mixedcos

Percentage Accurate: 66.9% → 97.3%

Time: 14.1s

Alternatives: 6

Speedup: 24.1×

• 32.1% 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: 66.9% 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: 97.3% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.2%

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

   1. associate-/l/ 65.2%

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

   2. remove-double-neg 65.2%

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

   3. distribute-frac-neg 65.2%

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

   4. distribute-neg-frac 65.2%

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

   5. remove-double-neg 65.2%

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

   6. *-commutative 65.2%

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

   7. associate-*r* 58.9%

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

   8. unpow2 58.9%

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

   9. associate-/r* 58.2%

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

   10. cos-neg 58.2%

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

   11. *-commutative 58.2%

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

   12. distribute-rgt-neg-in 58.2%

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

   13. metadata-eval 58.2%

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

3. Simplified 58.2%

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

   1. associate-/l/ 59.7%

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

   2. div-inv 59.7%

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

   3. associate-/l* 59.6%

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

   4. add-sqr-sqrt 26.6%

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

   5. sqrt-unprod 48.0%

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

   6. *-commutative 48.0%

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

   7. *-commutative 48.0%

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

   8. swap-sqr 48.0%

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

   9. metadata-eval 48.0%

      \[\leadsto \cos \left(\sqrt{\color{blue}{4} \cdot \left(x \cdot x\right)}\right) \cdot \frac{\frac{1}{{x}^{2}}}{{c}^{2} \cdot {s}^{2}} \]

   10. metadata-eval 48.0%

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

   11. swap-sqr 48.0%

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

   12. sqrt-unprod 30.5%

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

   13. add-sqr-sqrt 59.6%

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

   14. pow-flip 59.7%

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

   15. metadata-eval 59.7%

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

   16. pow-prod-down 76.7%

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

6. Applied egg-rr 76.7%

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

   1. add-sqr-sqrt 76.7%

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

   2. sqrt-div 76.7%

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

   3. sqrt-pow1 53.3%

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

   4. metadata-eval 53.3%

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

   5. inv-pow 53.3%

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

   6. sqrt-pow1 54.2%

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

   7. metadata-eval 54.2%

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

   8. pow1 54.2%

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

   9. sqrt-div 54.1%

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

   10. sqrt-pow1 61.7%

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

   11. metadata-eval 61.7%

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

   12. inv-pow 61.7%

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

   13. sqrt-pow1 98.5%

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

   14. metadata-eval 98.5%

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

   15. pow1 98.5%

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

8. Applied egg-rr 98.5%

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

9. Taylor expanded in x around 0 95.7%

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

    1. associate-*r* 98.6%

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

    2. associate-/r* 98.5%

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

11. Simplified 98.5%

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

Alternative 2: 97.5% accurate, 2.7× speedup

Math

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

FPCore

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

C

double code(double x, double c, double s) {
	double t_0 = x * (c * s);
	return (cos((2.0 * x)) / 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 = x * (c * s)
    code = (cos((2.0d0 * x)) / t_0) / t_0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.2%

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

   1. associate-/l/ 65.2%

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

   2. remove-double-neg 65.2%

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

   3. distribute-frac-neg 65.2%

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

   4. distribute-neg-frac 65.2%

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

   5. remove-double-neg 65.2%

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

   6. *-commutative 65.2%

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

   7. associate-*r* 58.9%

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

   8. unpow2 58.9%

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

   9. associate-/r* 58.2%

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

   10. cos-neg 58.2%

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

   11. *-commutative 58.2%

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

   12. distribute-rgt-neg-in 58.2%

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

   13. metadata-eval 58.2%

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

3. Simplified 58.2%

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

   1. associate-/l/ 59.7%

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

   2. div-inv 59.7%

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

   3. associate-/l* 59.6%

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

   4. add-sqr-sqrt 26.6%

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

   5. sqrt-unprod 48.0%

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

   6. *-commutative 48.0%

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

   7. *-commutative 48.0%

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

   8. swap-sqr 48.0%

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

   9. metadata-eval 48.0%

      \[\leadsto \cos \left(\sqrt{\color{blue}{4} \cdot \left(x \cdot x\right)}\right) \cdot \frac{\frac{1}{{x}^{2}}}{{c}^{2} \cdot {s}^{2}} \]

   10. metadata-eval 48.0%

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

   11. swap-sqr 48.0%

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

   12. sqrt-unprod 30.5%

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

   13. add-sqr-sqrt 59.6%

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

   14. pow-flip 59.7%

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

   15. metadata-eval 59.7%

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

   16. pow-prod-down 76.7%

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

6. Applied egg-rr 76.7%

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

   1. add-sqr-sqrt 76.7%

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

   2. sqrt-div 76.7%

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

   3. sqrt-pow1 53.3%

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

   4. metadata-eval 53.3%

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

   5. inv-pow 53.3%

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

   6. sqrt-pow1 54.2%

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

   7. metadata-eval 54.2%

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

   8. pow1 54.2%

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

   9. sqrt-div 54.1%

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

   10. sqrt-pow1 61.7%

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

   11. metadata-eval 61.7%

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

   12. inv-pow 61.7%

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

   13. sqrt-pow1 98.5%

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

   14. metadata-eval 98.5%

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

   15. pow1 98.5%

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

8. Applied egg-rr 98.5%

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

   1. associate-*r* 98.5%

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

   2. associate-/l/ 98.5%

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

   3. un-div-inv 98.6%

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

   4. associate-/l/ 98.5%

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

   5. un-div-inv 98.5%

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

   6. *-commutative 98.5%

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

10. Applied egg-rr 98.5%

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

11. Final simplification 98.5%

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

Alternative 3: 97.3% accurate, 2.7× speedup

Math

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

FPCore

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

C

double code(double x, double c, double s) {
	double t_0 = c * (x * s);
	return (cos((2.0 * x)) / 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 * (x * s)
    code = (cos((2.0d0 * x)) / t_0) / t_0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.2%

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

   1. associate-/l/ 65.2%

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

   2. remove-double-neg 65.2%

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

   3. distribute-frac-neg 65.2%

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

   4. distribute-neg-frac 65.2%

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

   5. remove-double-neg 65.2%

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

   6. *-commutative 65.2%

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

   7. associate-*r* 58.9%

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

   8. unpow2 58.9%

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

   9. associate-/r* 58.2%

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

   10. cos-neg 58.2%

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

   11. *-commutative 58.2%

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

   12. distribute-rgt-neg-in 58.2%

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

   13. metadata-eval 58.2%

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

3. Simplified 58.2%

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

   1. associate-/l/ 59.7%

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

   2. div-inv 59.7%

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

   3. associate-/l* 59.6%

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

   4. add-sqr-sqrt 26.6%

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

   5. sqrt-unprod 48.0%

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

   6. *-commutative 48.0%

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

   7. *-commutative 48.0%

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

   8. swap-sqr 48.0%

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

   9. metadata-eval 48.0%

      \[\leadsto \cos \left(\sqrt{\color{blue}{4} \cdot \left(x \cdot x\right)}\right) \cdot \frac{\frac{1}{{x}^{2}}}{{c}^{2} \cdot {s}^{2}} \]

   10. metadata-eval 48.0%

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

   11. swap-sqr 48.0%

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

   12. sqrt-unprod 30.5%

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

   13. add-sqr-sqrt 59.6%

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

   14. pow-flip 59.7%

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

   15. metadata-eval 59.7%

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

   16. pow-prod-down 76.7%

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

6. Applied egg-rr 76.7%

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

   1. add-sqr-sqrt 76.7%

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

   2. sqrt-div 76.7%

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

   3. sqrt-pow1 53.3%

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

   4. metadata-eval 53.3%

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

   5. inv-pow 53.3%

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

   6. sqrt-pow1 54.2%

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

   7. metadata-eval 54.2%

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

   8. pow1 54.2%

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

   9. sqrt-div 54.1%

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

   10. sqrt-pow1 61.7%

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

   11. metadata-eval 61.7%

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

   12. inv-pow 61.7%

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

   13. sqrt-pow1 98.5%

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

   14. metadata-eval 98.5%

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

   15. pow1 98.5%

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

8. Applied egg-rr 98.5%

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

   1. unpow2 98.5%

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

   2. associate-/l/ 98.5%

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

10. Simplified 98.5%

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

    1. *-commutative 98.5%

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

    2. associate-*l* 96.7%

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

    3. *-commutative 96.7%

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

    4. pow2 96.7%

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

    5. associate-*r* 96.7%

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

    6. div-inv 96.7%

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

    7. un-div-inv 96.7%

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

    8. *-commutative 96.7%

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

    9. *-commutative 96.7%

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

12. Applied egg-rr 96.7%

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

13. Final simplification 96.7%

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

Alternative 4: 58.3% accurate, 2.8× speedup

Math

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

FPCore

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

C

double code(double x, double c, double s) {
	return ((1.0 / c) / (x * s)) / (c * fabs((x * 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)) / (c * abs((x * s)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.2%

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

   1. associate-/l/ 65.2%

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

   2. remove-double-neg 65.2%

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

   3. distribute-frac-neg 65.2%

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

   4. distribute-neg-frac 65.2%

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

   5. remove-double-neg 65.2%

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

   6. *-commutative 65.2%

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

   7. associate-*r* 58.9%

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

   8. unpow2 58.9%

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

   9. associate-/r* 58.2%

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

   10. cos-neg 58.2%

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

   11. *-commutative 58.2%

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

   12. distribute-rgt-neg-in 58.2%

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

   13. metadata-eval 58.2%

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

3. Simplified 58.2%

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

   1. associate-/l/ 58.9%

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

   2. add-sqr-sqrt 26.6%

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

   3. sqrt-unprod 47.2%

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

   4. *-commutative 47.2%

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

   5. *-commutative 47.2%

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

   6. swap-sqr 47.2%

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

   7. metadata-eval 47.2%

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

   8. metadata-eval 47.2%

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

   9. swap-sqr 47.2%

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

   10. sqrt-unprod 29.8%

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

   11. add-sqr-sqrt 58.9%

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

   12. unpow2 58.9%

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

   13. associate-*r* 65.2%

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

   14. *-commutative 65.2%

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

   15. associate-/r* 65.2%

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

   16. *-commutative 65.2%

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

6. Applied egg-rr 85.8%

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

   1. associate-*l/ 85.8%

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

   2. *-lft-identity 85.8%

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

   3. *-commutative 85.8%

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

   4. unpow2 85.8%

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

   5. rem-sqrt-square 85.8%

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

   6. *-commutative 85.8%

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

   7. unpow2 85.8%

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

   8. rem-sqrt-square 96.7%

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

   9. *-commutative 96.7%

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

8. Simplified 96.7%

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

9. Taylor expanded in x around 0 78.8%

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

    1. associate-/r* 78.8%

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

    2. *-commutative 78.8%

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

    3. rem-square-sqrt 43.0%

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

    4. fabs-sqr 43.0%

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

    5. rem-square-sqrt 57.6%

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

    6. *-commutative 57.6%

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

11. Simplified 57.6%

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

12. Final simplification 57.6%

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

Alternative 5: 78.5% accurate, 3.0× speedup

Math

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

FPCore

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

C

double code(double x, double c, double s) {
	return pow((c * (x * s)), -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 * (x * s)) ** (-2.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.2%

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

   1. associate-/l/ 65.2%

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

   2. remove-double-neg 65.2%

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

   3. distribute-frac-neg 65.2%

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

   4. distribute-neg-frac 65.2%

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

   5. remove-double-neg 65.2%

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

   6. *-commutative 65.2%

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

   7. associate-*r* 58.9%

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

   8. unpow2 58.9%

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

   9. associate-/r* 58.2%

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

   10. cos-neg 58.2%

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

   11. *-commutative 58.2%

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

   12. distribute-rgt-neg-in 58.2%

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

   13. metadata-eval 58.2%

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

3. Simplified 58.2%

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

5. Taylor expanded in x around 0 53.6%

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

   1. associate-/r* 53.5%

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

   2. *-commutative 53.5%

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

   3. *-commutative 53.5%

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

   4. unpow2 53.5%

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

   5. unpow2 53.5%

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

   6. swap-sqr 65.2%

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

   7. unpow2 65.2%

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

   8. associate-/r* 65.2%

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

   9. unpow2 65.2%

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

   10. rem-square-sqrt 65.2%

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

   11. swap-sqr 72.1%

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

   12. unpow2 72.1%

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

   13. unpow2 72.1%

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

   14. rem-sqrt-square 78.8%

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

   15. *-commutative 78.8%

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

7. Simplified 78.8%

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

   1. pow1 78.8%

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

   2. metadata-eval 78.8%

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

   3. sqrt-pow1 78.8%

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

   4. pow2 78.8%

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

   5. add-sqr-sqrt 78.8%

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

   6. pow-flip 78.8%

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

   7. add-sqr-sqrt 43.0%

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

   8. fabs-sqr 43.0%

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

   9. add-sqr-sqrt 78.8%

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

   10. metadata-eval 78.8%

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

9. Applied egg-rr 78.8%

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

Alternative 6: 78.4% accurate, 24.1× speedup

Math

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

FPCore

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

C

double code(double x, double c, double s) {
	double t_0 = c * (x * s);
	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 * (x * s)
    code = 1.0d0 / (t_0 * t_0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.2%

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

   1. associate-/l/ 65.2%

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

   2. remove-double-neg 65.2%

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

   3. distribute-frac-neg 65.2%

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

   4. distribute-neg-frac 65.2%

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

   5. remove-double-neg 65.2%

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

   6. *-commutative 65.2%

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

   7. associate-*r* 58.9%

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

   8. unpow2 58.9%

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

   9. associate-/r* 58.2%

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

   10. cos-neg 58.2%

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

   11. *-commutative 58.2%

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

   12. distribute-rgt-neg-in 58.2%

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

   13. metadata-eval 58.2%

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

3. Simplified 58.2%

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

5. Taylor expanded in x around 0 53.6%

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

   1. associate-/r* 53.5%

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

   2. *-commutative 53.5%

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

   3. *-commutative 53.5%

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

   4. unpow2 53.5%

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

   5. unpow2 53.5%

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

   6. swap-sqr 65.2%

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

   7. unpow2 65.2%

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

   8. associate-/r* 65.2%

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

   9. unpow2 65.2%

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

   10. rem-square-sqrt 65.2%

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

   11. swap-sqr 72.1%

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

   12. unpow2 72.1%

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

   13. unpow2 72.1%

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

   14. rem-sqrt-square 78.8%

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

   15. *-commutative 78.8%

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

7. Simplified 78.8%

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

   1. pow1 78.8%

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

   2. metadata-eval 78.8%

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

   3. sqrt-pow1 78.8%

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

   4. pow2 78.8%

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

   5. add-sqr-sqrt 78.8%

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

   6. unpow2 78.8%

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

   7. add-sqr-sqrt 43.0%

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

   8. fabs-sqr 43.0%

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

   9. add-sqr-sqrt 57.6%

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

   10. add-sqr-sqrt 37.0%

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

   11. fabs-sqr 37.0%

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

   12. add-sqr-sqrt 78.8%

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

9. Applied egg-rr 78.8%

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

Reproduce

herbie shell --seed 2024137 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))