mixedcos

Percentage Accurate: 67.2% → 97.1%

Time: 15.7s

Alternatives: 13

Speedup: 24.1×

• 31.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 67.2% 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: 97.1% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.7%

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

   1. *-un-lft-identity 72.7%

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

   2. add-sqr-sqrt 72.7%

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

   3. times-frac 72.7%

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

3. Applied egg-rr 98.1%

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

   1. *-commutative 98.1%

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

   2. frac-2neg 98.1%

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

   3. frac-2neg 98.1%

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

   4. metadata-eval 98.1%

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

   5. frac-times 97.9%

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

   6. *-commutative 97.9%

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

   7. associate-*r* 97.2%

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

   8. distribute-rgt-neg-in 97.2%

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

   9. associate-*r* 98.7%

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

   10. distribute-rgt-neg-in 98.7%

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

5. Applied egg-rr 98.7%

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

6. Final simplification 98.7%

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

Alternative 2: 93.4% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.7%

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

   1. *-un-lft-identity 72.7%

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

   2. add-sqr-sqrt 72.7%

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

   3. times-frac 72.7%

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

3. Applied egg-rr 98.1%

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

   1. *-commutative 98.1%

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

   2. associate-/r* 98.1%

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

   3. frac-times 95.4%

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

   4. div-inv 95.4%

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

   5. *-commutative 95.4%

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

5. Applied egg-rr 95.4%

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

6. Final simplification 95.4%

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

Alternative 3: 97.1% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

s_m = N[Abs[s], $MachinePrecision]
NOTE: x, c, and s_m should be sorted in increasing order before calling this function.
code[x_, c_, s$95$m_] := Block[{t$95$0 = N[(c * 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 72.7%

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

   1. *-un-lft-identity 72.7%

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

   2. add-sqr-sqrt 72.7%

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

   3. times-frac 72.7%

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

3. Applied egg-rr 98.1%

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

   1. associate-*l/ 98.0%

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

   2. *-un-lft-identity 98.0%

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

   3. *-commutative 98.0%

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

5. Applied egg-rr 98.0%

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

6. Final simplification 98.0%

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

Alternative 4: 78.6% accurate, 18.3× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ [x, c, s_m] = \mathsf{sort}([x, c, s_m])\\ \\ \begin{array}{l} t_0 := c \cdot \left(x \cdot s_m\right)\\ \mathbf{if}\;x \leq 2.65 \cdot 10^{+14}:\\ \;\;\;\;\frac{1}{t_0 \cdot t_0}\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{+185}:\\ \;\;\;\;\frac{\frac{-1}{c \cdot s_m}}{x \cdot \left(\left(x \cdot c\right) \cdot s_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s_m \cdot \left(\left(x \cdot c\right) \cdot t_0\right)}\\ \end{array} \end{array} \]

FPCore

s_m = (fabs.f64 s)
NOTE: x, c, and s_m should be sorted in increasing order before calling this function.
(FPCore (x c s_m)
 :precision binary64
 (let* ((t_0 (* c (* x s_m))))
   (if (<= x 2.65e+14)
     (/ 1.0 (* t_0 t_0))
     (if (<= x 1.05e+185)
       (/ (/ -1.0 (* c s_m)) (* x (* (* x c) s_m)))
       (/ 1.0 (* s_m (* (* x c) t_0)))))))

C

s_m = fabs(s);
assert(x < c && c < s_m);
double code(double x, double c, double s_m) {
	double t_0 = c * (x * s_m);
	double tmp;
	if (x <= 2.65e+14) {
		tmp = 1.0 / (t_0 * t_0);
	} else if (x <= 1.05e+185) {
		tmp = (-1.0 / (c * s_m)) / (x * ((x * c) * s_m));
	} else {
		tmp = 1.0 / (s_m * ((x * c) * t_0));
	}
	return tmp;
}

Fortran

s_m = abs(s)
NOTE: x, c, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x, c, s_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c * (x * s_m)
    if (x <= 2.65d+14) then
        tmp = 1.0d0 / (t_0 * t_0)
    else if (x <= 1.05d+185) then
        tmp = ((-1.0d0) / (c * s_m)) / (x * ((x * c) * s_m))
    else
        tmp = 1.0d0 / (s_m * ((x * c) * t_0))
    end if
    code = tmp
end function

Java

s_m = Math.abs(s);
assert x < c && c < s_m;
public static double code(double x, double c, double s_m) {
	double t_0 = c * (x * s_m);
	double tmp;
	if (x <= 2.65e+14) {
		tmp = 1.0 / (t_0 * t_0);
	} else if (x <= 1.05e+185) {
		tmp = (-1.0 / (c * s_m)) / (x * ((x * c) * s_m));
	} else {
		tmp = 1.0 / (s_m * ((x * c) * t_0));
	}
	return tmp;
}

Python

s_m = math.fabs(s)
[x, c, s_m] = sort([x, c, s_m])
def code(x, c, s_m):
	t_0 = c * (x * s_m)
	tmp = 0
	if x <= 2.65e+14:
		tmp = 1.0 / (t_0 * t_0)
	elif x <= 1.05e+185:
		tmp = (-1.0 / (c * s_m)) / (x * ((x * c) * s_m))
	else:
		tmp = 1.0 / (s_m * ((x * c) * t_0))
	return tmp

Julia

s_m = abs(s)
x, c, s_m = sort([x, c, s_m])
function code(x, c, s_m)
	t_0 = Float64(c * Float64(x * s_m))
	tmp = 0.0
	if (x <= 2.65e+14)
		tmp = Float64(1.0 / Float64(t_0 * t_0));
	elseif (x <= 1.05e+185)
		tmp = Float64(Float64(-1.0 / Float64(c * s_m)) / Float64(x * Float64(Float64(x * c) * s_m)));
	else
		tmp = Float64(1.0 / Float64(s_m * Float64(Float64(x * c) * t_0)));
	end
	return tmp
end

MATLAB

s_m = abs(s);
x, c, s_m = num2cell(sort([x, c, s_m])){:}
function tmp_2 = code(x, c, s_m)
	t_0 = c * (x * s_m);
	tmp = 0.0;
	if (x <= 2.65e+14)
		tmp = 1.0 / (t_0 * t_0);
	elseif (x <= 1.05e+185)
		tmp = (-1.0 / (c * s_m)) / (x * ((x * c) * s_m));
	else
		tmp = 1.0 / (s_m * ((x * c) * t_0));
	end
	tmp_2 = tmp;
end

Wolfram

s_m = N[Abs[s], $MachinePrecision]
NOTE: x, c, and s_m should be sorted in increasing order before calling this function.
code[x_, c_, s$95$m_] := Block[{t$95$0 = N[(c * N[(x * s$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 2.65e+14], N[(1.0 / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.05e+185], N[(N[(-1.0 / N[(c * s$95$m), $MachinePrecision]), $MachinePrecision] / N[(x * N[(N[(x * c), $MachinePrecision] * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(s$95$m * N[(N[(x * c), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < 2.65e14

   1. Initial program 72.7%

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

   2. Taylor expanded in x around 0 58.4%

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

      1. associate-/r* 58.4%

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

      2. *-commutative 58.4%

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

      3. unpow2 58.4%

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

      4. unpow2 58.4%

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

      5. swap-sqr 68.3%

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

      6. unpow2 68.3%

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

      7. associate-/r* 68.3%

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

      8. unpow2 68.3%

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

      9. unpow2 68.3%

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

      10. swap-sqr 81.7%

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

      11. unpow2 81.7%

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

      12. *-commutative 81.7%

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

   4. Simplified 81.7%

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

      1. *-commutative 81.7%

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

      2. pow2 81.7%

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

   6. Applied egg-rr 81.7%

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

   if 2.65e14 < x < 1.05e185

   1. Initial program 77.1%

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

   2. Taylor expanded in x around 0 46.7%

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

      1. associate-/r* 46.6%

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

      2. *-commutative 46.6%

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

      3. unpow2 46.6%

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

      4. unpow2 46.6%

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

      5. swap-sqr 49.3%

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

      6. unpow2 49.3%

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

      7. associate-/r* 49.4%

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

      8. unpow2 49.4%

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

      9. unpow2 49.4%

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

      10. swap-sqr 51.6%

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

      11. unpow2 51.6%

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

      12. *-commutative 51.6%

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

   4. Simplified 51.6%

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

      1. unpow2 51.6%

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

      2. associate-*r* 51.6%

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

      3. *-commutative 51.6%

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

      4. associate-*l* 51.6%

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

   6. Applied egg-rr 51.6%

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

   8. Applied egg-rr 51.6%

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

   10. Applied egg-rr 61.9%

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

   if 1.05e185 < x

   1. Initial program 66.2%

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

   2. Taylor expanded in x around 0 62.4%

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

      1. associate-/r* 62.4%

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

      2. *-commutative 62.4%

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

      3. unpow2 62.4%

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

      4. unpow2 62.4%

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

      5. swap-sqr 65.9%

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

      6. unpow2 65.9%

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

      7. associate-/r* 65.9%

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

      8. unpow2 65.9%

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

      9. unpow2 65.9%

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

      10. swap-sqr 75.7%

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

      11. unpow2 75.7%

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

      12. *-commutative 75.7%

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

   4. Simplified 75.7%

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

      1. *-commutative 75.7%

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

      2. pow2 75.7%

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

      3. associate-*r* 75.7%

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

      4. associate-*r* 75.7%

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

   6. Applied egg-rr 75.7%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.65 \cdot 10^{+14}:\\ \;\;\;\;\frac{1}{\left(c \cdot \left(x \cdot s\right)\right) \cdot \left(c \cdot \left(x \cdot s\right)\right)}\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{+185}:\\ \;\;\;\;\frac{\frac{-1}{c \cdot s}}{x \cdot \left(\left(x \cdot c\right) \cdot s\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot \left(\left(x \cdot c\right) \cdot \left(c \cdot \left(x \cdot s\right)\right)\right)}\\ \end{array} \]

Alternative 5: 78.6% accurate, 18.3× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ [x, c, s_m] = \mathsf{sort}([x, c, s_m])\\ \\ \begin{array}{l} t_0 := c \cdot \left(x \cdot s_m\right)\\ \mathbf{if}\;x \leq 2.65 \cdot 10^{+14}:\\ \;\;\;\;\frac{\frac{1}{t_0}}{t_0}\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{+185}:\\ \;\;\;\;\frac{\frac{-1}{c \cdot s_m}}{x \cdot \left(\left(x \cdot c\right) \cdot s_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s_m \cdot \left(\left(x \cdot c\right) \cdot t_0\right)}\\ \end{array} \end{array} \]

FPCore

s_m = (fabs.f64 s)
NOTE: x, c, and s_m should be sorted in increasing order before calling this function.
(FPCore (x c s_m)
 :precision binary64
 (let* ((t_0 (* c (* x s_m))))
   (if (<= x 2.65e+14)
     (/ (/ 1.0 t_0) t_0)
     (if (<= x 1.05e+185)
       (/ (/ -1.0 (* c s_m)) (* x (* (* x c) s_m)))
       (/ 1.0 (* s_m (* (* x c) t_0)))))))

C

s_m = fabs(s);
assert(x < c && c < s_m);
double code(double x, double c, double s_m) {
	double t_0 = c * (x * s_m);
	double tmp;
	if (x <= 2.65e+14) {
		tmp = (1.0 / t_0) / t_0;
	} else if (x <= 1.05e+185) {
		tmp = (-1.0 / (c * s_m)) / (x * ((x * c) * s_m));
	} else {
		tmp = 1.0 / (s_m * ((x * c) * t_0));
	}
	return tmp;
}

Fortran

s_m = abs(s)
NOTE: x, c, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x, c, s_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c * (x * s_m)
    if (x <= 2.65d+14) then
        tmp = (1.0d0 / t_0) / t_0
    else if (x <= 1.05d+185) then
        tmp = ((-1.0d0) / (c * s_m)) / (x * ((x * c) * s_m))
    else
        tmp = 1.0d0 / (s_m * ((x * c) * t_0))
    end if
    code = tmp
end function

Java

s_m = Math.abs(s);
assert x < c && c < s_m;
public static double code(double x, double c, double s_m) {
	double t_0 = c * (x * s_m);
	double tmp;
	if (x <= 2.65e+14) {
		tmp = (1.0 / t_0) / t_0;
	} else if (x <= 1.05e+185) {
		tmp = (-1.0 / (c * s_m)) / (x * ((x * c) * s_m));
	} else {
		tmp = 1.0 / (s_m * ((x * c) * t_0));
	}
	return tmp;
}

Python

s_m = math.fabs(s)
[x, c, s_m] = sort([x, c, s_m])
def code(x, c, s_m):
	t_0 = c * (x * s_m)
	tmp = 0
	if x <= 2.65e+14:
		tmp = (1.0 / t_0) / t_0
	elif x <= 1.05e+185:
		tmp = (-1.0 / (c * s_m)) / (x * ((x * c) * s_m))
	else:
		tmp = 1.0 / (s_m * ((x * c) * t_0))
	return tmp

Julia

s_m = abs(s)
x, c, s_m = sort([x, c, s_m])
function code(x, c, s_m)
	t_0 = Float64(c * Float64(x * s_m))
	tmp = 0.0
	if (x <= 2.65e+14)
		tmp = Float64(Float64(1.0 / t_0) / t_0);
	elseif (x <= 1.05e+185)
		tmp = Float64(Float64(-1.0 / Float64(c * s_m)) / Float64(x * Float64(Float64(x * c) * s_m)));
	else
		tmp = Float64(1.0 / Float64(s_m * Float64(Float64(x * c) * t_0)));
	end
	return tmp
end

MATLAB

s_m = abs(s);
x, c, s_m = num2cell(sort([x, c, s_m])){:}
function tmp_2 = code(x, c, s_m)
	t_0 = c * (x * s_m);
	tmp = 0.0;
	if (x <= 2.65e+14)
		tmp = (1.0 / t_0) / t_0;
	elseif (x <= 1.05e+185)
		tmp = (-1.0 / (c * s_m)) / (x * ((x * c) * s_m));
	else
		tmp = 1.0 / (s_m * ((x * c) * t_0));
	end
	tmp_2 = tmp;
end

Wolfram

s_m = N[Abs[s], $MachinePrecision]
NOTE: x, c, and s_m should be sorted in increasing order before calling this function.
code[x_, c_, s$95$m_] := Block[{t$95$0 = N[(c * N[(x * s$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 2.65e+14], N[(N[(1.0 / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[x, 1.05e+185], N[(N[(-1.0 / N[(c * s$95$m), $MachinePrecision]), $MachinePrecision] / N[(x * N[(N[(x * c), $MachinePrecision] * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(s$95$m * N[(N[(x * c), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < 2.65e14

   1. Initial program 72.7%

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

      1. *-un-lft-identity 72.7%

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

      2. add-sqr-sqrt 72.6%

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

      3. times-frac 72.7%

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

   3. Applied egg-rr 97.9%

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

      1. associate-*l/ 97.9%

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

      2. *-un-lft-identity 97.9%

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

      3. *-commutative 97.9%

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

   5. Applied egg-rr 97.9%

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

   6. Taylor expanded in x around 0 81.9%

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

   if 2.65e14 < x < 1.05e185

   1. Initial program 77.1%

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

   2. Taylor expanded in x around 0 46.7%

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

      1. associate-/r* 46.6%

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

      2. *-commutative 46.6%

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

      3. unpow2 46.6%

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

      4. unpow2 46.6%

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

      5. swap-sqr 49.3%

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

      6. unpow2 49.3%

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

      7. associate-/r* 49.4%

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

      8. unpow2 49.4%

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

      9. unpow2 49.4%

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

      10. swap-sqr 51.6%

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

      11. unpow2 51.6%

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

      12. *-commutative 51.6%

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

   4. Simplified 51.6%

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

      1. unpow2 51.6%

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

      2. associate-*r* 51.6%

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

      3. *-commutative 51.6%

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

      4. associate-*l* 51.6%

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

   6. Applied egg-rr 51.6%

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

   8. Applied egg-rr 51.6%

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

   10. Applied egg-rr 61.9%

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

   if 1.05e185 < x

   1. Initial program 66.2%

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

   2. Taylor expanded in x around 0 62.4%

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

      1. associate-/r* 62.4%

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

      2. *-commutative 62.4%

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

      3. unpow2 62.4%

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

      4. unpow2 62.4%

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

      5. swap-sqr 65.9%

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

      6. unpow2 65.9%

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

      7. associate-/r* 65.9%

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

      8. unpow2 65.9%

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

      9. unpow2 65.9%

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

      10. swap-sqr 75.7%

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

      11. unpow2 75.7%

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

      12. *-commutative 75.7%

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

   4. Simplified 75.7%

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

      1. *-commutative 75.7%

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

      2. pow2 75.7%

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

      3. associate-*r* 75.7%

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

      4. associate-*r* 75.7%

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

   6. Applied egg-rr 75.7%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.65 \cdot 10^{+14}:\\ \;\;\;\;\frac{\frac{1}{c \cdot \left(x \cdot s\right)}}{c \cdot \left(x \cdot s\right)}\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{+185}:\\ \;\;\;\;\frac{\frac{-1}{c \cdot s}}{x \cdot \left(\left(x \cdot c\right) \cdot s\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot \left(\left(x \cdot c\right) \cdot \left(c \cdot \left(x \cdot s\right)\right)\right)}\\ \end{array} \]

Alternative 6: 78.7% accurate, 18.3× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ [x, c, s_m] = \mathsf{sort}([x, c, s_m])\\ \\ \begin{array}{l} t_0 := \left(x \cdot c\right) \cdot s_m\\ t_1 := c \cdot \left(x \cdot s_m\right)\\ \mathbf{if}\;x \leq 2.65 \cdot 10^{+14}:\\ \;\;\;\;\frac{\frac{1}{t_1}}{t_1}\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{+185}:\\ \;\;\;\;\frac{\frac{-1}{c \cdot s_m}}{x \cdot t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{t_0}}{t_0}\\ \end{array} \end{array} \]

FPCore

s_m = (fabs.f64 s)
NOTE: x, c, and s_m should be sorted in increasing order before calling this function.
(FPCore (x c s_m)
 :precision binary64
 (let* ((t_0 (* (* x c) s_m)) (t_1 (* c (* x s_m))))
   (if (<= x 2.65e+14)
     (/ (/ 1.0 t_1) t_1)
     (if (<= x 1.05e+185)
       (/ (/ -1.0 (* c s_m)) (* x t_0))
       (/ (/ 1.0 t_0) t_0)))))

C

s_m = fabs(s);
assert(x < c && c < s_m);
double code(double x, double c, double s_m) {
	double t_0 = (x * c) * s_m;
	double t_1 = c * (x * s_m);
	double tmp;
	if (x <= 2.65e+14) {
		tmp = (1.0 / t_1) / t_1;
	} else if (x <= 1.05e+185) {
		tmp = (-1.0 / (c * s_m)) / (x * t_0);
	} else {
		tmp = (1.0 / t_0) / t_0;
	}
	return tmp;
}

Fortran

s_m = abs(s)
NOTE: x, c, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x, c, s_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (x * c) * s_m
    t_1 = c * (x * s_m)
    if (x <= 2.65d+14) then
        tmp = (1.0d0 / t_1) / t_1
    else if (x <= 1.05d+185) then
        tmp = ((-1.0d0) / (c * s_m)) / (x * t_0)
    else
        tmp = (1.0d0 / t_0) / t_0
    end if
    code = tmp
end function

Java

s_m = Math.abs(s);
assert x < c && c < s_m;
public static double code(double x, double c, double s_m) {
	double t_0 = (x * c) * s_m;
	double t_1 = c * (x * s_m);
	double tmp;
	if (x <= 2.65e+14) {
		tmp = (1.0 / t_1) / t_1;
	} else if (x <= 1.05e+185) {
		tmp = (-1.0 / (c * s_m)) / (x * t_0);
	} else {
		tmp = (1.0 / t_0) / t_0;
	}
	return tmp;
}

Python

s_m = math.fabs(s)
[x, c, s_m] = sort([x, c, s_m])
def code(x, c, s_m):
	t_0 = (x * c) * s_m
	t_1 = c * (x * s_m)
	tmp = 0
	if x <= 2.65e+14:
		tmp = (1.0 / t_1) / t_1
	elif x <= 1.05e+185:
		tmp = (-1.0 / (c * s_m)) / (x * t_0)
	else:
		tmp = (1.0 / t_0) / t_0
	return tmp

Julia

s_m = abs(s)
x, c, s_m = sort([x, c, s_m])
function code(x, c, s_m)
	t_0 = Float64(Float64(x * c) * s_m)
	t_1 = Float64(c * Float64(x * s_m))
	tmp = 0.0
	if (x <= 2.65e+14)
		tmp = Float64(Float64(1.0 / t_1) / t_1);
	elseif (x <= 1.05e+185)
		tmp = Float64(Float64(-1.0 / Float64(c * s_m)) / Float64(x * t_0));
	else
		tmp = Float64(Float64(1.0 / t_0) / t_0);
	end
	return tmp
end

MATLAB

s_m = abs(s);
x, c, s_m = num2cell(sort([x, c, s_m])){:}
function tmp_2 = code(x, c, s_m)
	t_0 = (x * c) * s_m;
	t_1 = c * (x * s_m);
	tmp = 0.0;
	if (x <= 2.65e+14)
		tmp = (1.0 / t_1) / t_1;
	elseif (x <= 1.05e+185)
		tmp = (-1.0 / (c * s_m)) / (x * t_0);
	else
		tmp = (1.0 / t_0) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

s_m = N[Abs[s], $MachinePrecision]
NOTE: x, c, and s_m should be sorted in increasing order before calling this function.
code[x_, c_, s$95$m_] := Block[{t$95$0 = N[(N[(x * c), $MachinePrecision] * s$95$m), $MachinePrecision]}, Block[{t$95$1 = N[(c * N[(x * s$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 2.65e+14], N[(N[(1.0 / t$95$1), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[x, 1.05e+185], N[(N[(-1.0 / N[(c * s$95$m), $MachinePrecision]), $MachinePrecision] / N[(x * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < 2.65e14

   1. Initial program 72.7%

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

      1. *-un-lft-identity 72.7%

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

      2. add-sqr-sqrt 72.6%

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

      3. times-frac 72.7%

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

   3. Applied egg-rr 97.9%

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

      1. associate-*l/ 97.9%

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

      2. *-un-lft-identity 97.9%

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

      3. *-commutative 97.9%

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

   5. Applied egg-rr 97.9%

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

   6. Taylor expanded in x around 0 81.9%

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

   if 2.65e14 < x < 1.05e185

   1. Initial program 77.1%

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

   2. Taylor expanded in x around 0 46.7%

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

      1. associate-/r* 46.6%

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

      2. *-commutative 46.6%

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

      3. unpow2 46.6%

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

      4. unpow2 46.6%

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

      5. swap-sqr 49.3%

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

      6. unpow2 49.3%

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

      7. associate-/r* 49.4%

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

      8. unpow2 49.4%

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

      9. unpow2 49.4%

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

      10. swap-sqr 51.6%

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

      11. unpow2 51.6%

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

      12. *-commutative 51.6%

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

   4. Simplified 51.6%

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

      1. unpow2 51.6%

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

      2. associate-*r* 51.6%

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

      3. *-commutative 51.6%

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

      4. associate-*l* 51.6%

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

   6. Applied egg-rr 51.6%

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

   8. Applied egg-rr 51.6%

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

   10. Applied egg-rr 61.9%

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

   if 1.05e185 < x

   1. Initial program 66.2%

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

   2. Taylor expanded in x around 0 62.4%

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

      1. associate-/r* 62.4%

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

      2. *-commutative 62.4%

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

      3. unpow2 62.4%

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

      4. unpow2 62.4%

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

      5. swap-sqr 65.9%

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

      6. unpow2 65.9%

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

      7. associate-/r* 65.9%

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

      8. unpow2 65.9%

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

      9. unpow2 65.9%

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

      10. swap-sqr 75.7%

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

      11. unpow2 75.7%

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

      12. *-commutative 75.7%

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

   4. Simplified 75.7%

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

      1. unpow2 75.7%

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

      2. associate-*r* 74.9%

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

      3. *-commutative 74.9%

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

      4. associate-*l* 74.4%

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

   6. Applied egg-rr 74.4%

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

   8. Applied egg-rr 74.9%

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

      1. add-sqr-sqrt 70.0%

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

      2. sqrt-prod 74.7%

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

      3. frac-times 74.7%

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

      4. metadata-eval 74.7%

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

      5. associate-*l* 74.7%

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

      6. *-commutative 74.7%

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

      7. associate-*l* 74.7%

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

      8. *-commutative 74.7%

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

      9. associate-*r* 74.7%

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

      10. associate-*r* 74.7%

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

      11. sqrt-div 74.7%

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

      12. metadata-eval 74.7%

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

      13. associate-*r* 74.7%

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

      14. associate-*r* 74.7%

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

      15. sqrt-prod 28.9%

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

      16. add-sqr-sqrt 77.3%

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

   10. Applied egg-rr 75.7%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.65 \cdot 10^{+14}:\\ \;\;\;\;\frac{\frac{1}{c \cdot \left(x \cdot s\right)}}{c \cdot \left(x \cdot s\right)}\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{+185}:\\ \;\;\;\;\frac{\frac{-1}{c \cdot s}}{x \cdot \left(\left(x \cdot c\right) \cdot s\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\left(x \cdot c\right) \cdot s}}{\left(x \cdot c\right) \cdot s}\\ \end{array} \]

Alternative 7: 78.7% accurate, 18.3× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ [x, c, s_m] = \mathsf{sort}([x, c, s_m])\\ \\ \begin{array}{l} t_0 := \left(x \cdot c\right) \cdot s_m\\ \mathbf{if}\;x \leq 2.65 \cdot 10^{+14}:\\ \;\;\;\;\frac{\frac{\frac{1}{c}}{x \cdot s_m}}{c \cdot \left(x \cdot s_m\right)}\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{+185}:\\ \;\;\;\;\frac{\frac{-1}{c \cdot s_m}}{x \cdot t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{t_0}}{t_0}\\ \end{array} \end{array} \]

FPCore

s_m = (fabs.f64 s)
NOTE: x, c, and s_m should be sorted in increasing order before calling this function.
(FPCore (x c s_m)
 :precision binary64
 (let* ((t_0 (* (* x c) s_m)))
   (if (<= x 2.65e+14)
     (/ (/ (/ 1.0 c) (* x s_m)) (* c (* x s_m)))
     (if (<= x 1.05e+185)
       (/ (/ -1.0 (* c s_m)) (* x t_0))
       (/ (/ 1.0 t_0) t_0)))))

C

s_m = fabs(s);
assert(x < c && c < s_m);
double code(double x, double c, double s_m) {
	double t_0 = (x * c) * s_m;
	double tmp;
	if (x <= 2.65e+14) {
		tmp = ((1.0 / c) / (x * s_m)) / (c * (x * s_m));
	} else if (x <= 1.05e+185) {
		tmp = (-1.0 / (c * s_m)) / (x * t_0);
	} else {
		tmp = (1.0 / t_0) / t_0;
	}
	return tmp;
}

Fortran

s_m = abs(s)
NOTE: x, c, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x, c, s_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x * c) * s_m
    if (x <= 2.65d+14) then
        tmp = ((1.0d0 / c) / (x * s_m)) / (c * (x * s_m))
    else if (x <= 1.05d+185) then
        tmp = ((-1.0d0) / (c * s_m)) / (x * t_0)
    else
        tmp = (1.0d0 / t_0) / t_0
    end if
    code = tmp
end function

Java

s_m = Math.abs(s);
assert x < c && c < s_m;
public static double code(double x, double c, double s_m) {
	double t_0 = (x * c) * s_m;
	double tmp;
	if (x <= 2.65e+14) {
		tmp = ((1.0 / c) / (x * s_m)) / (c * (x * s_m));
	} else if (x <= 1.05e+185) {
		tmp = (-1.0 / (c * s_m)) / (x * t_0);
	} else {
		tmp = (1.0 / t_0) / t_0;
	}
	return tmp;
}

Python

s_m = math.fabs(s)
[x, c, s_m] = sort([x, c, s_m])
def code(x, c, s_m):
	t_0 = (x * c) * s_m
	tmp = 0
	if x <= 2.65e+14:
		tmp = ((1.0 / c) / (x * s_m)) / (c * (x * s_m))
	elif x <= 1.05e+185:
		tmp = (-1.0 / (c * s_m)) / (x * t_0)
	else:
		tmp = (1.0 / t_0) / t_0
	return tmp

Julia

s_m = abs(s)
x, c, s_m = sort([x, c, s_m])
function code(x, c, s_m)
	t_0 = Float64(Float64(x * c) * s_m)
	tmp = 0.0
	if (x <= 2.65e+14)
		tmp = Float64(Float64(Float64(1.0 / c) / Float64(x * s_m)) / Float64(c * Float64(x * s_m)));
	elseif (x <= 1.05e+185)
		tmp = Float64(Float64(-1.0 / Float64(c * s_m)) / Float64(x * t_0));
	else
		tmp = Float64(Float64(1.0 / t_0) / t_0);
	end
	return tmp
end

MATLAB

s_m = abs(s);
x, c, s_m = num2cell(sort([x, c, s_m])){:}
function tmp_2 = code(x, c, s_m)
	t_0 = (x * c) * s_m;
	tmp = 0.0;
	if (x <= 2.65e+14)
		tmp = ((1.0 / c) / (x * s_m)) / (c * (x * s_m));
	elseif (x <= 1.05e+185)
		tmp = (-1.0 / (c * s_m)) / (x * t_0);
	else
		tmp = (1.0 / t_0) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

s_m = N[Abs[s], $MachinePrecision]
NOTE: x, c, and s_m should be sorted in increasing order before calling this function.
code[x_, c_, s$95$m_] := Block[{t$95$0 = N[(N[(x * c), $MachinePrecision] * s$95$m), $MachinePrecision]}, If[LessEqual[x, 2.65e+14], N[(N[(N[(1.0 / c), $MachinePrecision] / N[(x * s$95$m), $MachinePrecision]), $MachinePrecision] / N[(c * N[(x * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.05e+185], N[(N[(-1.0 / N[(c * s$95$m), $MachinePrecision]), $MachinePrecision] / N[(x * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < 2.65e14

   1. Initial program 72.7%

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

      1. *-un-lft-identity 72.7%

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

      2. add-sqr-sqrt 72.6%

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

      3. times-frac 72.7%

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

   3. Applied egg-rr 97.9%

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

      1. associate-*l/ 97.9%

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

      2. *-un-lft-identity 97.9%

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

      3. *-commutative 97.9%

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

   5. Applied egg-rr 97.9%

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

   6. Taylor expanded in x around 0 81.9%

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

      1. associate-/r* 81.9%

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

   8. Simplified 81.9%

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

   if 2.65e14 < x < 1.05e185

   1. Initial program 77.1%

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

   2. Taylor expanded in x around 0 46.7%

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

      1. associate-/r* 46.6%

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

      2. *-commutative 46.6%

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

      3. unpow2 46.6%

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

      4. unpow2 46.6%

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

      5. swap-sqr 49.3%

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

      6. unpow2 49.3%

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

      7. associate-/r* 49.4%

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

      8. unpow2 49.4%

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

      9. unpow2 49.4%

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

      10. swap-sqr 51.6%

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

      11. unpow2 51.6%

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

      12. *-commutative 51.6%

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

   4. Simplified 51.6%

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

      1. unpow2 51.6%

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

      2. associate-*r* 51.6%

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

      3. *-commutative 51.6%

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

      4. associate-*l* 51.6%

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

   6. Applied egg-rr 51.6%

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

   8. Applied egg-rr 51.6%

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

   10. Applied egg-rr 61.9%

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

   if 1.05e185 < x

   1. Initial program 66.2%

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

   2. Taylor expanded in x around 0 62.4%

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

      1. associate-/r* 62.4%

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

      2. *-commutative 62.4%

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

      3. unpow2 62.4%

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

      4. unpow2 62.4%

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

      5. swap-sqr 65.9%

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

      6. unpow2 65.9%

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

      7. associate-/r* 65.9%

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

      8. unpow2 65.9%

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

      9. unpow2 65.9%

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

      10. swap-sqr 75.7%

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

      11. unpow2 75.7%

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

      12. *-commutative 75.7%

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

   4. Simplified 75.7%

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

      1. unpow2 75.7%

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

      2. associate-*r* 74.9%

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

      3. *-commutative 74.9%

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

      4. associate-*l* 74.4%

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

   6. Applied egg-rr 74.4%

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

   8. Applied egg-rr 74.9%

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

      1. add-sqr-sqrt 70.0%

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

      2. sqrt-prod 74.7%

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

      3. frac-times 74.7%

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

      4. metadata-eval 74.7%

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

      5. associate-*l* 74.7%

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

      6. *-commutative 74.7%

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

      7. associate-*l* 74.7%

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

      8. *-commutative 74.7%

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

      9. associate-*r* 74.7%

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

      10. associate-*r* 74.7%

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

      11. sqrt-div 74.7%

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

      12. metadata-eval 74.7%

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

      13. associate-*r* 74.7%

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

      14. associate-*r* 74.7%

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

      15. sqrt-prod 28.9%

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

      16. add-sqr-sqrt 77.3%

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

   10. Applied egg-rr 75.7%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.65 \cdot 10^{+14}:\\ \;\;\;\;\frac{\frac{\frac{1}{c}}{x \cdot s}}{c \cdot \left(x \cdot s\right)}\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{+185}:\\ \;\;\;\;\frac{\frac{-1}{c \cdot s}}{x \cdot \left(\left(x \cdot c\right) \cdot s\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\left(x \cdot c\right) \cdot s}}{\left(x \cdot c\right) \cdot s}\\ \end{array} \]

Alternative 8: 78.9% accurate, 18.3× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ [x, c, s_m] = \mathsf{sort}([x, c, s_m])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 2.65 \cdot 10^{+14}:\\ \;\;\;\;\frac{\frac{\frac{1}{c}}{x \cdot s_m}}{c \cdot \left(x \cdot s_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\left(x \cdot c\right) \cdot s_m}}{x \cdot c} \cdot \frac{-1}{s_m}\\ \end{array} \end{array} \]

FPCore

s_m = (fabs.f64 s)
NOTE: x, c, and s_m should be sorted in increasing order before calling this function.
(FPCore (x c s_m)
 :precision binary64
 (if (<= x 2.65e+14)
   (/ (/ (/ 1.0 c) (* x s_m)) (* c (* x s_m)))
   (* (/ (/ 1.0 (* (* x c) s_m)) (* x c)) (/ -1.0 s_m))))

C

s_m = fabs(s);
assert(x < c && c < s_m);
double code(double x, double c, double s_m) {
	double tmp;
	if (x <= 2.65e+14) {
		tmp = ((1.0 / c) / (x * s_m)) / (c * (x * s_m));
	} else {
		tmp = ((1.0 / ((x * c) * s_m)) / (x * c)) * (-1.0 / s_m);
	}
	return tmp;
}

Fortran

s_m = abs(s)
NOTE: x, c, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x, c, s_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s_m
    real(8) :: tmp
    if (x <= 2.65d+14) then
        tmp = ((1.0d0 / c) / (x * s_m)) / (c * (x * s_m))
    else
        tmp = ((1.0d0 / ((x * c) * s_m)) / (x * c)) * ((-1.0d0) / s_m)
    end if
    code = tmp
end function

Java

s_m = Math.abs(s);
assert x < c && c < s_m;
public static double code(double x, double c, double s_m) {
	double tmp;
	if (x <= 2.65e+14) {
		tmp = ((1.0 / c) / (x * s_m)) / (c * (x * s_m));
	} else {
		tmp = ((1.0 / ((x * c) * s_m)) / (x * c)) * (-1.0 / s_m);
	}
	return tmp;
}

Python

s_m = math.fabs(s)
[x, c, s_m] = sort([x, c, s_m])
def code(x, c, s_m):
	tmp = 0
	if x <= 2.65e+14:
		tmp = ((1.0 / c) / (x * s_m)) / (c * (x * s_m))
	else:
		tmp = ((1.0 / ((x * c) * s_m)) / (x * c)) * (-1.0 / s_m)
	return tmp

Julia

s_m = abs(s)
x, c, s_m = sort([x, c, s_m])
function code(x, c, s_m)
	tmp = 0.0
	if (x <= 2.65e+14)
		tmp = Float64(Float64(Float64(1.0 / c) / Float64(x * s_m)) / Float64(c * Float64(x * s_m)));
	else
		tmp = Float64(Float64(Float64(1.0 / Float64(Float64(x * c) * s_m)) / Float64(x * c)) * Float64(-1.0 / s_m));
	end
	return tmp
end

MATLAB

s_m = abs(s);
x, c, s_m = num2cell(sort([x, c, s_m])){:}
function tmp_2 = code(x, c, s_m)
	tmp = 0.0;
	if (x <= 2.65e+14)
		tmp = ((1.0 / c) / (x * s_m)) / (c * (x * s_m));
	else
		tmp = ((1.0 / ((x * c) * s_m)) / (x * c)) * (-1.0 / s_m);
	end
	tmp_2 = tmp;
end

Wolfram

s_m = N[Abs[s], $MachinePrecision]
NOTE: x, c, and s_m should be sorted in increasing order before calling this function.
code[x_, c_, s$95$m_] := If[LessEqual[x, 2.65e+14], N[(N[(N[(1.0 / c), $MachinePrecision] / N[(x * s$95$m), $MachinePrecision]), $MachinePrecision] / N[(c * N[(x * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / N[(N[(x * c), $MachinePrecision] * s$95$m), $MachinePrecision]), $MachinePrecision] / N[(x * c), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / s$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.65e14

   1. Initial program 72.7%

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

      1. *-un-lft-identity 72.7%

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

      2. add-sqr-sqrt 72.6%

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

      3. times-frac 72.7%

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

   3. Applied egg-rr 97.9%

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

      1. associate-*l/ 97.9%

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

      2. *-un-lft-identity 97.9%

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

      3. *-commutative 97.9%

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

   5. Applied egg-rr 97.9%

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

   6. Taylor expanded in x around 0 81.9%

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

      1. associate-/r* 81.9%

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

   8. Simplified 81.9%

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

   if 2.65e14 < x

   1. Initial program 72.7%

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

   2. Taylor expanded in x around 0 53.0%

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

      1. associate-/r* 53.0%

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

      2. *-commutative 53.0%

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

      3. unpow2 53.0%

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

      4. unpow2 53.0%

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

      5. swap-sqr 56.0%

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

      6. unpow2 56.0%

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

      7. associate-/r* 56.0%

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

      8. unpow2 56.0%

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

      9. unpow2 56.0%

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

      10. swap-sqr 61.3%

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

      11. unpow2 61.3%

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

      12. *-commutative 61.3%

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

   4. Simplified 61.3%

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

      1. *-commutative 61.3%

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

      2. pow2 61.3%

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

      3. associate-*r* 61.3%

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

      4. associate-*l* 61.1%

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

   6. Applied egg-rr 61.1%

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

      1. metadata-eval 61.1%

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

      2. associate-*r* 61.3%

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

      3. associate-*r* 61.3%

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

      4. frac-times 61.3%

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

      5. *-commutative 61.3%

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

      6. associate-*l* 61.0%

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

      7. /-rgt-identity 61.0%

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

      8. times-frac 61.0%

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

      9. *-un-lft-identity 61.0%

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

      10. associate-*r* 61.0%

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

      11. times-frac 60.7%

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

   8. Applied egg-rr 68.6%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.65 \cdot 10^{+14}:\\ \;\;\;\;\frac{\frac{\frac{1}{c}}{x \cdot s}}{c \cdot \left(x \cdot s\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\left(x \cdot c\right) \cdot s}}{x \cdot c} \cdot \frac{-1}{s}\\ \end{array} \]

Alternative 9: 78.9% accurate, 18.3× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ [x, c, s_m] = \mathsf{sort}([x, c, s_m])\\ \\ \begin{array}{l} t_0 := \left(x \cdot c\right) \cdot s_m\\ \mathbf{if}\;x \leq 2.65 \cdot 10^{+14}:\\ \;\;\;\;\frac{\frac{\frac{1}{c}}{x \cdot s_m}}{c \cdot \left(x \cdot s_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t_0} \cdot \frac{-1}{t_0}\\ \end{array} \end{array} \]

FPCore

s_m = (fabs.f64 s)
NOTE: x, c, and s_m should be sorted in increasing order before calling this function.
(FPCore (x c s_m)
 :precision binary64
 (let* ((t_0 (* (* x c) s_m)))
   (if (<= x 2.65e+14)
     (/ (/ (/ 1.0 c) (* x s_m)) (* c (* x s_m)))
     (* (/ 1.0 t_0) (/ -1.0 t_0)))))

C

s_m = fabs(s);
assert(x < c && c < s_m);
double code(double x, double c, double s_m) {
	double t_0 = (x * c) * s_m;
	double tmp;
	if (x <= 2.65e+14) {
		tmp = ((1.0 / c) / (x * s_m)) / (c * (x * s_m));
	} else {
		tmp = (1.0 / t_0) * (-1.0 / t_0);
	}
	return tmp;
}

Fortran

s_m = abs(s)
NOTE: x, c, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x, c, s_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x * c) * s_m
    if (x <= 2.65d+14) then
        tmp = ((1.0d0 / c) / (x * s_m)) / (c * (x * s_m))
    else
        tmp = (1.0d0 / t_0) * ((-1.0d0) / t_0)
    end if
    code = tmp
end function

Java

s_m = Math.abs(s);
assert x < c && c < s_m;
public static double code(double x, double c, double s_m) {
	double t_0 = (x * c) * s_m;
	double tmp;
	if (x <= 2.65e+14) {
		tmp = ((1.0 / c) / (x * s_m)) / (c * (x * s_m));
	} else {
		tmp = (1.0 / t_0) * (-1.0 / t_0);
	}
	return tmp;
}

Python

s_m = math.fabs(s)
[x, c, s_m] = sort([x, c, s_m])
def code(x, c, s_m):
	t_0 = (x * c) * s_m
	tmp = 0
	if x <= 2.65e+14:
		tmp = ((1.0 / c) / (x * s_m)) / (c * (x * s_m))
	else:
		tmp = (1.0 / t_0) * (-1.0 / t_0)
	return tmp

Julia

s_m = abs(s)
x, c, s_m = sort([x, c, s_m])
function code(x, c, s_m)
	t_0 = Float64(Float64(x * c) * s_m)
	tmp = 0.0
	if (x <= 2.65e+14)
		tmp = Float64(Float64(Float64(1.0 / c) / Float64(x * s_m)) / Float64(c * Float64(x * s_m)));
	else
		tmp = Float64(Float64(1.0 / t_0) * Float64(-1.0 / t_0));
	end
	return tmp
end

MATLAB

s_m = abs(s);
x, c, s_m = num2cell(sort([x, c, s_m])){:}
function tmp_2 = code(x, c, s_m)
	t_0 = (x * c) * s_m;
	tmp = 0.0;
	if (x <= 2.65e+14)
		tmp = ((1.0 / c) / (x * s_m)) / (c * (x * s_m));
	else
		tmp = (1.0 / t_0) * (-1.0 / t_0);
	end
	tmp_2 = tmp;
end

Wolfram

s_m = N[Abs[s], $MachinePrecision]
NOTE: x, c, and s_m should be sorted in increasing order before calling this function.
code[x_, c_, s$95$m_] := Block[{t$95$0 = N[(N[(x * c), $MachinePrecision] * s$95$m), $MachinePrecision]}, If[LessEqual[x, 2.65e+14], N[(N[(N[(1.0 / c), $MachinePrecision] / N[(x * s$95$m), $MachinePrecision]), $MachinePrecision] / N[(c * N[(x * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / t$95$0), $MachinePrecision] * N[(-1.0 / t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 2.65e14

   1. Initial program 72.7%

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

      1. *-un-lft-identity 72.7%

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

      2. add-sqr-sqrt 72.6%

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

      3. times-frac 72.7%

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

   3. Applied egg-rr 97.9%

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

      1. associate-*l/ 97.9%

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

      2. *-un-lft-identity 97.9%

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

      3. *-commutative 97.9%

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

   5. Applied egg-rr 97.9%

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

   6. Taylor expanded in x around 0 81.9%

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

      1. associate-/r* 81.9%

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

   8. Simplified 81.9%

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

   if 2.65e14 < x

   1. Initial program 72.7%

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

   2. Taylor expanded in x around 0 53.0%

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

      1. associate-/r* 53.0%

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

      2. *-commutative 53.0%

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

      3. unpow2 53.0%

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

      4. unpow2 53.0%

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

      5. swap-sqr 56.0%

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

      6. unpow2 56.0%

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

      7. associate-/r* 56.0%

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

      8. unpow2 56.0%

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

      9. unpow2 56.0%

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

      10. swap-sqr 61.3%

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

      11. unpow2 61.3%

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

      12. *-commutative 61.3%

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

   4. Simplified 61.3%

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

      1. *-commutative 61.3%

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

      2. pow2 61.3%

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

      3. associate-*r* 61.3%

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

      4. associate-*l* 61.1%

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

   6. Applied egg-rr 61.1%

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

      1. metadata-eval 61.1%

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

      2. associate-*r* 61.3%

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

      3. associate-*r* 61.3%

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

      4. frac-times 61.3%

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

      5. *-commutative 61.3%

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

      6. associate-*l* 61.0%

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

      7. /-rgt-identity 61.0%

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

      8. times-frac 61.0%

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

      9. *-un-lft-identity 61.0%

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

      10. div-inv 61.0%

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

   8. Applied egg-rr 68.5%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.65 \cdot 10^{+14}:\\ \;\;\;\;\frac{\frac{\frac{1}{c}}{x \cdot s}}{c \cdot \left(x \cdot s\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(x \cdot c\right) \cdot s} \cdot \frac{-1}{\left(x \cdot c\right) \cdot s}\\ \end{array} \]

Alternative 10: 78.9% accurate, 18.3× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ [x, c, s_m] = \mathsf{sort}([x, c, s_m])\\ \\ \begin{array}{l} t_0 := \left(x \cdot c\right) \cdot s_m\\ \mathbf{if}\;x \leq 2.65 \cdot 10^{+14}:\\ \;\;\;\;\frac{\frac{\frac{1}{c}}{x \cdot s_m}}{c \cdot \left(x \cdot s_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{t_0}{\frac{-1}{t_0}}}\\ \end{array} \end{array} \]

FPCore

s_m = (fabs.f64 s)
NOTE: x, c, and s_m should be sorted in increasing order before calling this function.
(FPCore (x c s_m)
 :precision binary64
 (let* ((t_0 (* (* x c) s_m)))
   (if (<= x 2.65e+14)
     (/ (/ (/ 1.0 c) (* x s_m)) (* c (* x s_m)))
     (/ 1.0 (/ t_0 (/ -1.0 t_0))))))

C

s_m = fabs(s);
assert(x < c && c < s_m);
double code(double x, double c, double s_m) {
	double t_0 = (x * c) * s_m;
	double tmp;
	if (x <= 2.65e+14) {
		tmp = ((1.0 / c) / (x * s_m)) / (c * (x * s_m));
	} else {
		tmp = 1.0 / (t_0 / (-1.0 / t_0));
	}
	return tmp;
}

Fortran

s_m = abs(s)
NOTE: x, c, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x, c, s_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x * c) * s_m
    if (x <= 2.65d+14) then
        tmp = ((1.0d0 / c) / (x * s_m)) / (c * (x * s_m))
    else
        tmp = 1.0d0 / (t_0 / ((-1.0d0) / t_0))
    end if
    code = tmp
end function

Java

s_m = Math.abs(s);
assert x < c && c < s_m;
public static double code(double x, double c, double s_m) {
	double t_0 = (x * c) * s_m;
	double tmp;
	if (x <= 2.65e+14) {
		tmp = ((1.0 / c) / (x * s_m)) / (c * (x * s_m));
	} else {
		tmp = 1.0 / (t_0 / (-1.0 / t_0));
	}
	return tmp;
}

Python

s_m = math.fabs(s)
[x, c, s_m] = sort([x, c, s_m])
def code(x, c, s_m):
	t_0 = (x * c) * s_m
	tmp = 0
	if x <= 2.65e+14:
		tmp = ((1.0 / c) / (x * s_m)) / (c * (x * s_m))
	else:
		tmp = 1.0 / (t_0 / (-1.0 / t_0))
	return tmp

Julia

s_m = abs(s)
x, c, s_m = sort([x, c, s_m])
function code(x, c, s_m)
	t_0 = Float64(Float64(x * c) * s_m)
	tmp = 0.0
	if (x <= 2.65e+14)
		tmp = Float64(Float64(Float64(1.0 / c) / Float64(x * s_m)) / Float64(c * Float64(x * s_m)));
	else
		tmp = Float64(1.0 / Float64(t_0 / Float64(-1.0 / t_0)));
	end
	return tmp
end

MATLAB

s_m = abs(s);
x, c, s_m = num2cell(sort([x, c, s_m])){:}
function tmp_2 = code(x, c, s_m)
	t_0 = (x * c) * s_m;
	tmp = 0.0;
	if (x <= 2.65e+14)
		tmp = ((1.0 / c) / (x * s_m)) / (c * (x * s_m));
	else
		tmp = 1.0 / (t_0 / (-1.0 / t_0));
	end
	tmp_2 = tmp;
end

Wolfram

s_m = N[Abs[s], $MachinePrecision]
NOTE: x, c, and s_m should be sorted in increasing order before calling this function.
code[x_, c_, s$95$m_] := Block[{t$95$0 = N[(N[(x * c), $MachinePrecision] * s$95$m), $MachinePrecision]}, If[LessEqual[x, 2.65e+14], N[(N[(N[(1.0 / c), $MachinePrecision] / N[(x * s$95$m), $MachinePrecision]), $MachinePrecision] / N[(c * N[(x * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(t$95$0 / N[(-1.0 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 2.65e14

   1. Initial program 72.7%

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

      1. *-un-lft-identity 72.7%

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

      2. add-sqr-sqrt 72.6%

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

      3. times-frac 72.7%

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

   3. Applied egg-rr 97.9%

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

      1. associate-*l/ 97.9%

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

      2. *-un-lft-identity 97.9%

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

      3. *-commutative 97.9%

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

   5. Applied egg-rr 97.9%

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

   6. Taylor expanded in x around 0 81.9%

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

      1. associate-/r* 81.9%

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

   8. Simplified 81.9%

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

   if 2.65e14 < x

   1. Initial program 72.7%

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

   2. Taylor expanded in x around 0 53.0%

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

      1. associate-/r* 53.0%

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

      2. *-commutative 53.0%

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

      3. unpow2 53.0%

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

      4. unpow2 53.0%

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

      5. swap-sqr 56.0%

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

      6. unpow2 56.0%

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

      7. associate-/r* 56.0%

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

      8. unpow2 56.0%

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

      9. unpow2 56.0%

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

      10. swap-sqr 61.3%

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

      11. unpow2 61.3%

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

      12. *-commutative 61.3%

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

   4. Simplified 61.3%

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

      1. unpow2 61.3%

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

      2. associate-*r* 61.0%

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

      3. *-commutative 61.0%

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

      4. associate-*l* 60.8%

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

   6. Applied egg-rr 60.8%

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

   8. Applied egg-rr 61.0%

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

      1. associate-*r/ 61.0%

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

      2. associate-*l* 61.0%

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

      3. *-commutative 61.0%

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

      4. clear-num 61.0%

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

      5. *-commutative 61.0%

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

      6. *-commutative 61.0%

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

      7. associate-*l* 61.0%

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

   10. Applied egg-rr 68.5%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.65 \cdot 10^{+14}:\\ \;\;\;\;\frac{\frac{\frac{1}{c}}{x \cdot s}}{c \cdot \left(x \cdot s\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\left(x \cdot c\right) \cdot s}{\frac{-1}{\left(x \cdot c\right) \cdot s}}}\\ \end{array} \]

Alternative 11: 78.8% accurate, 19.5× speedup

Math

\[\begin{array}{l} s_m = \left|s\right| \\ [x, c, s_m] = \mathsf{sort}([x, c, s_m])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 2.65 \cdot 10^{+14}:\\ \;\;\;\;\frac{\frac{\frac{1}{c}}{x \cdot s_m}}{c \cdot \left(x \cdot s_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{c \cdot \left(\left(x \cdot s_m\right) \cdot \left(c \cdot \left(x \cdot \left(-s_m\right)\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

s_m = (fabs.f64 s)
NOTE: x, c, and s_m should be sorted in increasing order before calling this function.
(FPCore (x c s_m)
 :precision binary64
 (if (<= x 2.65e+14)
   (/ (/ (/ 1.0 c) (* x s_m)) (* c (* x s_m)))
   (/ 1.0 (* c (* (* x s_m) (* c (* x (- s_m))))))))

C

s_m = fabs(s);
assert(x < c && c < s_m);
double code(double x, double c, double s_m) {
	double tmp;
	if (x <= 2.65e+14) {
		tmp = ((1.0 / c) / (x * s_m)) / (c * (x * s_m));
	} else {
		tmp = 1.0 / (c * ((x * s_m) * (c * (x * -s_m))));
	}
	return tmp;
}

Fortran

s_m = abs(s)
NOTE: x, c, and s_m should be sorted in increasing order before calling this function.
real(8) function code(x, c, s_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: c
    real(8), intent (in) :: s_m
    real(8) :: tmp
    if (x <= 2.65d+14) then
        tmp = ((1.0d0 / c) / (x * s_m)) / (c * (x * s_m))
    else
        tmp = 1.0d0 / (c * ((x * s_m) * (c * (x * -s_m))))
    end if
    code = tmp
end function

Java

s_m = Math.abs(s);
assert x < c && c < s_m;
public static double code(double x, double c, double s_m) {
	double tmp;
	if (x <= 2.65e+14) {
		tmp = ((1.0 / c) / (x * s_m)) / (c * (x * s_m));
	} else {
		tmp = 1.0 / (c * ((x * s_m) * (c * (x * -s_m))));
	}
	return tmp;
}

Python

s_m = math.fabs(s)
[x, c, s_m] = sort([x, c, s_m])
def code(x, c, s_m):
	tmp = 0
	if x <= 2.65e+14:
		tmp = ((1.0 / c) / (x * s_m)) / (c * (x * s_m))
	else:
		tmp = 1.0 / (c * ((x * s_m) * (c * (x * -s_m))))
	return tmp

Julia

s_m = abs(s)
x, c, s_m = sort([x, c, s_m])
function code(x, c, s_m)
	tmp = 0.0
	if (x <= 2.65e+14)
		tmp = Float64(Float64(Float64(1.0 / c) / Float64(x * s_m)) / Float64(c * Float64(x * s_m)));
	else
		tmp = Float64(1.0 / Float64(c * Float64(Float64(x * s_m) * Float64(c * Float64(x * Float64(-s_m))))));
	end
	return tmp
end

MATLAB

s_m = abs(s);
x, c, s_m = num2cell(sort([x, c, s_m])){:}
function tmp_2 = code(x, c, s_m)
	tmp = 0.0;
	if (x <= 2.65e+14)
		tmp = ((1.0 / c) / (x * s_m)) / (c * (x * s_m));
	else
		tmp = 1.0 / (c * ((x * s_m) * (c * (x * -s_m))));
	end
	tmp_2 = tmp;
end

Wolfram

s_m = N[Abs[s], $MachinePrecision]
NOTE: x, c, and s_m should be sorted in increasing order before calling this function.
code[x_, c_, s$95$m_] := If[LessEqual[x, 2.65e+14], N[(N[(N[(1.0 / c), $MachinePrecision] / N[(x * s$95$m), $MachinePrecision]), $MachinePrecision] / N[(c * N[(x * s$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(c * N[(N[(x * s$95$m), $MachinePrecision] * N[(c * N[(x * (-s$95$m)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.65e14

   1. Initial program 72.7%

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

      1. *-un-lft-identity 72.7%

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

      2. add-sqr-sqrt 72.6%

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

      3. times-frac 72.7%

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

   3. Applied egg-rr 97.9%

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

      1. associate-*l/ 97.9%

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

      2. *-un-lft-identity 97.9%

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

      3. *-commutative 97.9%

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

   5. Applied egg-rr 97.9%

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

   6. Taylor expanded in x around 0 81.9%

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

      1. associate-/r* 81.9%

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

   8. Simplified 81.9%

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

   if 2.65e14 < x

   1. Initial program 72.7%

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

   2. Taylor expanded in x around 0 53.0%

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

      1. associate-/r* 53.0%

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

      2. *-commutative 53.0%

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

      3. unpow2 53.0%

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

      4. unpow2 53.0%

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

      5. swap-sqr 56.0%

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

      6. unpow2 56.0%

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

      7. associate-/r* 56.0%

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

      8. unpow2 56.0%

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

      9. unpow2 56.0%

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

      10. swap-sqr 61.3%

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

      11. unpow2 61.3%

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

      12. *-commutative 61.3%

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

   4. Simplified 61.3%

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

      1. *-commutative 61.3%

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

      2. pow2 61.3%

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

      3. *-commutative 61.3%

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

      4. associate-*r* 60.9%

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

   6. Applied egg-rr 60.9%

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

      1. /-rgt-identity 60.9%

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

      2. clear-num 60.9%

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

      3. associate-/r* 60.9%

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

   8. Applied egg-rr 60.9%

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

      1. frac-2neg 60.9%

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

      2. metadata-eval 60.9%

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

      3. div-inv 60.9%

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

      4. associate-/l/ 60.9%

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

      5. *-commutative 60.9%

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

      6. distribute-neg-frac 60.9%

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

      7. metadata-eval 60.9%

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

      8. *-commutative 60.9%

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

      9. associate-*l* 60.6%

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

      10. add-sqr-sqrt 47.8%

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

      11. sqrt-prod 65.8%

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

      12. frac-times 65.8%

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

      13. metadata-eval 65.8%

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

      14. associate-*l* 65.8%

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

      15. *-commutative 65.8%

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

      16. associate-*l* 66.3%

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

      17. *-commutative 66.3%

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

      18. associate-*r* 66.3%

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

      19. associate-*r* 65.8%

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

   10. Applied egg-rr 68.4%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.65 \cdot 10^{+14}:\\ \;\;\;\;\frac{\frac{\frac{1}{c}}{x \cdot s}}{c \cdot \left(x \cdot s\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{c \cdot \left(\left(x \cdot s\right) \cdot \left(c \cdot \left(x \cdot \left(-s\right)\right)\right)\right)}\\ \end{array} \]

Alternative 12: 75.5% accurate, 24.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.7%

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

2. Taylor expanded in x around 0 56.9%

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

   1. associate-/r* 56.9%

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

   2. *-commutative 56.9%

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

   3. unpow2 56.9%

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

   4. unpow2 56.9%

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

   5. swap-sqr 64.9%

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

   6. unpow2 64.9%

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

   7. associate-/r* 64.9%

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

   8. unpow2 64.9%

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

   9. unpow2 64.9%

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

   10. swap-sqr 76.0%

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

   11. unpow2 76.0%

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

   12. *-commutative 76.0%

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

4. Simplified 76.0%

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

   1. unpow2 76.0%

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

   2. associate-*r* 75.2%

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

   3. *-commutative 75.2%

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

   4. associate-*l* 74.3%

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

6. Applied egg-rr 74.3%

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

7. Final simplification 74.3%

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

Alternative 13: 78.4% accurate, 24.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.7%

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

2. Taylor expanded in x around 0 56.9%

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

   1. associate-/r* 56.9%

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

   2. *-commutative 56.9%

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

   3. unpow2 56.9%

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

   4. unpow2 56.9%

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

   5. swap-sqr 64.9%

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

   6. unpow2 64.9%

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

   7. associate-/r* 64.9%

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

   8. unpow2 64.9%

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

   9. unpow2 64.9%

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

   10. swap-sqr 76.0%

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

   11. unpow2 76.0%

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

   12. *-commutative 76.0%

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

4. Simplified 76.0%

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

   1. *-commutative 76.0%

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

   2. pow2 76.0%

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

6. Applied egg-rr 76.0%

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

7. Final simplification 76.0%

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

Reproduce

herbie shell --seed 2023315 
(FPCore (x c s)
  :name "mixedcos"
  :precision binary64
  (/ (cos (* 2.0 x)) (* (pow c 2.0) (* (* x (pow s 2.0)) x))))