mixedcos

Percentage Accurate: 66.7% → 97.3%

Time: 21.7s

Alternatives: 8

Speedup: 24.1×

• 31.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 66.7% 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

\[\begin{array}{l} c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x, c_m, s_m] = \mathsf{sort}([x, c_m, s_m])\\ \\ \begin{array}{l} t_0 := s\_m \cdot \left(c\_m \cdot x\right)\\ \frac{\frac{\cos \left(x \cdot 2\right)}{t\_0}}{t\_0} \end{array} \end{array} \]

FPCore

c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x c_m s_m)
 :precision binary64
 (let* ((t_0 (* s_m (* c_m x)))) (/ (/ (cos (* x 2.0)) t_0) t_0)))

C

c_m = fabs(c);
s_m = fabs(s);
assert(x < c_m && c_m < s_m);
double code(double x, double c_m, double s_m) {
	double t_0 = s_m * (c_m * x);
	return (cos((x * 2.0)) / t_0) / t_0;
}

Fortran

c_m = abs(c)
s_m = abs(s)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x, c_m, s_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: t_0
    t_0 = s_m * (c_m * x)
    code = (cos((x * 2.0d0)) / t_0) / t_0
end function

Java

c_m = Math.abs(c);
s_m = Math.abs(s);
assert x < c_m && c_m < s_m;
public static double code(double x, double c_m, double s_m) {
	double t_0 = s_m * (c_m * x);
	return (Math.cos((x * 2.0)) / t_0) / t_0;
}

Python

c_m = math.fabs(c)
s_m = math.fabs(s)
[x, c_m, s_m] = sort([x, c_m, s_m])
def code(x, c_m, s_m):
	t_0 = s_m * (c_m * x)
	return (math.cos((x * 2.0)) / t_0) / t_0

Julia

c_m = abs(c)
s_m = abs(s)
x, c_m, s_m = sort([x, c_m, s_m])
function code(x, c_m, s_m)
	t_0 = Float64(s_m * Float64(c_m * x))
	return Float64(Float64(cos(Float64(x * 2.0)) / t_0) / t_0)
end

MATLAB

c_m = abs(c);
s_m = abs(s);
x, c_m, s_m = num2cell(sort([x, c_m, s_m])){:}
function tmp = code(x, c_m, s_m)
	t_0 = s_m * (c_m * x);
	tmp = (cos((x * 2.0)) / t_0) / t_0;
end

Wolfram

c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
code[x_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[(s$95$m * N[(c$95$m * x), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]]

Derivation

1. Initial program 65.7%

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

   1. add-sqr-sqrt 65.7%

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

   2. pow2 65.7%

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

   3. sqrt-prod 65.7%

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

   4. sqrt-pow1 72.5%

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

   5. metadata-eval 72.5%

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

   6. pow1 72.5%

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

   7. *-commutative 72.5%

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

   8. associate-*r* 65.0%

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

   9. unpow2 65.0%

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

   10. pow-prod-down 87.3%

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

   11. sqrt-pow1 97.6%

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

   12. metadata-eval 97.6%

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

   13. pow1 97.6%

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

   14. *-commutative 97.6%

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

4. Applied egg-rr 97.6%

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

   1. *-un-lft-identity 97.6%

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

   2. unpow2 97.6%

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

   3. frac-times 98.1%

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

   4. associate-*l/ 98.2%

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

   5. *-un-lft-identity 98.2%

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

   6. div-inv 98.2%

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

   7. associate-*r* 95.8%

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

   8. times-frac 90.7%

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

   9. *-commutative 90.7%

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

6. Applied egg-rr 90.7%

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

   1. inv-pow 90.7%

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

   2. associate-*r* 92.1%

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

   3. unpow-prod-down 92.1%

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

   4. inv-pow 92.1%

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

   5. inv-pow 92.1%

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

8. Applied egg-rr 92.1%

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

   1. frac-times 97.2%

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

10. Applied egg-rr 97.2%

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

Alternative 2: 96.9% accurate, 2.7× speedup

Math

\[\begin{array}{l} c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x, c_m, s_m] = \mathsf{sort}([x, c_m, s_m])\\ \\ \begin{array}{l} t_0 := c\_m \cdot \left(x \cdot s\_m\right)\\ \frac{\cos \left(2 \cdot x\right)}{t\_0 \cdot t\_0} \end{array} \end{array} \]

FPCore

c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x c_m s_m)
 :precision binary64
 (let* ((t_0 (* c_m (* x s_m)))) (/ (cos (* 2.0 x)) (* t_0 t_0))))

C

c_m = fabs(c);
s_m = fabs(s);
assert(x < c_m && c_m < s_m);
double code(double x, double c_m, double s_m) {
	double t_0 = c_m * (x * s_m);
	return cos((2.0 * x)) / (t_0 * t_0);
}

Fortran

c_m = abs(c)
s_m = abs(s)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x, c_m, s_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: t_0
    t_0 = c_m * (x * s_m)
    code = cos((2.0d0 * x)) / (t_0 * t_0)
end function

Java

c_m = Math.abs(c);
s_m = Math.abs(s);
assert x < c_m && c_m < s_m;
public static double code(double x, double c_m, double s_m) {
	double t_0 = c_m * (x * s_m);
	return Math.cos((2.0 * x)) / (t_0 * t_0);
}

Python

c_m = math.fabs(c)
s_m = math.fabs(s)
[x, c_m, s_m] = sort([x, c_m, s_m])
def code(x, c_m, s_m):
	t_0 = c_m * (x * s_m)
	return math.cos((2.0 * x)) / (t_0 * t_0)

Julia

c_m = abs(c)
s_m = abs(s)
x, c_m, s_m = sort([x, c_m, s_m])
function code(x, c_m, s_m)
	t_0 = Float64(c_m * Float64(x * s_m))
	return Float64(cos(Float64(2.0 * x)) / Float64(t_0 * t_0))
end

MATLAB

c_m = abs(c);
s_m = abs(s);
x, c_m, s_m = num2cell(sort([x, c_m, s_m])){:}
function tmp = code(x, c_m, s_m)
	t_0 = c_m * (x * s_m);
	tmp = cos((2.0 * x)) / (t_0 * t_0);
end

Wolfram

c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
code[x_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[(c$95$m * N[(x * s$95$m), $MachinePrecision]), $MachinePrecision]}, N[(N[Cos[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 65.7%

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

   1. add-sqr-sqrt 65.7%

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

   2. pow2 65.7%

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

   3. sqrt-prod 65.7%

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

   4. sqrt-pow1 72.5%

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

   5. metadata-eval 72.5%

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

   6. pow1 72.5%

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

   7. *-commutative 72.5%

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

   8. associate-*r* 65.0%

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

   9. unpow2 65.0%

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

   10. pow-prod-down 87.3%

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

   11. sqrt-pow1 97.6%

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

   12. metadata-eval 97.6%

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

   13. pow1 97.6%

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

   14. *-commutative 97.6%

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

4. Applied egg-rr 97.6%

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

   1. unpow2 97.6%

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

6. Applied egg-rr 97.6%

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

Alternative 3: 97.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x, c_m, s_m] = \mathsf{sort}([x, c_m, s_m])\\ \\ \begin{array}{l} t_0 := c\_m \cdot \left(x \cdot s\_m\right)\\ \frac{\frac{\cos \left(x \cdot 2\right)}{t\_0}}{t\_0} \end{array} \end{array} \]

FPCore

c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x c_m s_m)
 :precision binary64
 (let* ((t_0 (* c_m (* x s_m)))) (/ (/ (cos (* x 2.0)) t_0) t_0)))

C

c_m = fabs(c);
s_m = fabs(s);
assert(x < c_m && c_m < s_m);
double code(double x, double c_m, double s_m) {
	double t_0 = c_m * (x * s_m);
	return (cos((x * 2.0)) / t_0) / t_0;
}

Fortran

c_m = abs(c)
s_m = abs(s)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x, c_m, s_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: t_0
    t_0 = c_m * (x * s_m)
    code = (cos((x * 2.0d0)) / t_0) / t_0
end function

Java

c_m = Math.abs(c);
s_m = Math.abs(s);
assert x < c_m && c_m < s_m;
public static double code(double x, double c_m, double s_m) {
	double t_0 = c_m * (x * s_m);
	return (Math.cos((x * 2.0)) / t_0) / t_0;
}

Python

c_m = math.fabs(c)
s_m = math.fabs(s)
[x, c_m, s_m] = sort([x, c_m, s_m])
def code(x, c_m, s_m):
	t_0 = c_m * (x * s_m)
	return (math.cos((x * 2.0)) / t_0) / t_0

Julia

c_m = abs(c)
s_m = abs(s)
x, c_m, s_m = sort([x, c_m, s_m])
function code(x, c_m, s_m)
	t_0 = Float64(c_m * Float64(x * s_m))
	return Float64(Float64(cos(Float64(x * 2.0)) / t_0) / t_0)
end

MATLAB

c_m = abs(c);
s_m = abs(s);
x, c_m, s_m = num2cell(sort([x, c_m, s_m])){:}
function tmp = code(x, c_m, s_m)
	t_0 = c_m * (x * s_m);
	tmp = (cos((x * 2.0)) / t_0) / t_0;
end

Wolfram

c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
code[x_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[(c$95$m * N[(x * s$95$m), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[Cos[N[(x * 2.0), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]]

Derivation

1. Initial program 65.7%

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

   1. *-un-lft-identity 65.7%

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

   2. add-sqr-sqrt 65.7%

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

   3. times-frac 65.7%

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

   4. sqrt-prod 65.7%

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

   5. sqrt-pow1 46.6%

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

   6. metadata-eval 46.6%

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

   7. pow1 46.6%

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

   8. *-commutative 46.6%

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

   9. associate-*r* 41.9%

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

   10. unpow2 41.9%

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

   11. pow-prod-down 46.6%

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

   12. sqrt-pow1 46.7%

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

   13. metadata-eval 46.7%

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

   14. pow1 46.7%

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

   15. *-commutative 46.7%

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

4. Applied egg-rr 98.1%

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

   1. associate-*l/ 98.2%

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

   2. *-un-lft-identity 98.2%

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

   3. *-commutative 98.2%

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

6. Applied egg-rr 98.2%

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

Alternative 4: 79.9% accurate, 18.4× speedup

Math

\[\begin{array}{l} c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x, c_m, s_m] = \mathsf{sort}([x, c_m, s_m])\\ \\ \frac{1}{c\_m \cdot \left(x \cdot s\_m\right)} \cdot \left(\frac{\frac{1}{x}}{s\_m} \cdot \frac{1}{c\_m}\right) \end{array} \]

FPCore

c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x c_m s_m)
 :precision binary64
 (* (/ 1.0 (* c_m (* x s_m))) (* (/ (/ 1.0 x) s_m) (/ 1.0 c_m))))

C

c_m = fabs(c);
s_m = fabs(s);
assert(x < c_m && c_m < s_m);
double code(double x, double c_m, double s_m) {
	return (1.0 / (c_m * (x * s_m))) * (((1.0 / x) / s_m) * (1.0 / c_m));
}

Fortran

c_m = abs(c)
s_m = abs(s)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x, c_m, s_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    code = (1.0d0 / (c_m * (x * s_m))) * (((1.0d0 / x) / s_m) * (1.0d0 / c_m))
end function

Java

c_m = Math.abs(c);
s_m = Math.abs(s);
assert x < c_m && c_m < s_m;
public static double code(double x, double c_m, double s_m) {
	return (1.0 / (c_m * (x * s_m))) * (((1.0 / x) / s_m) * (1.0 / c_m));
}

Python

c_m = math.fabs(c)
s_m = math.fabs(s)
[x, c_m, s_m] = sort([x, c_m, s_m])
def code(x, c_m, s_m):
	return (1.0 / (c_m * (x * s_m))) * (((1.0 / x) / s_m) * (1.0 / c_m))

Julia

c_m = abs(c)
s_m = abs(s)
x, c_m, s_m = sort([x, c_m, s_m])
function code(x, c_m, s_m)
	return Float64(Float64(1.0 / Float64(c_m * Float64(x * s_m))) * Float64(Float64(Float64(1.0 / x) / s_m) * Float64(1.0 / c_m)))
end

MATLAB

c_m = abs(c);
s_m = abs(s);
x, c_m, s_m = num2cell(sort([x, c_m, s_m])){:}
function tmp = code(x, c_m, s_m)
	tmp = (1.0 / (c_m * (x * s_m))) * (((1.0 / x) / s_m) * (1.0 / c_m));
end

Wolfram

c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
code[x_, c$95$m_, s$95$m_] := N[(N[(1.0 / N[(c$95$m * N[(x * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(1.0 / x), $MachinePrecision] / s$95$m), $MachinePrecision] * N[(1.0 / c$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 65.7%

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

   1. *-un-lft-identity 65.7%

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

   2. add-sqr-sqrt 65.7%

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

   3. times-frac 65.7%

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

   4. sqrt-prod 65.7%

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

   5. sqrt-pow1 46.6%

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

   6. metadata-eval 46.6%

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

   7. pow1 46.6%

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

   8. *-commutative 46.6%

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

   9. associate-*r* 41.9%

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

   10. unpow2 41.9%

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

   11. pow-prod-down 46.6%

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

   12. sqrt-pow1 46.7%

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

   13. metadata-eval 46.7%

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

   14. pow1 46.7%

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

   15. *-commutative 46.7%

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

4. Applied egg-rr 98.1%

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

5. Taylor expanded in x around 0 78.4%

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

   1. inv-pow 78.4%

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

   2. *-commutative 78.4%

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

   3. *-commutative 78.4%

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

   4. unpow-prod-down 78.4%

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

   5. inv-pow 78.4%

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

   6. associate-/r* 78.4%

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

   7. inv-pow 78.4%

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

7. Applied egg-rr 78.4%

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

Alternative 5: 37.1% accurate, 24.1× speedup

Math

\[\begin{array}{l} c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x, c_m, s_m] = \mathsf{sort}([x, c_m, s_m])\\ \\ \frac{\frac{-1}{c\_m \cdot x}}{\left(s\_m \cdot \left(c\_m \cdot x\right)\right) \cdot s\_m} \end{array} \]

FPCore

c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x c_m s_m)
 :precision binary64
 (/ (/ -1.0 (* c_m x)) (* (* s_m (* c_m x)) s_m)))

C

c_m = fabs(c);
s_m = fabs(s);
assert(x < c_m && c_m < s_m);
double code(double x, double c_m, double s_m) {
	return (-1.0 / (c_m * x)) / ((s_m * (c_m * x)) * s_m);
}

Fortran

c_m = abs(c)
s_m = abs(s)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x, c_m, s_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    code = ((-1.0d0) / (c_m * x)) / ((s_m * (c_m * x)) * s_m)
end function

Java

c_m = Math.abs(c);
s_m = Math.abs(s);
assert x < c_m && c_m < s_m;
public static double code(double x, double c_m, double s_m) {
	return (-1.0 / (c_m * x)) / ((s_m * (c_m * x)) * s_m);
}

Python

c_m = math.fabs(c)
s_m = math.fabs(s)
[x, c_m, s_m] = sort([x, c_m, s_m])
def code(x, c_m, s_m):
	return (-1.0 / (c_m * x)) / ((s_m * (c_m * x)) * s_m)

Julia

c_m = abs(c)
s_m = abs(s)
x, c_m, s_m = sort([x, c_m, s_m])
function code(x, c_m, s_m)
	return Float64(Float64(-1.0 / Float64(c_m * x)) / Float64(Float64(s_m * Float64(c_m * x)) * s_m))
end

MATLAB

c_m = abs(c);
s_m = abs(s);
x, c_m, s_m = num2cell(sort([x, c_m, s_m])){:}
function tmp = code(x, c_m, s_m)
	tmp = (-1.0 / (c_m * x)) / ((s_m * (c_m * x)) * s_m);
end

Wolfram

c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
code[x_, c$95$m_, s$95$m_] := N[(N[(-1.0 / N[(c$95$m * x), $MachinePrecision]), $MachinePrecision] / N[(N[(s$95$m * N[(c$95$m * x), $MachinePrecision]), $MachinePrecision] * s$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 65.7%

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

4. Applied egg-rr 84.5%

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

5. Taylor expanded in x around 0 69.0%

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

7. Applied egg-rr 35.8%

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

Alternative 6: 37.2% accurate, 24.1× speedup

Math

\[\begin{array}{l} c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x, c_m, s_m] = \mathsf{sort}([x, c_m, s_m])\\ \\ \begin{array}{l} t_0 := s\_m \cdot \left(c\_m \cdot x\right)\\ \frac{\frac{-1}{t\_0}}{t\_0} \end{array} \end{array} \]

FPCore

c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x c_m s_m)
 :precision binary64
 (let* ((t_0 (* s_m (* c_m x)))) (/ (/ -1.0 t_0) t_0)))

C

c_m = fabs(c);
s_m = fabs(s);
assert(x < c_m && c_m < s_m);
double code(double x, double c_m, double s_m) {
	double t_0 = s_m * (c_m * x);
	return (-1.0 / t_0) / t_0;
}

Fortran

c_m = abs(c)
s_m = abs(s)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x, c_m, s_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    real(8) :: t_0
    t_0 = s_m * (c_m * x)
    code = ((-1.0d0) / t_0) / t_0
end function

Java

c_m = Math.abs(c);
s_m = Math.abs(s);
assert x < c_m && c_m < s_m;
public static double code(double x, double c_m, double s_m) {
	double t_0 = s_m * (c_m * x);
	return (-1.0 / t_0) / t_0;
}

Python

c_m = math.fabs(c)
s_m = math.fabs(s)
[x, c_m, s_m] = sort([x, c_m, s_m])
def code(x, c_m, s_m):
	t_0 = s_m * (c_m * x)
	return (-1.0 / t_0) / t_0

Julia

c_m = abs(c)
s_m = abs(s)
x, c_m, s_m = sort([x, c_m, s_m])
function code(x, c_m, s_m)
	t_0 = Float64(s_m * Float64(c_m * x))
	return Float64(Float64(-1.0 / t_0) / t_0)
end

MATLAB

c_m = abs(c);
s_m = abs(s);
x, c_m, s_m = num2cell(sort([x, c_m, s_m])){:}
function tmp = code(x, c_m, s_m)
	t_0 = s_m * (c_m * x);
	tmp = (-1.0 / t_0) / t_0;
end

Wolfram

c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
code[x_, c$95$m_, s$95$m_] := Block[{t$95$0 = N[(s$95$m * N[(c$95$m * x), $MachinePrecision]), $MachinePrecision]}, N[(N[(-1.0 / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]]

Derivation

1. Initial program 65.7%

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

4. Applied egg-rr 84.5%

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

5. Taylor expanded in x around 0 69.0%

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

7. Applied egg-rr 35.9%

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

Alternative 7: 75.0% accurate, 24.1× speedup

Math

\[\begin{array}{l} c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x, c_m, s_m] = \mathsf{sort}([x, c_m, s_m])\\ \\ \frac{\frac{1}{s\_m}}{\left(s\_m \cdot \left(c\_m \cdot x\right)\right) \cdot \left(c\_m \cdot x\right)} \end{array} \]

FPCore

c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x c_m s_m)
 :precision binary64
 (/ (/ 1.0 s_m) (* (* s_m (* c_m x)) (* c_m x))))

C

c_m = fabs(c);
s_m = fabs(s);
assert(x < c_m && c_m < s_m);
double code(double x, double c_m, double s_m) {
	return (1.0 / s_m) / ((s_m * (c_m * x)) * (c_m * x));
}

Fortran

c_m = abs(c)
s_m = abs(s)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x, c_m, s_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    code = (1.0d0 / s_m) / ((s_m * (c_m * x)) * (c_m * x))
end function

Java

c_m = Math.abs(c);
s_m = Math.abs(s);
assert x < c_m && c_m < s_m;
public static double code(double x, double c_m, double s_m) {
	return (1.0 / s_m) / ((s_m * (c_m * x)) * (c_m * x));
}

Python

c_m = math.fabs(c)
s_m = math.fabs(s)
[x, c_m, s_m] = sort([x, c_m, s_m])
def code(x, c_m, s_m):
	return (1.0 / s_m) / ((s_m * (c_m * x)) * (c_m * x))

Julia

c_m = abs(c)
s_m = abs(s)
x, c_m, s_m = sort([x, c_m, s_m])
function code(x, c_m, s_m)
	return Float64(Float64(1.0 / s_m) / Float64(Float64(s_m * Float64(c_m * x)) * Float64(c_m * x)))
end

MATLAB

c_m = abs(c);
s_m = abs(s);
x, c_m, s_m = num2cell(sort([x, c_m, s_m])){:}
function tmp = code(x, c_m, s_m)
	tmp = (1.0 / s_m) / ((s_m * (c_m * x)) * (c_m * x));
end

Wolfram

c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
code[x_, c$95$m_, s$95$m_] := N[(N[(1.0 / s$95$m), $MachinePrecision] / N[(N[(s$95$m * N[(c$95$m * x), $MachinePrecision]), $MachinePrecision] * N[(c$95$m * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 65.7%

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

4. Applied egg-rr 84.5%

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

5. Taylor expanded in x around 0 69.0%

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

7. Applied egg-rr 77.0%

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

Alternative 8: 79.8% accurate, 24.1× speedup

Math

\[\begin{array}{l} c_m = \left|c\right| \\ s_m = \left|s\right| \\ [x, c_m, s_m] = \mathsf{sort}([x, c_m, s_m])\\ \\ \frac{\frac{1}{c\_m \cdot \left(s\_m \cdot x\right)}}{c\_m \cdot \left(x \cdot s\_m\right)} \end{array} \]

FPCore

c_m = (fabs.f64 c)
s_m = (fabs.f64 s)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
(FPCore (x c_m s_m)
 :precision binary64
 (/ (/ 1.0 (* c_m (* s_m x))) (* c_m (* x s_m))))

C

c_m = fabs(c);
s_m = fabs(s);
assert(x < c_m && c_m < s_m);
double code(double x, double c_m, double s_m) {
	return (1.0 / (c_m * (s_m * x))) / (c_m * (x * s_m));
}

Fortran

c_m = abs(c)
s_m = abs(s)
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x, c_m, s_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: c_m
    real(8), intent (in) :: s_m
    code = (1.0d0 / (c_m * (s_m * x))) / (c_m * (x * s_m))
end function

Java

c_m = Math.abs(c);
s_m = Math.abs(s);
assert x < c_m && c_m < s_m;
public static double code(double x, double c_m, double s_m) {
	return (1.0 / (c_m * (s_m * x))) / (c_m * (x * s_m));
}

Python

c_m = math.fabs(c)
s_m = math.fabs(s)
[x, c_m, s_m] = sort([x, c_m, s_m])
def code(x, c_m, s_m):
	return (1.0 / (c_m * (s_m * x))) / (c_m * (x * s_m))

Julia

c_m = abs(c)
s_m = abs(s)
x, c_m, s_m = sort([x, c_m, s_m])
function code(x, c_m, s_m)
	return Float64(Float64(1.0 / Float64(c_m * Float64(s_m * x))) / Float64(c_m * Float64(x * s_m)))
end

MATLAB

c_m = abs(c);
s_m = abs(s);
x, c_m, s_m = num2cell(sort([x, c_m, s_m])){:}
function tmp = code(x, c_m, s_m)
	tmp = (1.0 / (c_m * (s_m * x))) / (c_m * (x * s_m));
end

Wolfram

c_m = N[Abs[c], $MachinePrecision]
s_m = N[Abs[s], $MachinePrecision]
NOTE: x, c_m, and s_m should be sorted in increasing order before calling this function.
code[x_, c$95$m_, s$95$m_] := N[(N[(1.0 / N[(c$95$m * N[(s$95$m * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(c$95$m * N[(x * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 65.7%

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

   1. *-un-lft-identity 65.7%

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

   2. add-sqr-sqrt 65.7%

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

   3. times-frac 65.7%

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

   4. sqrt-prod 65.7%

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

   5. sqrt-pow1 46.6%

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

   6. metadata-eval 46.6%

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

   7. pow1 46.6%

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

   8. *-commutative 46.6%

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

   9. associate-*r* 41.9%

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

   10. unpow2 41.9%

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

   11. pow-prod-down 46.6%

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

   12. sqrt-pow1 46.7%

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

   13. metadata-eval 46.7%

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

   14. pow1 46.7%

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

   15. *-commutative 46.7%

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

4. Applied egg-rr 98.1%

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

   1. associate-*l/ 98.2%

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

   2. *-un-lft-identity 98.2%

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

   3. *-commutative 98.2%

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

6. Applied egg-rr 98.2%

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

7. Taylor expanded in x around 0 78.4%

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

Reproduce

herbie shell --seed 2024076 -o generate:simplify
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))