Linear.Quaternion:$csin from linear-1.19.1.3

Percentage Accurate: 100.0% → 100.0%

Time: 6.8s

Alternatives: 7

Speedup: 1.0×

• 49.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \cos x \cdot \frac{\sinh y}{y} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* (cos x) (/ (sinh y) y)))

C

double code(double x, double y) {
	return cos(x) * (sinh(y) / y);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = cos(x) * (sinh(y) / y)
end function

Java

public static double code(double x, double y) {
	return Math.cos(x) * (Math.sinh(y) / y);
}

Python

def code(x, y):
	return math.cos(x) * (math.sinh(y) / y)

Julia

function code(x, y)
	return Float64(cos(x) * Float64(sinh(y) / y))
end

MATLAB

function tmp = code(x, y)
	tmp = cos(x) * (sinh(y) / y);
end

Wolfram

code[x_, y_] := N[(N[Cos[x], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \cos x \cdot \frac{\sinh y}{y} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* (cos x) (/ (sinh y) y)))

C

double code(double x, double y) {
	return cos(x) * (sinh(y) / y);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = cos(x) * (sinh(y) / y)
end function

Java

public static double code(double x, double y) {
	return Math.cos(x) * (Math.sinh(y) / y);
}

Python

def code(x, y):
	return math.cos(x) * (math.sinh(y) / y)

Julia

function code(x, y)
	return Float64(cos(x) * Float64(sinh(y) / y))
end

MATLAB

function tmp = code(x, y)
	tmp = cos(x) * (sinh(y) / y);
end

Wolfram

code[x_, y_] := N[(N[Cos[x], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \cos x \cdot \frac{\sinh y}{y} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* (cos x) (/ (sinh y) y)))

C

double code(double x, double y) {
	return cos(x) * (sinh(y) / y);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = cos(x) * (sinh(y) / y)
end function

Java

public static double code(double x, double y) {
	return Math.cos(x) * (Math.sinh(y) / y);
}

Python

def code(x, y):
	return math.cos(x) * (math.sinh(y) / y)

Julia

function code(x, y)
	return Float64(cos(x) * Float64(sinh(y) / y))
end

MATLAB

function tmp = code(x, y)
	tmp = cos(x) * (sinh(y) / y);
end

Wolfram

code[x_, y_] := N[(N[Cos[x], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\cos x \cdot \frac{\sinh y}{y} \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 2: 72.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 0.45:\\ \;\;\;\;\cos x\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\sinh y \cdot \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y \cdot y}{\frac{y}{\cos x}}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 0.45)
   (cos x)
   (if (<= y 1.35e+154)
     (* (sinh y) (/ 1.0 y))
     (/ (/ (* y y) (/ y (cos x))) y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 0.45) {
		tmp = cos(x);
	} else if (y <= 1.35e+154) {
		tmp = sinh(y) * (1.0 / y);
	} else {
		tmp = ((y * y) / (y / cos(x))) / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 0.45d0) then
        tmp = cos(x)
    else if (y <= 1.35d+154) then
        tmp = sinh(y) * (1.0d0 / y)
    else
        tmp = ((y * y) / (y / cos(x))) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 0.45) {
		tmp = Math.cos(x);
	} else if (y <= 1.35e+154) {
		tmp = Math.sinh(y) * (1.0 / y);
	} else {
		tmp = ((y * y) / (y / Math.cos(x))) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 0.45:
		tmp = math.cos(x)
	elif y <= 1.35e+154:
		tmp = math.sinh(y) * (1.0 / y)
	else:
		tmp = ((y * y) / (y / math.cos(x))) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 0.45)
		tmp = cos(x);
	elseif (y <= 1.35e+154)
		tmp = Float64(sinh(y) * Float64(1.0 / y));
	else
		tmp = Float64(Float64(Float64(y * y) / Float64(y / cos(x))) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 0.45)
		tmp = cos(x);
	elseif (y <= 1.35e+154)
		tmp = sinh(y) * (1.0 / y);
	else
		tmp = ((y * y) / (y / cos(x))) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 0.45], N[Cos[x], $MachinePrecision], If[LessEqual[y, 1.35e+154], N[(N[Sinh[y], $MachinePrecision] * N[(1.0 / y), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y * y), $MachinePrecision] / N[(y / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 0.450000000000000011

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 64.6%

      \[\leadsto \color{blue}{\cos x} \]

   if 0.450000000000000011 < y < 1.35000000000000003e154

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto \color{blue}{\frac{\sinh y}{y} \cdot \cos x} \]

      2. associate-*l/ 100.0%

         \[\leadsto \color{blue}{\frac{\sinh y \cdot \cos x}{y}} \]

      3. associate-/l* 100.0%

         \[\leadsto \color{blue}{\sinh y \cdot \frac{\cos x}{y}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\sinh y \cdot \frac{\cos x}{y}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 83.9%

      \[\leadsto \sinh y \cdot \color{blue}{\frac{1}{y}} \]

   if 1.35000000000000003e154 < y

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto \color{blue}{\frac{\sinh y}{y} \cdot \cos x} \]

      2. associate-*l/ 100.0%

         \[\leadsto \color{blue}{\frac{\sinh y \cdot \cos x}{y}} \]

      3. associate-/l* 100.0%

         \[\leadsto \color{blue}{\sinh y \cdot \frac{\cos x}{y}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\sinh y \cdot \frac{\cos x}{y}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 3.1%

      \[\leadsto \color{blue}{y} \cdot \frac{\cos x}{y} \]
   6. Step-by-step derivation

      1. associate-*r/ 3.1%

         \[\leadsto \color{blue}{\frac{y \cdot \cos x}{y}} \]

   7. Applied egg-rr 3.1%

      \[\leadsto \color{blue}{\frac{y \cdot \cos x}{y}} \]
   8. Step-by-step derivation

      1. *-commutative 3.1%

         \[\leadsto \frac{\color{blue}{\cos x \cdot y}}{y} \]

      2. *-un-lft-identity 3.1%

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

      3. rgt-mult-inverse 3.1%

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

      4. un-div-inv 3.1%

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

      5. associate-/r/ 3.1%

         \[\leadsto \frac{\color{blue}{\frac{y}{\frac{y}{\cos x}}} \cdot y}{y} \]

      6. frac-2neg 3.1%

         \[\leadsto \frac{\color{blue}{\frac{-y}{-\frac{y}{\cos x}}} \cdot y}{y} \]

      7. associate-*l/ 100.0%

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

      8. distribute-neg-frac2 100.0%

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

   9. Applied egg-rr 100.0%

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

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.45:\\ \;\;\;\;\cos x\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\sinh y \cdot \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y \cdot y}{\frac{y}{\cos x}}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 69.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 0.45:\\ \;\;\;\;\cos x\\ \mathbf{else}:\\ \;\;\;\;\frac{\sinh y}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 0.45) (cos x) (/ (sinh y) y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 0.45) {
		tmp = cos(x);
	} else {
		tmp = sinh(y) / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 0.45d0) then
        tmp = cos(x)
    else
        tmp = sinh(y) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 0.45) {
		tmp = Math.cos(x);
	} else {
		tmp = Math.sinh(y) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 0.45:
		tmp = math.cos(x)
	else:
		tmp = math.sinh(y) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 0.45)
		tmp = cos(x);
	else
		tmp = Float64(sinh(y) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 0.45)
		tmp = cos(x);
	else
		tmp = sinh(y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 0.45], N[Cos[x], $MachinePrecision], N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 0.450000000000000011

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 64.6%

      \[\leadsto \color{blue}{\cos x} \]

   if 0.450000000000000011 < y

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-*r/ 100.0%

         \[\leadsto \color{blue}{\frac{\cos x \cdot \sinh y}{y}} \]

      2. associate-*l/ 100.0%

         \[\leadsto \color{blue}{\frac{\cos x}{y} \cdot \sinh y} \]

      3. *-commutative 100.0%

         \[\leadsto \color{blue}{\sinh y \cdot \frac{\cos x}{y}} \]

      4. clear-num 100.0%

         \[\leadsto \sinh y \cdot \color{blue}{\frac{1}{\frac{y}{\cos x}}} \]

      5. un-div-inv 100.0%

         \[\leadsto \color{blue}{\frac{\sinh y}{\frac{y}{\cos x}}} \]

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{\sinh y}{\frac{y}{\cos x}}} \]

   5. Taylor expanded in x around 0 78.1%

      \[\leadsto \frac{\sinh y}{\color{blue}{y}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 61.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.2 \cdot 10^{+29}:\\ \;\;\;\;\cos x\\ \mathbf{elif}\;y \leq 8.4 \cdot 10^{+147}:\\ \;\;\;\;\frac{y + -0.5 \cdot \left(x \cdot \left(x \cdot y\right)\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y \cdot y}{y}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 4.2e+29)
   (cos x)
   (if (<= y 8.4e+147)
     (/ (+ y (* -0.5 (* x (* x y)))) y)
     (/ (/ (* y y) y) y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 4.2e+29) {
		tmp = cos(x);
	} else if (y <= 8.4e+147) {
		tmp = (y + (-0.5 * (x * (x * y)))) / y;
	} else {
		tmp = ((y * y) / y) / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 4.2d+29) then
        tmp = cos(x)
    else if (y <= 8.4d+147) then
        tmp = (y + ((-0.5d0) * (x * (x * y)))) / y
    else
        tmp = ((y * y) / y) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 4.2e+29) {
		tmp = Math.cos(x);
	} else if (y <= 8.4e+147) {
		tmp = (y + (-0.5 * (x * (x * y)))) / y;
	} else {
		tmp = ((y * y) / y) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 4.2e+29:
		tmp = math.cos(x)
	elif y <= 8.4e+147:
		tmp = (y + (-0.5 * (x * (x * y)))) / y
	else:
		tmp = ((y * y) / y) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 4.2e+29)
		tmp = cos(x);
	elseif (y <= 8.4e+147)
		tmp = Float64(Float64(y + Float64(-0.5 * Float64(x * Float64(x * y)))) / y);
	else
		tmp = Float64(Float64(Float64(y * y) / y) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 4.2e+29)
		tmp = cos(x);
	elseif (y <= 8.4e+147)
		tmp = (y + (-0.5 * (x * (x * y)))) / y;
	else
		tmp = ((y * y) / y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 4.2e+29], N[Cos[x], $MachinePrecision], If[LessEqual[y, 8.4e+147], N[(N[(y + N[(-0.5 * N[(x * N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], N[(N[(N[(y * y), $MachinePrecision] / y), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 4.2000000000000003e29

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 62.7%

      \[\leadsto \color{blue}{\cos x} \]

   if 4.2000000000000003e29 < y < 8.40000000000000024e147

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto \color{blue}{\frac{\sinh y}{y} \cdot \cos x} \]

      2. associate-*l/ 100.0%

         \[\leadsto \color{blue}{\frac{\sinh y \cdot \cos x}{y}} \]

      3. associate-/l* 100.0%

         \[\leadsto \color{blue}{\sinh y \cdot \frac{\cos x}{y}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\sinh y \cdot \frac{\cos x}{y}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 3.1%

      \[\leadsto \color{blue}{y} \cdot \frac{\cos x}{y} \]
   6. Step-by-step derivation

      1. associate-*r/ 3.1%

         \[\leadsto \color{blue}{\frac{y \cdot \cos x}{y}} \]

   7. Applied egg-rr 3.1%

      \[\leadsto \color{blue}{\frac{y \cdot \cos x}{y}} \]

   8. Taylor expanded in x around 0 10.3%

      \[\leadsto \frac{\color{blue}{y + -0.5 \cdot \left({x}^{2} \cdot y\right)}}{y} \]
   9. Step-by-step derivation

      1. *-commutative 10.3%

         \[\leadsto \frac{y + -0.5 \cdot \color{blue}{\left(y \cdot {x}^{2}\right)}}{y} \]

      2. unpow2 10.3%

         \[\leadsto \frac{y + -0.5 \cdot \left(y \cdot \color{blue}{\left(x \cdot x\right)}\right)}{y} \]

      3. associate-*r* 10.3%

         \[\leadsto \frac{y + -0.5 \cdot \color{blue}{\left(\left(y \cdot x\right) \cdot x\right)}}{y} \]

   10. Applied egg-rr 10.3%

       \[\leadsto \frac{y + -0.5 \cdot \color{blue}{\left(\left(y \cdot x\right) \cdot x\right)}}{y} \]

   if 8.40000000000000024e147 < y

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto \color{blue}{\frac{\sinh y}{y} \cdot \cos x} \]

      2. associate-*l/ 100.0%

         \[\leadsto \color{blue}{\frac{\sinh y \cdot \cos x}{y}} \]

      3. associate-/l* 100.0%

         \[\leadsto \color{blue}{\sinh y \cdot \frac{\cos x}{y}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\sinh y \cdot \frac{\cos x}{y}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 3.1%

      \[\leadsto \color{blue}{y} \cdot \frac{\cos x}{y} \]
   6. Step-by-step derivation

      1. associate-*r/ 3.1%

         \[\leadsto \color{blue}{\frac{y \cdot \cos x}{y}} \]

   7. Applied egg-rr 3.1%

      \[\leadsto \color{blue}{\frac{y \cdot \cos x}{y}} \]
   8. Step-by-step derivation

      1. *-commutative 3.1%

         \[\leadsto \frac{\color{blue}{\cos x \cdot y}}{y} \]

      2. *-un-lft-identity 3.1%

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

      3. rgt-mult-inverse 3.1%

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

      4. un-div-inv 3.1%

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

      5. associate-/r/ 3.1%

         \[\leadsto \frac{\color{blue}{\frac{y}{\frac{y}{\cos x}}} \cdot y}{y} \]

      6. frac-2neg 3.1%

         \[\leadsto \frac{\color{blue}{\frac{-y}{-\frac{y}{\cos x}}} \cdot y}{y} \]

      7. associate-*l/ 97.2%

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

      8. distribute-neg-frac2 97.2%

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

   9. Applied egg-rr 97.2%

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

   10. Taylor expanded in x around 0 70.7%

       \[\leadsto \frac{\frac{\left(-y\right) \cdot y}{\color{blue}{-1 \cdot y}}}{y} \]
   11. Step-by-step derivation

       1. mul-1-neg 70.7%

          \[\leadsto \frac{\frac{\left(-y\right) \cdot y}{\color{blue}{-y}}}{y} \]

   12. Simplified 70.7%

       \[\leadsto \frac{\frac{\left(-y\right) \cdot y}{\color{blue}{-y}}}{y} \]
3. Recombined 3 regimes into one program.

4. Final simplification 58.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.2 \cdot 10^{+29}:\\ \;\;\;\;\cos x\\ \mathbf{elif}\;y \leq 8.4 \cdot 10^{+147}:\\ \;\;\;\;\frac{y + -0.5 \cdot \left(x \cdot \left(x \cdot y\right)\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y \cdot y}{y}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 40.1% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 40000000000000:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 8.8 \cdot 10^{+147}:\\ \;\;\;\;\frac{y + -0.5 \cdot \left(x \cdot \left(x \cdot y\right)\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y \cdot y}{y}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 40000000000000.0)
   1.0
   (if (<= y 8.8e+147)
     (/ (+ y (* -0.5 (* x (* x y)))) y)
     (/ (/ (* y y) y) y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 40000000000000.0) {
		tmp = 1.0;
	} else if (y <= 8.8e+147) {
		tmp = (y + (-0.5 * (x * (x * y)))) / y;
	} else {
		tmp = ((y * y) / y) / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 40000000000000.0d0) then
        tmp = 1.0d0
    else if (y <= 8.8d+147) then
        tmp = (y + ((-0.5d0) * (x * (x * y)))) / y
    else
        tmp = ((y * y) / y) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 40000000000000.0) {
		tmp = 1.0;
	} else if (y <= 8.8e+147) {
		tmp = (y + (-0.5 * (x * (x * y)))) / y;
	} else {
		tmp = ((y * y) / y) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 40000000000000.0:
		tmp = 1.0
	elif y <= 8.8e+147:
		tmp = (y + (-0.5 * (x * (x * y)))) / y
	else:
		tmp = ((y * y) / y) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 40000000000000.0)
		tmp = 1.0;
	elseif (y <= 8.8e+147)
		tmp = Float64(Float64(y + Float64(-0.5 * Float64(x * Float64(x * y)))) / y);
	else
		tmp = Float64(Float64(Float64(y * y) / y) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 40000000000000.0)
		tmp = 1.0;
	elseif (y <= 8.8e+147)
		tmp = (y + (-0.5 * (x * (x * y)))) / y;
	else
		tmp = ((y * y) / y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 40000000000000.0], 1.0, If[LessEqual[y, 8.8e+147], N[(N[(y + N[(-0.5 * N[(x * N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], N[(N[(N[(y * y), $MachinePrecision] / y), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 4e13

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto \color{blue}{\frac{\sinh y}{y} \cdot \cos x} \]

      2. associate-*l/ 99.9%

         \[\leadsto \color{blue}{\frac{\sinh y \cdot \cos x}{y}} \]

      3. associate-/l* 99.9%

         \[\leadsto \color{blue}{\sinh y \cdot \frac{\cos x}{y}} \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\sinh y \cdot \frac{\cos x}{y}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 63.9%

      \[\leadsto \color{blue}{y} \cdot \frac{\cos x}{y} \]

   6. Taylor expanded in x around 0 35.3%

      \[\leadsto \color{blue}{1} \]

   if 4e13 < y < 8.8000000000000007e147

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto \color{blue}{\frac{\sinh y}{y} \cdot \cos x} \]

      2. associate-*l/ 100.0%

         \[\leadsto \color{blue}{\frac{\sinh y \cdot \cos x}{y}} \]

      3. associate-/l* 100.0%

         \[\leadsto \color{blue}{\sinh y \cdot \frac{\cos x}{y}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\sinh y \cdot \frac{\cos x}{y}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 3.1%

      \[\leadsto \color{blue}{y} \cdot \frac{\cos x}{y} \]
   6. Step-by-step derivation

      1. associate-*r/ 3.1%

         \[\leadsto \color{blue}{\frac{y \cdot \cos x}{y}} \]

   7. Applied egg-rr 3.1%

      \[\leadsto \color{blue}{\frac{y \cdot \cos x}{y}} \]

   8. Taylor expanded in x around 0 9.2%

      \[\leadsto \frac{\color{blue}{y + -0.5 \cdot \left({x}^{2} \cdot y\right)}}{y} \]
   9. Step-by-step derivation

      1. *-commutative 9.2%

         \[\leadsto \frac{y + -0.5 \cdot \color{blue}{\left(y \cdot {x}^{2}\right)}}{y} \]

      2. unpow2 9.2%

         \[\leadsto \frac{y + -0.5 \cdot \left(y \cdot \color{blue}{\left(x \cdot x\right)}\right)}{y} \]

      3. associate-*r* 9.2%

         \[\leadsto \frac{y + -0.5 \cdot \color{blue}{\left(\left(y \cdot x\right) \cdot x\right)}}{y} \]

   10. Applied egg-rr 9.2%

       \[\leadsto \frac{y + -0.5 \cdot \color{blue}{\left(\left(y \cdot x\right) \cdot x\right)}}{y} \]

   if 8.8000000000000007e147 < y

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto \color{blue}{\frac{\sinh y}{y} \cdot \cos x} \]

      2. associate-*l/ 100.0%

         \[\leadsto \color{blue}{\frac{\sinh y \cdot \cos x}{y}} \]

      3. associate-/l* 100.0%

         \[\leadsto \color{blue}{\sinh y \cdot \frac{\cos x}{y}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\sinh y \cdot \frac{\cos x}{y}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 3.1%

      \[\leadsto \color{blue}{y} \cdot \frac{\cos x}{y} \]
   6. Step-by-step derivation

      1. associate-*r/ 3.1%

         \[\leadsto \color{blue}{\frac{y \cdot \cos x}{y}} \]

   7. Applied egg-rr 3.1%

      \[\leadsto \color{blue}{\frac{y \cdot \cos x}{y}} \]
   8. Step-by-step derivation

      1. *-commutative 3.1%

         \[\leadsto \frac{\color{blue}{\cos x \cdot y}}{y} \]

      2. *-un-lft-identity 3.1%

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

      3. rgt-mult-inverse 3.1%

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

      4. un-div-inv 3.1%

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

      5. associate-/r/ 3.1%

         \[\leadsto \frac{\color{blue}{\frac{y}{\frac{y}{\cos x}}} \cdot y}{y} \]

      6. frac-2neg 3.1%

         \[\leadsto \frac{\color{blue}{\frac{-y}{-\frac{y}{\cos x}}} \cdot y}{y} \]

      7. associate-*l/ 97.2%

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

      8. distribute-neg-frac2 97.2%

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

   9. Applied egg-rr 97.2%

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

   10. Taylor expanded in x around 0 70.7%

       \[\leadsto \frac{\frac{\left(-y\right) \cdot y}{\color{blue}{-1 \cdot y}}}{y} \]
   11. Step-by-step derivation

       1. mul-1-neg 70.7%

          \[\leadsto \frac{\frac{\left(-y\right) \cdot y}{\color{blue}{-y}}}{y} \]

   12. Simplified 70.7%

       \[\leadsto \frac{\frac{\left(-y\right) \cdot y}{\color{blue}{-y}}}{y} \]
3. Recombined 3 regimes into one program.

4. Final simplification 37.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 40000000000000:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 8.8 \cdot 10^{+147}:\\ \;\;\;\;\frac{y + -0.5 \cdot \left(x \cdot \left(x \cdot y\right)\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y \cdot y}{y}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 38.6% accurate, 17.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{-15}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y \cdot y}{y}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 5e-15) 1.0 (/ (/ (* y y) y) y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 5e-15) {
		tmp = 1.0;
	} else {
		tmp = ((y * y) / y) / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 5d-15) then
        tmp = 1.0d0
    else
        tmp = ((y * y) / y) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 5e-15) {
		tmp = 1.0;
	} else {
		tmp = ((y * y) / y) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 5e-15:
		tmp = 1.0
	else:
		tmp = ((y * y) / y) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 5e-15)
		tmp = 1.0;
	else
		tmp = Float64(Float64(Float64(y * y) / y) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 5e-15)
		tmp = 1.0;
	else
		tmp = ((y * y) / y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 5e-15], 1.0, N[(N[(N[(y * y), $MachinePrecision] / y), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.99999999999999999e-15

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto \color{blue}{\frac{\sinh y}{y} \cdot \cos x} \]

      2. associate-*l/ 99.9%

         \[\leadsto \color{blue}{\frac{\sinh y \cdot \cos x}{y}} \]

      3. associate-/l* 99.9%

         \[\leadsto \color{blue}{\sinh y \cdot \frac{\cos x}{y}} \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\sinh y \cdot \frac{\cos x}{y}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 64.6%

      \[\leadsto \color{blue}{y} \cdot \frac{\cos x}{y} \]

   6. Taylor expanded in x around 0 36.0%

      \[\leadsto \color{blue}{1} \]

   if 4.99999999999999999e-15 < y

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto \color{blue}{\frac{\sinh y}{y} \cdot \cos x} \]

      2. associate-*l/ 100.0%

         \[\leadsto \color{blue}{\frac{\sinh y \cdot \cos x}{y}} \]

      3. associate-/l* 100.0%

         \[\leadsto \color{blue}{\sinh y \cdot \frac{\cos x}{y}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\sinh y \cdot \frac{\cos x}{y}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 6.6%

      \[\leadsto \color{blue}{y} \cdot \frac{\cos x}{y} \]
   6. Step-by-step derivation

      1. associate-*r/ 6.6%

         \[\leadsto \color{blue}{\frac{y \cdot \cos x}{y}} \]

   7. Applied egg-rr 6.6%

      \[\leadsto \color{blue}{\frac{y \cdot \cos x}{y}} \]
   8. Step-by-step derivation

      1. *-commutative 6.6%

         \[\leadsto \frac{\color{blue}{\cos x \cdot y}}{y} \]

      2. *-un-lft-identity 6.6%

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

      3. rgt-mult-inverse 6.6%

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

      4. un-div-inv 6.6%

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

      5. associate-/r/ 6.6%

         \[\leadsto \frac{\color{blue}{\frac{y}{\frac{y}{\cos x}}} \cdot y}{y} \]

      6. frac-2neg 6.6%

         \[\leadsto \frac{\color{blue}{\frac{-y}{-\frac{y}{\cos x}}} \cdot y}{y} \]

      7. associate-*l/ 53.6%

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

      8. distribute-neg-frac2 53.6%

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

   9. Applied egg-rr 53.6%

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

   10. Taylor expanded in x around 0 37.8%

       \[\leadsto \frac{\frac{\left(-y\right) \cdot y}{\color{blue}{-1 \cdot y}}}{y} \]
   11. Step-by-step derivation

       1. mul-1-neg 37.8%

          \[\leadsto \frac{\frac{\left(-y\right) \cdot y}{\color{blue}{-y}}}{y} \]

   12. Simplified 37.8%

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

4. Final simplification 36.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{-15}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y \cdot y}{y}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 29.5% accurate, 205.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 1.0)

C

double code(double x, double y) {
	return 1.0;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0
end function

Java

public static double code(double x, double y) {
	return 1.0;
}

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0;
end

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 100.0%

   \[\cos x \cdot \frac{\sinh y}{y} \]
2. Step-by-step derivation

   1. *-commutative 100.0%

      \[\leadsto \color{blue}{\frac{\sinh y}{y} \cdot \cos x} \]

   2. associate-*l/ 99.9%

      \[\leadsto \color{blue}{\frac{\sinh y \cdot \cos x}{y}} \]

   3. associate-/l* 99.9%

      \[\leadsto \color{blue}{\sinh y \cdot \frac{\cos x}{y}} \]

3. Simplified 99.9%

   \[\leadsto \color{blue}{\sinh y \cdot \frac{\cos x}{y}} \]
4. Add Preprocessing

5. Taylor expanded in y around 0 49.2%

   \[\leadsto \color{blue}{y} \cdot \frac{\cos x}{y} \]

6. Taylor expanded in x around 0 27.4%

   \[\leadsto \color{blue}{1} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024096 
(FPCore (x y)
  :name "Linear.Quaternion:$csin from linear-1.19.1.3"
  :precision binary64
  (* (cos x) (/ (sinh y) y)))