mixedcos

Percentage Accurate: 67.3% → 97.4%

Time: 13.0s

Alternatives: 10

Speedup: 24.1×

• 31.8% 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: 67.3% 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: 97.4% accurate, 2.6× speedup

Math

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

FPCore

(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (/ (/ (/ 1.0 c) s) x))) (* t_0 (* t_0 (cos (* x 2.0))))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 66.4%

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

   1. associate-/r* 66.0%

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

   2. cos-neg 66.0%

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

   3. distribute-rgt-neg-out 66.0%

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

   4. distribute-rgt-neg-out 66.0%

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

   5. *-commutative 66.0%

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

   6. distribute-rgt-neg-in 66.0%

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

   7. metadata-eval 66.0%

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

   8. *-commutative 66.0%

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

   9. associate-*l* 62.1%

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

   10. unpow2 62.1%

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

3. Simplified 62.1%

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

5. Taylor expanded in x around inf 62.5%

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

   1. associate-/r* 62.1%

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

   2. *-commutative 62.1%

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

   3. unpow2 62.1%

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

   4. unpow2 62.1%

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

   5. swap-sqr 76.3%

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

   6. unpow2 76.3%

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

   7. associate-/r* 76.7%

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

   8. *-commutative 76.7%

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

   9. unpow2 76.7%

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

   10. unpow2 76.7%

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

   11. swap-sqr 96.1%

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

   12. unpow2 96.1%

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

   13. *-commutative 96.1%

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

7. Simplified 96.1%

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

   1. unpow2 96.1%

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

9. Applied egg-rr 96.1%

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

    1. clear-num 96.1%

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

    2. associate-/r/ 96.1%

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

    3. add-sqr-sqrt 96.0%

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

    4. add-sqr-sqrt 44.2%

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

    5. sqrt-unprod 74.1%

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

    6. swap-sqr 74.1%

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

    7. metadata-eval 74.1%

       \[\leadsto \left(\sqrt{\frac{1}{\left(c \cdot \left(s \cdot x\right)\right) \cdot \left(c \cdot \left(s \cdot x\right)\right)}} \cdot \sqrt{\frac{1}{\left(c \cdot \left(s \cdot x\right)\right) \cdot \left(c \cdot \left(s \cdot x\right)\right)}}\right) \cdot \cos \left(\sqrt{\left(x \cdot x\right) \cdot \color{blue}{4}}\right) \]

    8. metadata-eval 74.1%

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

    9. swap-sqr 74.1%

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

    10. sqrt-unprod 45.4%

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

    11. add-sqr-sqrt 96.0%

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

    12. associate-*l* 96.0%

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

11. Applied egg-rr 98.4%

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

12. Final simplification 98.4%

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

Alternative 2: 86.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{1}{c}}{s \cdot x}\\ \mathbf{if}\;x \leq 1.75 \cdot 10^{-208}:\\ \;\;\;\;t\_0 \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot -2\right)}{\left(c \cdot s\right) \cdot \left(x \cdot \left(c \cdot \left(s \cdot x\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (/ (/ 1.0 c) (* s x))))
   (if (<= x 1.75e-208)
     (* t_0 t_0)
     (/ (cos (* x -2.0)) (* (* c s) (* x (* c (* s x))))))))

C

double code(double x, double c, double s) {
	double t_0 = (1.0 / c) / (s * x);
	double tmp;
	if (x <= 1.75e-208) {
		tmp = t_0 * t_0;
	} else {
		tmp = cos((x * -2.0)) / ((c * s) * (x * (c * (s * x))));
	}
	return tmp;
}

Fortran

real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 / c) / (s * x)
    if (x <= 1.75d-208) then
        tmp = t_0 * t_0
    else
        tmp = cos((x * (-2.0d0))) / ((c * s) * (x * (c * (s * x))))
    end if
    code = tmp
end function

Java

public static double code(double x, double c, double s) {
	double t_0 = (1.0 / c) / (s * x);
	double tmp;
	if (x <= 1.75e-208) {
		tmp = t_0 * t_0;
	} else {
		tmp = Math.cos((x * -2.0)) / ((c * s) * (x * (c * (s * x))));
	}
	return tmp;
}

Python

def code(x, c, s):
	t_0 = (1.0 / c) / (s * x)
	tmp = 0
	if x <= 1.75e-208:
		tmp = t_0 * t_0
	else:
		tmp = math.cos((x * -2.0)) / ((c * s) * (x * (c * (s * x))))
	return tmp

Julia

function code(x, c, s)
	t_0 = Float64(Float64(1.0 / c) / Float64(s * x))
	tmp = 0.0
	if (x <= 1.75e-208)
		tmp = Float64(t_0 * t_0);
	else
		tmp = Float64(cos(Float64(x * -2.0)) / Float64(Float64(c * s) * Float64(x * Float64(c * Float64(s * x)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	t_0 = (1.0 / c) / (s * x);
	tmp = 0.0;
	if (x <= 1.75e-208)
		tmp = t_0 * t_0;
	else
		tmp = cos((x * -2.0)) / ((c * s) * (x * (c * (s * x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := Block[{t$95$0 = N[(N[(1.0 / c), $MachinePrecision] / N[(s * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 1.75e-208], N[(t$95$0 * t$95$0), $MachinePrecision], N[(N[Cos[N[(x * -2.0), $MachinePrecision]], $MachinePrecision] / N[(N[(c * s), $MachinePrecision] * N[(x * N[(c * N[(s * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.74999999999999996e-208

   1. Initial program 64.7%

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

   3. Taylor expanded in x around 0 57.2%

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

      1. associate-/r* 57.2%

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

      2. *-commutative 57.2%

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

      3. unpow2 57.2%

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

      4. unpow2 57.2%

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

      5. swap-sqr 65.8%

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

      6. unpow2 65.8%

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

      7. associate-/r* 65.8%

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

      8. unpow2 65.8%

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

      9. unpow2 65.8%

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

      10. swap-sqr 80.7%

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

      11. unpow2 80.7%

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

      12. *-commutative 80.7%

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

   5. Simplified 80.7%

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

      1. unpow2 80.7%

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

      2. associate-*r* 78.8%

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

      3. associate-*l* 73.3%

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

   7. Applied egg-rr 73.3%

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

      1. inv-pow 73.3%

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

      2. associate-*r* 78.8%

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

      3. associate-*r* 80.7%

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

      4. unpow-prod-down 80.6%

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

      5. metadata-eval 80.6%

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

      6. metadata-eval 80.6%

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

      7. sqr-pow 80.7%

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

      8. unpow-prod-down 65.9%

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

      9. metadata-eval 65.9%

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

      10. pow-prod-up 65.9%

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

      11. inv-pow 65.9%

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

      12. inv-pow 65.9%

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

      13. metadata-eval 65.9%

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

      14. pow-flip 65.9%

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

      15. div-inv 65.8%

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

      16. unpow2 65.8%

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

      17. times-frac 80.7%

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

   9. Applied egg-rr 80.7%

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

   if 1.74999999999999996e-208 < x

   1. Initial program 68.7%

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

      1. associate-/r* 67.8%

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

      2. cos-neg 67.8%

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

      3. distribute-rgt-neg-out 67.8%

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

      4. distribute-rgt-neg-out 67.8%

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

      5. *-commutative 67.8%

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

      6. distribute-rgt-neg-in 67.8%

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

      7. metadata-eval 67.8%

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

      8. *-commutative 67.8%

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

      9. associate-*l* 63.0%

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

      10. unpow2 63.0%

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

   3. Simplified 63.0%

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

   5. Taylor expanded in x around inf 63.9%

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

      1. associate-/r* 63.0%

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

      2. *-commutative 63.0%

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

      3. unpow2 63.0%

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

      4. unpow2 63.0%

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

      5. swap-sqr 77.7%

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

      6. unpow2 77.7%

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

      7. associate-/r* 78.7%

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

      8. *-commutative 78.7%

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

      9. unpow2 78.7%

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

      10. unpow2 78.7%

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

      11. swap-sqr 97.0%

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

      12. unpow2 97.0%

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

      13. *-commutative 97.0%

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

   7. Simplified 97.0%

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

      1. unpow2 79.0%

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

      2. associate-*r* 78.9%

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

      3. associate-*l* 78.8%

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

   9. Applied egg-rr 94.4%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.75 \cdot 10^{-208}:\\ \;\;\;\;\frac{\frac{1}{c}}{s \cdot x} \cdot \frac{\frac{1}{c}}{s \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(x \cdot -2\right)}{\left(c \cdot s\right) \cdot \left(x \cdot \left(c \cdot \left(s \cdot x\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1}{c}}{s \cdot x} \cdot \frac{\cos \left(x \cdot -2\right)}{c \cdot \left(s \cdot x\right)} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (* (/ (/ 1.0 c) (* s x)) (/ (cos (* x -2.0)) (* c (* s x)))))

C

double code(double x, double c, double s) {
	return ((1.0 / c) / (s * x)) * (cos((x * -2.0)) / (c * (s * 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 = ((1.0d0 / c) / (s * x)) * (cos((x * (-2.0d0))) / (c * (s * x)))
end function

Java

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

Python

def code(x, c, s):
	return ((1.0 / c) / (s * x)) * (math.cos((x * -2.0)) / (c * (s * x)))

Julia

function code(x, c, s)
	return Float64(Float64(Float64(1.0 / c) / Float64(s * x)) * Float64(cos(Float64(x * -2.0)) / Float64(c * Float64(s * x))))
end

MATLAB

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

Wolfram

code[x_, c_, s_] := N[(N[(N[(1.0 / c), $MachinePrecision] / N[(s * x), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[N[(x * -2.0), $MachinePrecision]], $MachinePrecision] / N[(c * N[(s * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 66.4%

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

   1. associate-/r* 66.0%

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

   2. cos-neg 66.0%

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

   3. distribute-rgt-neg-out 66.0%

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

   4. distribute-rgt-neg-out 66.0%

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

   5. *-commutative 66.0%

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

   6. distribute-rgt-neg-in 66.0%

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

   7. metadata-eval 66.0%

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

   8. *-commutative 66.0%

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

   9. associate-*l* 62.1%

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

   10. unpow2 62.1%

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

3. Simplified 62.1%

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

5. Taylor expanded in x around inf 62.5%

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

   1. associate-/r* 62.1%

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

   2. *-commutative 62.1%

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

   3. unpow2 62.1%

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

   4. unpow2 62.1%

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

   5. swap-sqr 76.3%

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

   6. unpow2 76.3%

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

   7. associate-/r* 76.7%

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

   8. *-commutative 76.7%

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

   9. unpow2 76.7%

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

   10. unpow2 76.7%

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

   11. swap-sqr 96.1%

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

   12. unpow2 96.1%

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

   13. *-commutative 96.1%

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

7. Simplified 96.1%

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

   1. associate-*r* 98.1%

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

   2. unpow-prod-down 76.5%

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

   3. pow2 76.5%

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

   4. associate-*l* 84.9%

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

   5. add-sqr-sqrt 40.9%

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

   6. associate-*l* 40.9%

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

   7. sqrt-prod 40.9%

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

   8. sqrt-pow1 27.4%

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

   9. metadata-eval 27.4%

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

   10. pow1 27.4%

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

   11. sqrt-prod 27.4%

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

   12. sqrt-pow1 49.0%

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

   13. metadata-eval 49.0%

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

   14. pow1 49.0%

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

9. Applied egg-rr 49.0%

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

11. Applied egg-rr 96.1%

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

12. Final simplification 96.1%

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

Alternative 4: 97.0% accurate, 2.7× speedup

Math

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

FPCore

(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (* c (* s x)))) (/ 1.0 (* t_0 (/ t_0 (cos (* x 2.0)))))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 66.4%

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

4. Applied egg-rr 83.9%

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

   1. *-commutative 83.9%

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

   2. div-inv 83.8%

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

   3. associate-*r* 76.3%

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

   4. inv-pow 76.3%

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

   5. inv-pow 76.3%

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

   6. pow-sqr 76.3%

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

   7. metadata-eval 76.3%

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

   8. associate-*l* 76.3%

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

   9. *-commutative 76.3%

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

   10. unpow-prod-down 96.1%

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

   11. *-commutative 96.1%

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

   12. metadata-eval 96.1%

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

   13. pow-flip 96.1%

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

   14. associate-*r* 98.1%

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

   15. unpow-prod-down 76.5%

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

   16. pow2 76.5%

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

   17. associate-*l* 84.9%

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

   18. div-inv 84.9%

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

6. Applied egg-rr 96.1%

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

   1. pow2 96.1%

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

   2. associate-/l* 96.1%

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

8. Applied egg-rr 96.1%

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

9. Final simplification 96.1%

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

Alternative 5: 97.0% accurate, 2.7× speedup

Math

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

FPCore

(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (* c (* s x)))) (/ (cos (* x -2.0)) (* t_0 t_0))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, c_, s_] := Block[{t$95$0 = N[(c * N[(s * x), $MachinePrecision]), $MachinePrecision]}, N[(N[Cos[N[(x * -2.0), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 66.4%

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

   1. associate-/r* 66.0%

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

   2. cos-neg 66.0%

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

   3. distribute-rgt-neg-out 66.0%

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

   4. distribute-rgt-neg-out 66.0%

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

   5. *-commutative 66.0%

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

   6. distribute-rgt-neg-in 66.0%

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

   7. metadata-eval 66.0%

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

   8. *-commutative 66.0%

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

   9. associate-*l* 62.1%

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

   10. unpow2 62.1%

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

3. Simplified 62.1%

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

5. Taylor expanded in x around inf 62.5%

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

   1. associate-/r* 62.1%

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

   2. *-commutative 62.1%

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

   3. unpow2 62.1%

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

   4. unpow2 62.1%

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

   5. swap-sqr 76.3%

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

   6. unpow2 76.3%

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

   7. associate-/r* 76.7%

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

   8. *-commutative 76.7%

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

   9. unpow2 76.7%

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

   10. unpow2 76.7%

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

   11. swap-sqr 96.1%

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

   12. unpow2 96.1%

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

   13. *-commutative 96.1%

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

7. Simplified 96.1%

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

   1. unpow2 96.1%

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

9. Applied egg-rr 96.1%

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

10. Final simplification 96.1%

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

Alternative 6: 78.8% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    code = (c * (s * x)) ** (-2.0d0)
end function

Java

public static double code(double x, double c, double s) {
	return Math.pow((c * (s * x)), -2.0);
}

Python

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

Julia

function code(x, c, s)
	return Float64(c * Float64(s * x)) ^ -2.0
end

MATLAB

function tmp = code(x, c, s)
	tmp = (c * (s * x)) ^ -2.0;
end

Wolfram

code[x_, c_, s_] := N[Power[N[(c * N[(s * x), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]

Derivation

1. Initial program 66.4%

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

3. Taylor expanded in x around 0 56.7%

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

   1. associate-/r* 56.3%

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

   2. *-commutative 56.3%

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

   3. unpow2 56.3%

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

   4. unpow2 56.3%

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

   5. swap-sqr 66.0%

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

   6. unpow2 66.0%

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

   7. associate-/r* 66.4%

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

   8. unpow2 66.4%

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

   9. unpow2 66.4%

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

   10. swap-sqr 80.0%

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

   11. unpow2 80.0%

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

   12. *-commutative 80.0%

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

5. Simplified 80.0%

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

   1. *-un-lft-identity 80.0%

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

   2. pow-flip 80.0%

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

   3. metadata-eval 80.0%

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

7. Applied egg-rr 80.0%

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

   1. *-lft-identity 80.0%

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

9. Simplified 80.0%

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

10. Final simplification 80.0%

    \[\leadsto {\left(c \cdot \left(s \cdot x\right)\right)}^{-2} \]
11. Add Preprocessing

Alternative 7: 76.0% accurate, 17.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.2 \cdot 10^{-210}:\\ \;\;\;\;\frac{1}{x \cdot \left(c \cdot \left(\left(s \cdot x\right) \cdot \left(c \cdot s\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(c \cdot s\right) \cdot \left(x \cdot \left(c \cdot \left(s \cdot x\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x c s)
 :precision binary64
 (if (<= x 2.2e-210)
   (/ 1.0 (* x (* c (* (* s x) (* c s)))))
   (/ 1.0 (* (* c s) (* x (* c (* s x)))))))

C

double code(double x, double c, double s) {
	double tmp;
	if (x <= 2.2e-210) {
		tmp = 1.0 / (x * (c * ((s * x) * (c * s))));
	} else {
		tmp = 1.0 / ((c * s) * (x * (c * (s * x))));
	}
	return tmp;
}

Fortran

real(8) function code(x, c, s)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s
    real(8) :: tmp
    if (x <= 2.2d-210) then
        tmp = 1.0d0 / (x * (c * ((s * x) * (c * s))))
    else
        tmp = 1.0d0 / ((c * s) * (x * (c * (s * x))))
    end if
    code = tmp
end function

Java

public static double code(double x, double c, double s) {
	double tmp;
	if (x <= 2.2e-210) {
		tmp = 1.0 / (x * (c * ((s * x) * (c * s))));
	} else {
		tmp = 1.0 / ((c * s) * (x * (c * (s * x))));
	}
	return tmp;
}

Python

def code(x, c, s):
	tmp = 0
	if x <= 2.2e-210:
		tmp = 1.0 / (x * (c * ((s * x) * (c * s))))
	else:
		tmp = 1.0 / ((c * s) * (x * (c * (s * x))))
	return tmp

Julia

function code(x, c, s)
	tmp = 0.0
	if (x <= 2.2e-210)
		tmp = Float64(1.0 / Float64(x * Float64(c * Float64(Float64(s * x) * Float64(c * s)))));
	else
		tmp = Float64(1.0 / Float64(Float64(c * s) * Float64(x * Float64(c * Float64(s * x)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, c, s)
	tmp = 0.0;
	if (x <= 2.2e-210)
		tmp = 1.0 / (x * (c * ((s * x) * (c * s))));
	else
		tmp = 1.0 / ((c * s) * (x * (c * (s * x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, c_, s_] := If[LessEqual[x, 2.2e-210], N[(1.0 / N[(x * N[(c * N[(N[(s * x), $MachinePrecision] * N[(c * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(c * s), $MachinePrecision] * N[(x * N[(c * N[(s * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.19999999999999989e-210

   1. Initial program 64.7%

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

   4. Applied egg-rr 82.1%

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

   5. Taylor expanded in x around 0 61.1%

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

      1. associate-/r* 61.1%

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

      2. unpow2 61.1%

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

      3. unpow2 61.1%

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

      4. swap-sqr 71.2%

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

      5. unpow2 71.2%

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

   7. Simplified 71.2%

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

      1. div-inv 71.2%

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

      2. associate-*r* 65.9%

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

      3. inv-pow 65.9%

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

      4. inv-pow 65.9%

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

      5. pow-prod-up 65.9%

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

      6. metadata-eval 65.9%

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

      7. pow-flip 65.9%

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

      8. metadata-eval 65.9%

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

      9. unpow-prod-down 80.7%

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

      10. sqr-pow 80.6%

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

      11. metadata-eval 80.6%

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

      12. metadata-eval 80.6%

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

      13. unpow-prod-down 80.7%

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

      14. associate-*r* 78.8%

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

      15. associate-*r* 73.3%

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

      16. inv-pow 73.3%

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

      17. associate-/r* 73.2%

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

      18. clear-num 73.3%

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

      19. associate-*r* 73.0%

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

   9. Applied egg-rr 73.0%

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

       1. associate-/l* 74.8%

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

       2. associate-*l* 76.4%

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

       3. div-inv 76.4%

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

       4. remove-double-div 76.4%

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

   11. Applied egg-rr 76.4%

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

   if 2.19999999999999989e-210 < x

   1. Initial program 68.7%

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

   3. Taylor expanded in x around 0 55.9%

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

      1. associate-/r* 54.9%

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

      2. *-commutative 54.9%

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

      3. unpow2 54.9%

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

      4. unpow2 54.9%

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

      5. swap-sqr 66.4%

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

      6. unpow2 66.4%

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

      7. associate-/r* 67.3%

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

      8. unpow2 67.3%

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

      9. unpow2 67.3%

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

      10. swap-sqr 79.0%

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

      11. unpow2 79.0%

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

      12. *-commutative 79.0%

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

   5. Simplified 79.0%

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

      1. unpow2 79.0%

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

      2. associate-*r* 78.9%

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

      3. associate-*l* 78.8%

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

   7. Applied egg-rr 78.8%

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

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.2 \cdot 10^{-210}:\\ \;\;\;\;\frac{1}{x \cdot \left(c \cdot \left(\left(s \cdot x\right) \cdot \left(c \cdot s\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(c \cdot s\right) \cdot \left(x \cdot \left(c \cdot \left(s \cdot x\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 75.2% accurate, 24.1× speedup

Math

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

FPCore

(FPCore (x c s) :precision binary64 (/ 1.0 (* c (* x (* (* s x) (* c s))))))

C

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

Fortran

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

Java

public static double code(double x, double c, double s) {
	return 1.0 / (c * (x * ((s * x) * (c * s))));
}

Python

def code(x, c, s):
	return 1.0 / (c * (x * ((s * x) * (c * s))))

Julia

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

MATLAB

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

Wolfram

code[x_, c_, s_] := N[(1.0 / N[(c * N[(x * N[(N[(s * x), $MachinePrecision] * N[(c * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 66.4%

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

4. Applied egg-rr 83.9%

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

5. Taylor expanded in x around 0 61.3%

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

   1. associate-/r* 61.3%

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

   2. unpow2 61.3%

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

   3. unpow2 61.3%

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

   4. swap-sqr 72.7%

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

   5. unpow2 72.7%

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

7. Simplified 72.7%

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

   1. div-inv 72.7%

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

   2. associate-*r* 66.1%

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

   3. inv-pow 66.1%

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

   4. inv-pow 66.1%

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

   5. pow-prod-up 66.1%

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

   6. metadata-eval 66.1%

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

   7. pow-flip 66.1%

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

   8. metadata-eval 66.1%

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

   9. unpow-prod-down 80.0%

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

   10. sqr-pow 79.9%

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

   11. metadata-eval 79.9%

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

   12. metadata-eval 79.9%

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

   13. unpow-prod-down 80.0%

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

   14. associate-*r* 78.9%

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

   15. associate-*r* 75.6%

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

   16. inv-pow 75.6%

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

   17. associate-/r* 75.6%

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

   18. clear-num 75.5%

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

   19. associate-*r* 74.6%

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

9. Applied egg-rr 74.6%

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

    1. associate-/l* 73.9%

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

    2. *-commutative 73.9%

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

    3. associate-*l* 73.8%

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

    4. div-inv 73.8%

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

    5. remove-double-div 73.8%

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

11. Applied egg-rr 73.8%

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

12. Final simplification 73.8%

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

Alternative 9: 75.2% accurate, 24.1× speedup

Math

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

FPCore

(FPCore (x c s) :precision binary64 (/ 1.0 (* x (* c (* (* s x) (* c s))))))

C

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

Fortran

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

Java

public static double code(double x, double c, double s) {
	return 1.0 / (x * (c * ((s * x) * (c * s))));
}

Python

def code(x, c, s):
	return 1.0 / (x * (c * ((s * x) * (c * s))))

Julia

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

MATLAB

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

Wolfram

code[x_, c_, s_] := N[(1.0 / N[(x * N[(c * N[(N[(s * x), $MachinePrecision] * N[(c * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 66.4%

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

4. Applied egg-rr 83.9%

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

5. Taylor expanded in x around 0 61.3%

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

   1. associate-/r* 61.3%

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

   2. unpow2 61.3%

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

   3. unpow2 61.3%

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

   4. swap-sqr 72.7%

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

   5. unpow2 72.7%

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

7. Simplified 72.7%

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

   1. div-inv 72.7%

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

   2. associate-*r* 66.1%

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

   3. inv-pow 66.1%

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

   4. inv-pow 66.1%

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

   5. pow-prod-up 66.1%

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

   6. metadata-eval 66.1%

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

   7. pow-flip 66.1%

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

   8. metadata-eval 66.1%

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

   9. unpow-prod-down 80.0%

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

   10. sqr-pow 79.9%

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

   11. metadata-eval 79.9%

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

   12. metadata-eval 79.9%

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

   13. unpow-prod-down 80.0%

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

   14. associate-*r* 78.9%

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

   15. associate-*r* 75.6%

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

   16. inv-pow 75.6%

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

   17. associate-/r* 75.6%

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

   18. clear-num 75.5%

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

   19. associate-*r* 74.6%

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

9. Applied egg-rr 74.6%

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

    1. associate-/l* 73.9%

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

    2. associate-*l* 74.9%

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

    3. div-inv 74.9%

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

    4. remove-double-div 74.9%

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

11. Applied egg-rr 74.9%

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

12. Final simplification 74.9%

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

Alternative 10: 78.7% accurate, 24.1× speedup

Math

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

FPCore

(FPCore (x c s)
 :precision binary64
 (let* ((t_0 (* c (* s x)))) (/ 1.0 (* t_0 t_0))))

C

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

Fortran

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

Java

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

Python

def code(x, c, s):
	t_0 = c * (s * x)
	return 1.0 / (t_0 * t_0)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 66.4%

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

3. Taylor expanded in x around 0 56.7%

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

   1. associate-/r* 56.3%

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

   2. *-commutative 56.3%

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

   3. unpow2 56.3%

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

   4. unpow2 56.3%

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

   5. swap-sqr 66.0%

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

   6. unpow2 66.0%

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

   7. associate-/r* 66.4%

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

   8. unpow2 66.4%

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

   9. unpow2 66.4%

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

   10. swap-sqr 80.0%

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

   11. unpow2 80.0%

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

   12. *-commutative 80.0%

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

5. Simplified 80.0%

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

   1. unpow2 96.1%

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

7. Applied egg-rr 80.0%

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

8. Final simplification 80.0%

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

Reproduce

herbie shell --seed 2024080 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))