mixedcos

Percentage Accurate: 66.9% → 97.3%

Time: 12.9s

Alternatives: 8

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, 2.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, c_, s_] := N[(N[(N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision] / N[(x / N[(1.0 / N[(c * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * N[(c * s), $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. Add Preprocessing
3. Step-by-step derivation

   1. *-un-lft-identity 65.2%

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

   2. associate-*r* 66.8%

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

   3. times-frac 67.0%

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

   4. *-commutative 67.0%

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

   5. associate-*r* 66.3%

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

   6. pow-prod-down 87.9%

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

4. Applied egg-rr 87.9%

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

   1. frac-times 87.7%

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

   2. *-un-lft-identity 87.7%

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

   3. add-sqr-sqrt 43.8%

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

   4. add-sqr-sqrt 43.8%

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

   5. unswap-sqr 43.7%

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

   6. sqrt-prod 43.7%

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

   7. sqrt-pow1 31.4%

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

   8. metadata-eval 31.4%

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

   9. pow1 31.4%

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

   10. associate-*r* 31.4%

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

   11. add-sqr-sqrt 31.4%

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

   12. associate-*r* 30.8%

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

   13. add-sqr-sqrt 16.5%

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

   14. fabs-sqr 16.5%

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

   15. add-sqr-sqrt 32.1%

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

6. Applied egg-rr 98.5%

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

   1. *-commutative 98.5%

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

   2. *-commutative 98.5%

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

   3. /-rgt-identity 98.5%

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

   4. clear-num 98.5%

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

   5. div-inv 98.5%

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

   6. clear-num 98.5%

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

   7. div-inv 98.5%

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

   8. associate-/l/ 98.5%

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

   9. *-commutative 98.5%

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

8. Applied egg-rr 98.5%

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

9. Final simplification 98.5%

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

Alternative 2: 97.3% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, c_, s_] := N[(N[(N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision] / N[(x * N[(c * s), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x / N[(1.0 / 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. Add Preprocessing
3. Step-by-step derivation

   1. *-un-lft-identity 65.2%

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

   2. associate-*r* 66.8%

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

   3. times-frac 67.0%

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

   4. *-commutative 67.0%

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

   5. associate-*r* 66.3%

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

   6. pow-prod-down 87.9%

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

4. Applied egg-rr 87.9%

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

   1. frac-times 87.7%

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

   2. *-un-lft-identity 87.7%

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

   3. add-sqr-sqrt 43.8%

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

   4. add-sqr-sqrt 43.8%

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

   5. unswap-sqr 43.7%

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

   6. sqrt-prod 43.7%

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

   7. sqrt-pow1 31.4%

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

   8. metadata-eval 31.4%

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

   9. pow1 31.4%

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

   10. associate-*r* 31.4%

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

   11. add-sqr-sqrt 31.4%

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

   12. associate-*r* 30.8%

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

   13. add-sqr-sqrt 16.5%

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

   14. fabs-sqr 16.5%

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

   15. add-sqr-sqrt 32.1%

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

6. Applied egg-rr 98.5%

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

   1. *-commutative 98.5%

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

   2. *-commutative 98.5%

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

   3. /-rgt-identity 98.5%

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

   4. clear-num 98.5%

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

   5. div-inv 98.5%

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

   6. clear-num 98.5%

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

   7. div-inv 98.5%

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

   8. associate-/l/ 98.5%

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

   9. *-commutative 98.5%

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

8. Applied egg-rr 98.5%

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

9. Final simplification 98.5%

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

Alternative 3: 97.5% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

function tmp = code(x, c, s)
	t_0 = x * (c * s);
	tmp = (cos((x * 2.0)) / 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[(x * 2.0), $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. Add Preprocessing
3. Step-by-step derivation

   1. *-un-lft-identity 65.2%

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

   2. associate-*r* 66.8%

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

   3. times-frac 67.0%

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

   4. *-commutative 67.0%

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

   5. associate-*r* 66.3%

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

   6. pow-prod-down 87.9%

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

4. Applied egg-rr 87.9%

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

   1. frac-times 87.7%

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

   2. *-un-lft-identity 87.7%

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

   3. add-sqr-sqrt 43.8%

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

   4. add-sqr-sqrt 43.8%

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

   5. unswap-sqr 43.7%

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

   6. sqrt-prod 43.7%

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

   7. sqrt-pow1 31.4%

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

   8. metadata-eval 31.4%

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

   9. pow1 31.4%

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

   10. associate-*r* 31.4%

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

   11. add-sqr-sqrt 31.4%

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

   12. associate-*r* 30.8%

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

   13. add-sqr-sqrt 16.5%

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

   14. fabs-sqr 16.5%

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

   15. add-sqr-sqrt 32.1%

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

6. Applied egg-rr 98.5%

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

7. Final simplification 98.5%

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

Alternative 4: 97.3% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

function tmp = code(x, c, s)
	t_0 = c * (x * s);
	tmp = (cos((x * 2.0)) / 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[(x * 2.0), $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. Add Preprocessing
3. Step-by-step derivation

   1. *-un-lft-identity 65.2%

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

   2. associate-*r* 66.8%

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

   3. times-frac 67.0%

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

   4. *-commutative 67.0%

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

   5. associate-*r* 66.3%

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

   6. pow-prod-down 87.9%

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

4. Applied egg-rr 87.9%

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

   1. unpow2 87.9%

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

   2. *-commutative 87.9%

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

   3. associate-*r* 84.8%

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

6. Applied egg-rr 84.8%

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

   1. frac-times 84.6%

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

   2. *-un-lft-identity 84.6%

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

   3. *-commutative 84.6%

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

   4. associate-*r* 75.6%

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

   5. associate-*l* 77.5%

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

   6. *-commutative 77.5%

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

   7. swap-sqr 98.3%

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

   8. associate-/l/ 98.5%

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

   9. associate-*l* 95.8%

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

   10. associate-*l* 96.7%

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

8. Applied egg-rr 96.7%

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

9. Final simplification 96.7%

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

Alternative 5: 78.4% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, c_, s_] := N[Power[N[(N[(N[(1.0 / s), $MachinePrecision] / x), $MachinePrecision] / c), $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. Add Preprocessing
3. Step-by-step derivation

   1. *-un-lft-identity 65.2%

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

   2. add-sqr-sqrt 65.2%

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

   3. times-frac 65.2%

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

   4. sqrt-prod 65.2%

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

   5. sqrt-pow1 45.0%

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

   6. metadata-eval 45.0%

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

   7. pow1 45.0%

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

   8. *-commutative 45.0%

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

   9. associate-*r* 41.8%

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

   10. unpow2 41.8%

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

   11. pow-prod-down 45.0%

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

   12. sqrt-prod 45.0%

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

4. Applied egg-rr 85.8%

   \[\leadsto \color{blue}{\frac{1}{c \cdot \sqrt{{\left(s \cdot x\right)}^{2}}} \cdot \frac{\cos \left(2 \cdot x\right)}{c \cdot \sqrt{{\left(s \cdot x\right)}^{2}}}} \]
5. 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. unpow2 85.8%

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

   4. rem-sqrt-square 85.8%

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

   5. unpow2 85.8%

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

   6. rem-sqrt-square 96.7%

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

6. Simplified 96.7%

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

7. Taylor expanded in x around 0 65.2%

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

   1. unpow2 65.2%

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

   2. unpow2 65.2%

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

   3. sqr-abs 65.2%

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

   4. swap-sqr 78.8%

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

   5. associate-*r* 78.1%

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

   6. associate-*r* 79.9%

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

   7. associate-/l/ 79.9%

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

   8. *-rgt-identity 79.9%

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

   9. associate-*r/ 79.9%

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

   10. unpow2 79.9%

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

   11. associate-/r* 79.9%

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

   12. associate-/r* 79.9%

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

9. Simplified 79.9%

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

    1. div-inv 79.8%

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

    2. associate-/l/ 79.8%

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

    3. *-commutative 79.8%

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

11. Applied egg-rr 79.8%

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

    1. associate-*l/ 79.9%

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

    2. *-un-lft-identity 79.9%

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

    3. *-commutative 79.9%

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

    4. associate-/r* 78.8%

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

    5. associate-/r* 78.8%

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

    6. *-commutative 78.8%

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

    7. associate-/r* 78.8%

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

13. Applied egg-rr 78.8%

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

Alternative 6: 78.5% accurate, 24.1× 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 (* x s))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, c_, s_] := N[(N[(N[(1.0 / c), $MachinePrecision] / N[(x * s), $MachinePrecision]), $MachinePrecision] / N[(c * N[(x * s), $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. Add Preprocessing
3. Step-by-step derivation

   1. *-un-lft-identity 65.2%

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

   2. add-sqr-sqrt 65.2%

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

   3. times-frac 65.2%

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

   4. sqrt-prod 65.2%

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

   5. sqrt-pow1 45.0%

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

   6. metadata-eval 45.0%

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

   7. pow1 45.0%

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

   8. *-commutative 45.0%

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

   9. associate-*r* 41.8%

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

   10. unpow2 41.8%

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

   11. pow-prod-down 45.0%

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

   12. sqrt-prod 45.0%

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

4. Applied egg-rr 85.8%

   \[\leadsto \color{blue}{\frac{1}{c \cdot \sqrt{{\left(s \cdot x\right)}^{2}}} \cdot \frac{\cos \left(2 \cdot x\right)}{c \cdot \sqrt{{\left(s \cdot x\right)}^{2}}}} \]
5. 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. unpow2 85.8%

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

   4. rem-sqrt-square 85.8%

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

   5. unpow2 85.8%

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

   6. rem-sqrt-square 96.7%

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

6. Simplified 96.7%

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

7. Taylor expanded in x around 0 65.2%

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

   1. unpow2 65.2%

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

   2. unpow2 65.2%

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

   3. sqr-abs 65.2%

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

   4. swap-sqr 78.8%

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

   5. associate-*r* 78.1%

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

   6. associate-*r* 79.9%

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

   7. associate-/l/ 79.9%

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

   8. *-rgt-identity 79.9%

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

   9. associate-*r/ 79.9%

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

   10. unpow2 79.9%

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

   11. associate-/r* 79.9%

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

   12. associate-/r* 79.9%

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

9. Simplified 79.9%

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

    1. div-inv 79.8%

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

    2. associate-/l/ 79.8%

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

    3. *-commutative 79.8%

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

    4. frac-times 79.9%

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

    5. metadata-eval 79.9%

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

    6. metadata-eval 79.9%

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

    7. pow-sqr 79.9%

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

    8. pow1 79.9%

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

    9. pow1 79.9%

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

    10. div-inv 79.9%

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

    11. metadata-eval 79.9%

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

    12. frac-times 79.9%

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

    13. *-commutative 79.9%

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

    14. associate-/l/ 79.9%

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

    15. div-inv 79.9%

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

    16. associate-/l/ 78.1%

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

    17. associate-*l* 78.8%

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

11. Applied egg-rr 78.8%

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

12. Final simplification 78.8%

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

Alternative 7: 76.9% accurate, 24.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, c_, s_] := N[(N[(1.0 / c), $MachinePrecision] / N[(N[(c * N[(x * s), $MachinePrecision]), $MachinePrecision] * N[(x * s), $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. Add Preprocessing
3. Step-by-step derivation

   1. *-un-lft-identity 65.2%

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

   2. add-sqr-sqrt 65.2%

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

   3. times-frac 65.2%

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

   4. sqrt-prod 65.2%

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

   5. sqrt-pow1 45.0%

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

   6. metadata-eval 45.0%

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

   7. pow1 45.0%

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

   8. *-commutative 45.0%

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

   9. associate-*r* 41.8%

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

   10. unpow2 41.8%

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

   11. pow-prod-down 45.0%

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

   12. sqrt-prod 45.0%

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

4. Applied egg-rr 85.8%

   \[\leadsto \color{blue}{\frac{1}{c \cdot \sqrt{{\left(s \cdot x\right)}^{2}}} \cdot \frac{\cos \left(2 \cdot x\right)}{c \cdot \sqrt{{\left(s \cdot x\right)}^{2}}}} \]
5. 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. unpow2 85.8%

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

   4. rem-sqrt-square 85.8%

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

   5. unpow2 85.8%

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

   6. rem-sqrt-square 96.7%

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

6. Simplified 96.7%

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

7. Taylor expanded in x around 0 65.2%

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

   1. unpow2 65.2%

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

   2. unpow2 65.2%

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

   3. sqr-abs 65.2%

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

   4. swap-sqr 78.8%

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

   5. associate-*r* 78.1%

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

   6. associate-*r* 79.9%

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

   7. associate-/l/ 79.9%

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

   8. *-rgt-identity 79.9%

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

   9. associate-*r/ 79.9%

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

   10. unpow2 79.9%

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

   11. associate-/r* 79.9%

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

   12. associate-/r* 79.9%

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

9. Simplified 79.9%

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

    1. unpow2 79.9%

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

    2. clear-num 79.9%

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

    3. associate-/l/ 78.1%

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

    4. frac-times 76.9%

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

    5. *-un-lft-identity 76.9%

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

    6. div-inv 76.9%

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

    7. clear-num 76.9%

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

    8. div-inv 76.9%

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

    9. clear-num 76.9%

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

    10. /-rgt-identity 76.9%

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

    11. *-commutative 76.9%

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

    12. *-commutative 76.9%

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

    13. associate-*l* 77.6%

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

11. Applied egg-rr 77.6%

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

12. Final simplification 77.6%

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

Alternative 8: 78.5% accurate, 24.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, c_, s_] := Block[{t$95$0 = N[(x * N[(c * 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. Add Preprocessing

3. Taylor expanded in x around 0 53.6%

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

   1. associate-/r* 53.5%

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

   2. unpow2 53.5%

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

   3. unpow2 53.5%

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

   4. swap-sqr 65.2%

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

   5. unpow2 65.2%

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

   6. associate-/r* 65.2%

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

   7. unpow2 65.2%

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

   8. rem-square-sqrt 65.2%

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

   9. swap-sqr 72.1%

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

   10. unpow2 72.1%

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

   11. unpow2 72.1%

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

   12. rem-sqrt-square 78.8%

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

5. Simplified 78.8%

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

   1. pow1 78.8%

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

   2. metadata-eval 78.8%

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

   3. sqrt-pow1 78.8%

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

   4. pow2 78.8%

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

   5. add-sqr-sqrt 78.8%

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

   6. unpow2 78.8%

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

   7. add-sqr-sqrt 43.0%

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

   8. fabs-sqr 43.0%

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

   9. add-sqr-sqrt 57.6%

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

   10. associate-*r* 57.2%

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

   11. add-sqr-sqrt 36.7%

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

   12. fabs-sqr 36.7%

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

   13. add-sqr-sqrt 78.1%

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

   14. associate-*r* 79.9%

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

7. Applied egg-rr 79.9%

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

8. Final simplification 79.9%

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