. metadata-eval 79.5%

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

      6. frac-times 79.9%

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

      7. associate-/r* 79.9%

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

      8. associate-/r* 79.9%

         \[\leadsto \frac{\frac{1}{c}}{x \cdot s} \cdot \color{blue}{\frac{\frac{1}{c}}{x \cdot s}} \]

   7. Applied egg-rr 79.9%

      \[\leadsto \color{blue}{\frac{\frac{1}{c}}{x \cdot s} \cdot \frac{\frac{1}{c}}{x \cdot s}} \]

   if 1.02000000000000001e-20 < x

   1. Initial program 70.2%

      \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-un-lft-identity 70.2%

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

      2. add-sqr-sqrt 70.2%

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

      3. times-frac 70.2%

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

      4. sqrt-prod 70.2%

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

      5. sqrt-pow1 58.2%

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

      6. metadata-eval 58.2%

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

      7. pow1 58.2%

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

      8. *-commutative 58.2%

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

      9. associate-*r* 52.7%

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

      10. unpow2 52.7%

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

      11. pow-prod-down 58.2%

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

      12. sqrt-pow1 58.7%

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

      13. metadata-eval 58.7%

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

      14. pow1 58.7%

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

      15. *-commutative 58.7%

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

   4. Applied egg-rr 97.3%

      \[\leadsto \color{blue}{\frac{1}{c \cdot \left(x \cdot s\right)} \cdot \frac{\cos \left(2 \cdot x\right)}{c \cdot \left(x \cdot s\right)}} \]
   5. Step-by-step derivation

      1. associate-*l/ 97.3%

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

      2. *-un-lft-identity 97.3%

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

      3. associate-*r* 96.2%

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

      4. associate-/r* 94.9%

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

      5. *-commutative 94.9%

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

   6. Applied egg-rr 94.9%

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

   7. Taylor expanded in x around inf 94.9%

      \[\leadsto \frac{\frac{\color{blue}{\frac{\cos \left(2 \cdot x\right)}{c \cdot \left(s \cdot x\right)}}}{c \cdot x}}{s} \]
   8. Step-by-step derivation

      1. associate-*r* 92.7%

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

      2. *-commutative 92.7%

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

   9. Simplified 92.7%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.02 \cdot 10^{-20}:\\ \;\;\;\;\frac{\frac{1}{c}}{x \cdot s} \cdot \frac{\frac{1}{c}}{x \cdot s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\cos \left(2 \cdot x\right)}{x \cdot \left(c \cdot s\right)}}{x \cdot c}}{s}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.2% accurate, 2.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := c \cdot \left(x\_m \cdot s\right)\\ \frac{1}{t\_0} \cdot \frac{\cos \left(2 \cdot x\_m\right)}{t\_0} \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m c s)
 :precision binary64
 (let* ((t_0 (* c (* x_m s)))) (* (/ 1.0 t_0) (/ (cos (* 2.0 x_m)) t_0))))

C

x_m = fabs(x);
double code(double x_m, double c, double s) {
	double t_0 = c * (x_m * s);
	return (1.0 / t_0) * (cos((2.0 * x_m)) / t_0);
}

Fortran

x_m = abs(x)
real(8) function code(x_m, c, s)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: t_0
    t_0 = c * (x_m * s)
    code = (1.0d0 / t_0) * (cos((2.0d0 * x_m)) / t_0)
end function

Java

x_m = Math.abs(x);
public static double code(double x_m, double c, double s) {
	double t_0 = c * (x_m * s);
	return (1.0 / t_0) * (Math.cos((2.0 * x_m)) / t_0);
}

Python

x_m = math.fabs(x)
def code(x_m, c, s):
	t_0 = c * (x_m * s)
	return (1.0 / t_0) * (math.cos((2.0 * x_m)) / t_0)

Julia

x_m = abs(x)
function code(x_m, c, s)
	t_0 = Float64(c * Float64(x_m * s))
	return Float64(Float64(1.0 / t_0) * Float64(cos(Float64(2.0 * x_m)) / t_0))
end

MATLAB

x_m = abs(x);
function tmp = code(x_m, c, s)
	t_0 = c * (x_m * s);
	tmp = (1.0 / t_0) * (cos((2.0 * x_m)) / t_0);
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, c_, s_] := Block[{t$95$0 = N[(c * N[(x$95$m * s), $MachinePrecision]), $MachinePrecision]}, N[(N[(1.0 / t$95$0), $MachinePrecision] * N[(N[Cos[N[(2.0 * x$95$m), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 66.5%

   \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. *-un-lft-identity 66.5%

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

   2. add-sqr-sqrt 66.5%

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

   3. times-frac 66.4%

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

   4. sqrt-prod 66.4%

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

   5. sqrt-pow1 49.5%

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

   6. metadata-eval 49.5%

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

   7. pow1 49.5%

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

   8. *-commutative 49.5%

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

   9. associate-*r* 45.0%

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

   10. unpow2 45.0%

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

   11. pow-prod-down 49.5%

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

   12. sqrt-pow1 47.0%

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

   13. metadata-eval 47.0%

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

   14. pow1 47.0%

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

   15. *-commutative 47.0%

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

4. Applied egg-rr 95.6%

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

5. Final simplification 95.6%

   \[\leadsto \frac{1}{c \cdot \left(x \cdot s\right)} \cdot \frac{\cos \left(2 \cdot x\right)}{c \cdot \left(x \cdot s\right)} \]
6. Add Preprocessing

Alternative 7: 93.7% accurate, 2.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \frac{\frac{\cos \left(2 \cdot x\_m\right)}{c}}{\left(x\_m \cdot s\right) \cdot \left(c \cdot \left(x\_m \cdot s\right)\right)} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m c s)
 :precision binary64
 (/ (/ (cos (* 2.0 x_m)) c) (* (* x_m s) (* c (* x_m s)))))

C

x_m = fabs(x);
double code(double x_m, double c, double s) {
	return (cos((2.0 * x_m)) / c) / ((x_m * s) * (c * (x_m * s)));
}

Fortran

x_m = abs(x)
real(8) function code(x_m, c, s)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    code = (cos((2.0d0 * x_m)) / c) / ((x_m * s) * (c * (x_m * s)))
end function

Java

x_m = Math.abs(x);
public static double code(double x_m, double c, double s) {
	return (Math.cos((2.0 * x_m)) / c) / ((x_m * s) * (c * (x_m * s)));
}

Python

x_m = math.fabs(x)
def code(x_m, c, s):
	return (math.cos((2.0 * x_m)) / c) / ((x_m * s) * (c * (x_m * s)))

Julia

x_m = abs(x)
function code(x_m, c, s)
	return Float64(Float64(cos(Float64(2.0 * x_m)) / c) / Float64(Float64(x_m * s) * Float64(c * Float64(x_m * s))))
end

MATLAB

x_m = abs(x);
function tmp = code(x_m, c, s)
	tmp = (cos((2.0 * x_m)) / c) / ((x_m * s) * (c * (x_m * s)));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, c_, s_] := N[(N[(N[Cos[N[(2.0 * x$95$m), $MachinePrecision]], $MachinePrecision] / c), $MachinePrecision] / N[(N[(x$95$m * s), $MachinePrecision] * N[(c * N[(x$95$m * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 66.5%

   \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. *-un-lft-identity 66.5%

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

   2. add-sqr-sqrt 66.5%

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

   3. times-frac 66.4%

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

   4. sqrt-prod 66.4%

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

   5. sqrt-pow1 49.5%

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

   6. metadata-eval 49.5%

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

   7. pow1 49.5%

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

   8. *-commutative 49.5%

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

   9. associate-*r* 45.0%

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

   10. unpow2 45.0%

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

   11. pow-prod-down 49.5%

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

   12. sqrt-pow1 47.0%

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

   13. metadata-eval 47.0%

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

   14. pow1 47.0%

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

   15. *-commutative 47.0%

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

4. Applied egg-rr 95.6%

   \[\leadsto \color{blue}{\frac{1}{c \cdot \left(x \cdot s\right)} \cdot \frac{\cos \left(2 \cdot x\right)}{c \cdot \left(x \cdot s\right)}} \]
5. Step-by-step derivation

   1. *-commutative 95.6%

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

   2. associate-/r* 95.6%

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

   3. frac-times 92.5%

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

   4. div-inv 92.5%

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

   5. *-commutative 92.5%

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

6. Applied egg-rr 92.5%

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

7. Final simplification 92.5%

   \[\leadsto \frac{\frac{\cos \left(2 \cdot x\right)}{c}}{\left(x \cdot s\right) \cdot \left(c \cdot \left(x \cdot s\right)\right)} \]
8. Add Preprocessing

Alternative 8: 76.1% accurate, 17.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;s \leq 1.2 \cdot 10^{+224}:\\ \;\;\;\;\frac{1}{\left(x\_m \cdot c\right) \cdot \left(s \cdot \left(c \cdot \left(x\_m \cdot s\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(c \cdot s\right) \cdot \left(x\_m \cdot \left(x\_m \cdot \left(c \cdot s\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m c s)
 :precision binary64
 (if (<= s 1.2e+224)
   (/ 1.0 (* (* x_m c) (* s (* c (* x_m s)))))
   (/ 1.0 (* (* c s) (* x_m (* x_m (* c s)))))))

C

x_m = fabs(x);
double code(double x_m, double c, double s) {
	double tmp;
	if (s <= 1.2e+224) {
		tmp = 1.0 / ((x_m * c) * (s * (c * (x_m * s))));
	} else {
		tmp = 1.0 / ((c * s) * (x_m * (x_m * (c * s))));
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m, c, s)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: tmp
    if (s <= 1.2d+224) then
        tmp = 1.0d0 / ((x_m * c) * (s * (c * (x_m * s))))
    else
        tmp = 1.0d0 / ((c * s) * (x_m * (x_m * (c * s))))
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m, double c, double s) {
	double tmp;
	if (s <= 1.2e+224) {
		tmp = 1.0 / ((x_m * c) * (s * (c * (x_m * s))));
	} else {
		tmp = 1.0 / ((c * s) * (x_m * (x_m * (c * s))));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m, c, s):
	tmp = 0
	if s <= 1.2e+224:
		tmp = 1.0 / ((x_m * c) * (s * (c * (x_m * s))))
	else:
		tmp = 1.0 / ((c * s) * (x_m * (x_m * (c * s))))
	return tmp

Julia

x_m = abs(x)
function code(x_m, c, s)
	tmp = 0.0
	if (s <= 1.2e+224)
		tmp = Float64(1.0 / Float64(Float64(x_m * c) * Float64(s * Float64(c * Float64(x_m * s)))));
	else
		tmp = Float64(1.0 / Float64(Float64(c * s) * Float64(x_m * Float64(x_m * Float64(c * s)))));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m, c, s)
	tmp = 0.0;
	if (s <= 1.2e+224)
		tmp = 1.0 / ((x_m * c) * (s * (c * (x_m * s))));
	else
		tmp = 1.0 / ((c * s) * (x_m * (x_m * (c * s))));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, c_, s_] := If[LessEqual[s, 1.2e+224], N[(1.0 / N[(N[(x$95$m * c), $MachinePrecision] * N[(s * N[(c * N[(x$95$m * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(c * s), $MachinePrecision] * N[(x$95$m * N[(x$95$m * N[(c * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if s < 1.2e224

   1. Initial program 67.0%

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

   3. Taylor expanded in x around 0 52.8%

      \[\leadsto \color{blue}{\frac{1}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)}} \]
   4. Step-by-step derivation

      1. associate-/r* 52.7%

         \[\leadsto \color{blue}{\frac{\frac{1}{{c}^{2}}}{{s}^{2} \cdot {x}^{2}}} \]

      2. *-commutative 52.7%

         \[\leadsto \frac{\frac{1}{{c}^{2}}}{\color{blue}{{x}^{2} \cdot {s}^{2}}} \]

      3. unpow2 52.7%

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

      4. unpow2 52.7%

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

      5. swap-sqr 65.1%

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

      6. unpow2 65.1%

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

      7. associate-/r* 65.3%

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

      8. unpow2 65.3%

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

      9. unpow2 65.3%

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

      10. swap-sqr 73.5%

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

      11. unpow2 73.5%

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

      12. *-commutative 73.5%

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

   5. Simplified 73.5%

      \[\leadsto \color{blue}{\frac{1}{{\left(c \cdot \left(s \cdot x\right)\right)}^{2}}} \]
   6. Step-by-step derivation

      1. *-commutative 73.5%

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

      2. *-commutative 73.5%

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

      3. *-commutative 73.5%

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

      4. unpow2 73.5%

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

      5. associate-*r* 73.1%

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

      6. associate-*l* 72.6%

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

   7. Applied egg-rr 72.6%

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

   if 1.2e224 < s

   1. Initial program 61.2%

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

   3. Taylor expanded in x around 0 52.5%

      \[\leadsto \color{blue}{\frac{1}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)}} \]
   4. Step-by-step derivation

      1. associate-/r* 52.5%

         \[\leadsto \color{blue}{\frac{\frac{1}{{c}^{2}}}{{s}^{2} \cdot {x}^{2}}} \]

      2. *-commutative 52.5%

         \[\leadsto \frac{\frac{1}{{c}^{2}}}{\color{blue}{{x}^{2} \cdot {s}^{2}}} \]

      3. unpow2 52.5%

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

      4. unpow2 52.5%

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

      5. swap-sqr 61.4%

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

      6. unpow2 61.4%

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

      7. associate-/r* 61.4%

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

      8. unpow2 61.4%

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

      9. unpow2 61.4%

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

      10. swap-sqr 88.3%

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

      11. unpow2 88.3%

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

      12. *-commutative 88.3%

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

   5. Simplified 88.3%

      \[\leadsto \color{blue}{\frac{1}{{\left(c \cdot \left(s \cdot x\right)\right)}^{2}}} \]
   6. Step-by-step derivation

      1. unpow2 88.3%

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

      2. associate-*r* 88.3%

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

      3. *-commutative 88.3%

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

      4. associate-*l* 88.4%

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

   7. Applied egg-rr 88.4%

      \[\leadsto \frac{1}{\color{blue}{\left(c \cdot s\right) \cdot \left(x \cdot \left(c \cdot \left(x \cdot s\right)\right)\right)}} \]
   8. Step-by-step derivation

      1. pow1 88.4%

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

      2. *-commutative 88.4%

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

   9. Applied egg-rr 88.4%

      \[\leadsto \frac{1}{\left(c \cdot s\right) \cdot \color{blue}{{\left(x \cdot \left(c \cdot \left(s \cdot x\right)\right)\right)}^{1}}} \]
   10. Step-by-step derivation

       1. unpow1 88.4%

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

       2. associate-*r* 88.3%

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

       3. *-commutative 88.3%

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

   11. Simplified 88.3%

       \[\leadsto \frac{1}{\left(c \cdot s\right) \cdot \color{blue}{\left(x \cdot \left(x \cdot \left(c \cdot s\right)\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;s \leq 1.2 \cdot 10^{+224}:\\ \;\;\;\;\frac{1}{\left(x \cdot c\right) \cdot \left(s \cdot \left(c \cdot \left(x \cdot s\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(c \cdot s\right) \cdot \left(x \cdot \left(x \cdot \left(c \cdot s\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 78.5% accurate, 20.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \frac{\frac{1}{c}}{x\_m \cdot s}\\ t\_0 \cdot t\_0 \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m c s)
 :precision binary64
 (let* ((t_0 (/ (/ 1.0 c) (* x_m s)))) (* t_0 t_0)))

C

x_m = fabs(x);
double code(double x_m, double c, double s) {
	double t_0 = (1.0 / c) / (x_m * s);
	return t_0 * t_0;
}

Fortran

x_m = abs(x)
real(8) function code(x_m, c, s)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: t_0
    t_0 = (1.0d0 / c) / (x_m * s)
    code = t_0 * t_0
end function

Java

x_m = Math.abs(x);
public static double code(double x_m, double c, double s) {
	double t_0 = (1.0 / c) / (x_m * s);
	return t_0 * t_0;
}

Python

x_m = math.fabs(x)
def code(x_m, c, s):
	t_0 = (1.0 / c) / (x_m * s)
	return t_0 * t_0

Julia

x_m = abs(x)
function code(x_m, c, s)
	t_0 = Float64(Float64(1.0 / c) / Float64(x_m * s))
	return Float64(t_0 * t_0)
end

MATLAB

x_m = abs(x);
function tmp = code(x_m, c, s)
	t_0 = (1.0 / c) / (x_m * s);
	tmp = t_0 * t_0;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, c_, s_] := Block[{t$95$0 = N[(N[(1.0 / c), $MachinePrecision] / N[(x$95$m * s), $MachinePrecision]), $MachinePrecision]}, N[(t$95$0 * t$95$0), $MachinePrecision]]

Derivation

1. Initial program 66.5%

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

3. Taylor expanded in x around 0 52.8%

   \[\leadsto \color{blue}{\frac{1}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)}} \]
4. Step-by-step derivation

   1. associate-/r* 52.6%

      \[\leadsto \color{blue}{\frac{\frac{1}{{c}^{2}}}{{s}^{2} \cdot {x}^{2}}} \]

   2. *-commutative 52.6%

      \[\leadsto \frac{\frac{1}{{c}^{2}}}{\color{blue}{{x}^{2} \cdot {s}^{2}}} \]

   3. unpow2 52.6%

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

   4. unpow2 52.6%

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

   5. swap-sqr 64.8%

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

   6. unpow2 64.8%

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

   7. associate-/r* 65.0%

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

   8. unpow2 65.0%

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

   9. unpow2 65.0%

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

   10. swap-sqr 74.9%

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

   11. unpow2 74.9%

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

   12. *-commutative 74.9%

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

5. Simplified 74.9%

   \[\leadsto \color{blue}{\frac{1}{{\left(c \cdot \left(s \cdot x\right)\right)}^{2}}} \]
6. Step-by-step derivation

   1. pow-flip 75.1%

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

   2. *-commutative 75.1%

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

   3. pow-flip 74.9%

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

   4. unpow2 74.9%

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

   5. metadata-eval 74.9%

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

   6. frac-times 75.1%

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

   7. associate-/r* 75.1%

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

   8. associate-/r* 75.1%

      \[\leadsto \frac{\frac{1}{c}}{x \cdot s} \cdot \color{blue}{\frac{\frac{1}{c}}{x \cdot s}} \]

7. Applied egg-rr 75.1%

   \[\leadsto \color{blue}{\frac{\frac{1}{c}}{x \cdot s} \cdot \frac{\frac{1}{c}}{x \cdot s}} \]

8. Final simplification 75.1%

   \[\leadsto \frac{\frac{1}{c}}{x \cdot s} \cdot \frac{\frac{1}{c}}{x \cdot s} \]
9. Add Preprocessing

Alternative 10: 75.5% accurate, 24.1× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m c s)
 :precision binary64
 (/ 1.0 (* (* c s) (* x_m (* c (* x_m s))))))

C

x_m = fabs(x);
double code(double x_m, double c, double s) {
	return 1.0 / ((c * s) * (x_m * (c * (x_m * s))));
}

Fortran

x_m = abs(x)
real(8) function code(x_m, c, s)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    code = 1.0d0 / ((c * s) * (x_m * (c * (x_m * s))))
end function

Java

x_m = Math.abs(x);
public static double code(double x_m, double c, double s) {
	return 1.0 / ((c * s) * (x_m * (c * (x_m * s))));
}

Python

x_m = math.fabs(x)
def code(x_m, c, s):
	return 1.0 / ((c * s) * (x_m * (c * (x_m * s))))

Julia

x_m = abs(x)
function code(x_m, c, s)
	return Float64(1.0 / Float64(Float64(c * s) * Float64(x_m * Float64(c * Float64(x_m * s)))))
end

MATLAB

x_m = abs(x);
function tmp = code(x_m, c, s)
	tmp = 1.0 / ((c * s) * (x_m * (c * (x_m * s))));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, c_, s_] := N[(1.0 / N[(N[(c * s), $MachinePrecision] * N[(x$95$m * N[(c * N[(x$95$m * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 66.5%

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

3. Taylor expanded in x around 0 52.8%

   \[\leadsto \color{blue}{\frac{1}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)}} \]
4. Step-by-step derivation

   1. associate-/r* 52.6%

      \[\leadsto \color{blue}{\frac{\frac{1}{{c}^{2}}}{{s}^{2} \cdot {x}^{2}}} \]

   2. *-commutative 52.6%

      \[\leadsto \frac{\frac{1}{{c}^{2}}}{\color{blue}{{x}^{2} \cdot {s}^{2}}} \]

   3. unpow2 52.6%

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

   4. unpow2 52.6%

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

   5. swap-sqr 64.8%

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

   6. unpow2 64.8%

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

   7. associate-/r* 65.0%

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

   8. unpow2 65.0%

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

   9. unpow2 65.0%

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

   10. swap-sqr 74.9%

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

   11. unpow2 74.9%

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

   12. *-commutative 74.9%

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

5. Simplified 74.9%

   \[\leadsto \color{blue}{\frac{1}{{\left(c \cdot \left(s \cdot x\right)\right)}^{2}}} \]
6. Step-by-step derivation

   1. unpow2 74.9%

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

   2. associate-*r* 74.1%

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

   3. *-commutative 74.1%

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

   4. associate-*l* 72.4%

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

7. Applied egg-rr 72.4%

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

8. Final simplification 72.4%

   \[\leadsto \frac{1}{\left(c \cdot s\right) \cdot \left(x \cdot \left(c \cdot \left(x \cdot s\right)\right)\right)} \]
9. Add Preprocessing

Alternative 11: 76.1% accurate, 24.1× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m c s)
 :precision binary64
 (/ 1.0 (* (* c s) (* x_m (* x_m (* c s))))))

C

x_m = fabs(x);
double code(double x_m, double c, double s) {
	return 1.0 / ((c * s) * (x_m * (x_m * (c * s))));
}

Fortran

x_m = abs(x)
real(8) function code(x_m, c, s)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    code = 1.0d0 / ((c * s) * (x_m * (x_m * (c * s))))
end function

Java

x_m = Math.abs(x);
public static double code(double x_m, double c, double s) {
	return 1.0 / ((c * s) * (x_m * (x_m * (c * s))));
}

Python

x_m = math.fabs(x)
def code(x_m, c, s):
	return 1.0 / ((c * s) * (x_m * (x_m * (c * s))))

Julia

x_m = abs(x)
function code(x_m, c, s)
	return Float64(1.0 / Float64(Float64(c * s) * Float64(x_m * Float64(x_m * Float64(c * s)))))
end

MATLAB

x_m = abs(x);
function tmp = code(x_m, c, s)
	tmp = 1.0 / ((c * s) * (x_m * (x_m * (c * s))));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, c_, s_] := N[(1.0 / N[(N[(c * s), $MachinePrecision] * N[(x$95$m * N[(x$95$m * N[(c * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 66.5%

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

3. Taylor expanded in x around 0 52.8%

   \[\leadsto \color{blue}{\frac{1}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)}} \]
4. Step-by-step derivation

   1. associate-/r* 52.6%

      \[\leadsto \color{blue}{\frac{\frac{1}{{c}^{2}}}{{s}^{2} \cdot {x}^{2}}} \]

   2. *-commutative 52.6%

      \[\leadsto \frac{\frac{1}{{c}^{2}}}{\color{blue}{{x}^{2} \cdot {s}^{2}}} \]

   3. unpow2 52.6%

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

   4. unpow2 52.6%

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

   5. swap-sqr 64.8%

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

   6. unpow2 64.8%

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

   7. associate-/r* 65.0%

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

   8. unpow2 65.0%

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

   9. unpow2 65.0%

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

   10. swap-sqr 74.9%

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

   11. unpow2 74.9%

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

   12. *-commutative 74.9%

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

5. Simplified 74.9%

   \[\leadsto \color{blue}{\frac{1}{{\left(c \cdot \left(s \cdot x\right)\right)}^{2}}} \]
6. Step-by-step derivation

   1. unpow2 74.9%

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

   2. associate-*r* 74.1%

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

   3. *-commutative 74.1%

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

   4. associate-*l* 72.4%

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

7. Applied egg-rr 72.4%

   \[\leadsto \frac{1}{\color{blue}{\left(c \cdot s\right) \cdot \left(x \cdot \left(c \cdot \left(x \cdot s\right)\right)\right)}} \]
8. Step-by-step derivation

   1. pow1 72.4%

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

   2. *-commutative 72.4%

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

9. Applied egg-rr 72.4%

   \[\leadsto \frac{1}{\left(c \cdot s\right) \cdot \color{blue}{{\left(x \cdot \left(c \cdot \left(s \cdot x\right)\right)\right)}^{1}}} \]
10. Step-by-step derivation

    1. unpow1 72.4%

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

    2. associate-*r* 73.9%

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

    3. *-commutative 73.9%

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

11. Simplified 73.9%

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

12. Final simplification 73.9%

    \[\leadsto \frac{1}{\left(c \cdot s\right) \cdot \left(x \cdot \left(x \cdot \left(c \cdot s\right)\right)\right)} \]
13. Add Preprocessing

Alternative 12: 76.4% accurate, 24.1× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m c s)
 :precision binary64
 (/ 1.0 (* (* x_m s) (* c (* c (* x_m s))))))

C

x_m = fabs(x);
double code(double x_m, double c, double s) {
	return 1.0 / ((x_m * s) * (c * (c * (x_m * s))));
}

Fortran

x_m = abs(x)
real(8) function code(x_m, c, s)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    code = 1.0d0 / ((x_m * s) * (c * (c * (x_m * s))))
end function

Java

x_m = Math.abs(x);
public static double code(double x_m, double c, double s) {
	return 1.0 / ((x_m * s) * (c * (c * (x_m * s))));
}

Python

x_m = math.fabs(x)
def code(x_m, c, s):
	return 1.0 / ((x_m * s) * (c * (c * (x_m * s))))

Julia

x_m = abs(x)
function code(x_m, c, s)
	return Float64(1.0 / Float64(Float64(x_m * s) * Float64(c * Float64(c * Float64(x_m * s)))))
end

MATLAB

x_m = abs(x);
function tmp = code(x_m, c, s)
	tmp = 1.0 / ((x_m * s) * (c * (c * (x_m * s))));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, c_, s_] := N[(1.0 / N[(N[(x$95$m * s), $MachinePrecision] * N[(c * N[(c * N[(x$95$m * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 66.5%

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

3. Taylor expanded in x around 0 52.8%

   \[\leadsto \color{blue}{\frac{1}{{c}^{2} \cdot \left({s}^{2} \cdot {x}^{2}\right)}} \]
4. Step-by-step derivation

   1. associate-/r* 52.6%

      \[\leadsto \color{blue}{\frac{\frac{1}{{c}^{2}}}{{s}^{2} \cdot {x}^{2}}} \]

   2. *-commutative 52.6%

      \[\leadsto \frac{\frac{1}{{c}^{2}}}{\color{blue}{{x}^{2} \cdot {s}^{2}}} \]

   3. unpow2 52.6%

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

   4. unpow2 52.6%

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

   5. swap-sqr 64.8%

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

   6. unpow2 64.8%

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

   7. associate-/r* 65.0%

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

   8. unpow2 65.0%

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

   9. unpow2 65.0%

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

   10. swap-sqr 74.9%

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

   11. unpow2 74.9%

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

   12. *-commutative 74.9%

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

5. Simplified 74.9%

   \[\leadsto \color{blue}{\frac{1}{{\left(c \cdot \left(s \cdot x\right)\right)}^{2}}} \]
6. Step-by-step derivation

   1. *-commutative 74.9%

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

   2. *-commutative 74.9%

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

   3. *-commutative 74.9%

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

   4. unpow2 74.9%

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

   5. associate-*r* 74.2%

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

7. Applied egg-rr 74.2%

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

8. Final simplification 74.2%

   \[\leadsto \frac{1}{\left(x \cdot s\right) \cdot \left(c \cdot \left(c \cdot \left(x \cdot s\right)\right)\right)} \]
9. Add Preprocessing

Alternative 13: 78.5% accurate, 24.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := c \cdot \left(x\_m \cdot s\right)\\ \frac{\frac{1}{t\_0}}{t\_0} \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m c s)
 :precision binary64
 (let* ((t_0 (* c (* x_m s)))) (/ (/ 1.0 t_0) t_0)))

C

x_m = fabs(x);
double code(double x_m, double c, double s) {
	double t_0 = c * (x_m * s);
	return (1.0 / t_0) / t_0;
}

Fortran

x_m = abs(x)
real(8) function code(x_m, c, s)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: t_0
    t_0 = c * (x_m * s)
    code = (1.0d0 / t_0) / t_0
end function

Java

x_m = Math.abs(x);
public static double code(double x_m, double c, double s) {
	double t_0 = c * (x_m * s);
	return (1.0 / t_0) / t_0;
}

Python

x_m = math.fabs(x)
def code(x_m, c, s):
	t_0 = c * (x_m * s)
	return (1.0 / t_0) / t_0

Julia

x_m = abs(x)
function code(x_m, c, s)
	t_0 = Float64(c * Float64(x_m * s))
	return Float64(Float64(1.0 / t_0) / t_0)
end

MATLAB

x_m = abs(x);
function tmp = code(x_m, c, s)
	t_0 = c * (x_m * s);
	tmp = (1.0 / t_0) / t_0;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, c_, s_] := Block[{t$95$0 = N[(c * N[(x$95$m * s), $MachinePrecision]), $MachinePrecision]}, N[(N[(1.0 / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]]

Derivation

1. Initial program 66.5%

   \[\frac{\cos \left(2 \cdot x\right)}{{c}^{2} \cdot \left(\left(x \cdot {s}^{2}\right) \cdot x\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. *-un-lft-identity 66.5%

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

   2. add-sqr-sqrt 66.5%

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

   3. times-frac 66.4%

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

   4. sqrt-prod 66.4%

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

   5. sqrt-pow1 49.5%

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

   6. metadata-eval 49.5%

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

   7. pow1 49.5%

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

   8. *-commutative 49.5%

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

   9. associate-*r* 45.0%

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

   10. unpow2 45.0%

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

   11. pow-prod-down 49.5%

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

   12. sqrt-pow1 47.0%

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

   13. metadata-eval 47.0%

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

   14. pow1 47.0%

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

   15. *-commutative 47.0%

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

4. Applied egg-rr 95.6%

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

5. Taylor expanded in x around 0 75.1%

   \[\leadsto \frac{1}{c \cdot \left(x \cdot s\right)} \cdot \color{blue}{\frac{1}{c \cdot \left(s \cdot x\right)}} \]
6. Step-by-step derivation

   1. associate-/r* 75.1%

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

7. Simplified 75.1%

   \[\leadsto \frac{1}{c \cdot \left(x \cdot s\right)} \cdot \color{blue}{\frac{\frac{1}{c}}{s \cdot x}} \]
8. Step-by-step derivation

   1. associate-*l/ 75.2%

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

   2. *-un-lft-identity 75.2%

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

   3. *-commutative 75.2%

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

   4. associate-/r* 75.1%

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

   5. *-commutative 75.1%

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

   6. *-commutative 75.1%

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

9. Applied egg-rr 75.1%

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

10. Final simplification 75.1%

    \[\leadsto \frac{\frac{1}{c \cdot \left(x \cdot s\right)}}{c \cdot \left(x \cdot s\right)} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024095 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))