Linear.Quaternion:$csin from linear-1.19.1.3

Percentage Accurate: 100.0% → 100.0%

Time: 7.3s

Alternatives: 9

Speedup: 1.0×

• 50.4% 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. Final simplification 100.0%

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

Alternative 2: 71.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 0.000112:\\ \;\;\;\;\cos x\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{\sinh y}{y}\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot \left(\cos x \cdot \left(y \cdot y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 0.000112)
   (cos x)
   (if (<= y 1.35e+154)
     (/ (sinh y) y)
     (* 0.16666666666666666 (* (cos x) (* y y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 0.000112) {
		tmp = cos(x);
	} else if (y <= 1.35e+154) {
		tmp = sinh(y) / y;
	} else {
		tmp = 0.16666666666666666 * (cos(x) * (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 <= 0.000112d0) then
        tmp = cos(x)
    else if (y <= 1.35d+154) then
        tmp = sinh(y) / y
    else
        tmp = 0.16666666666666666d0 * (cos(x) * (y * y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 0.000112) {
		tmp = Math.cos(x);
	} else if (y <= 1.35e+154) {
		tmp = Math.sinh(y) / y;
	} else {
		tmp = 0.16666666666666666 * (Math.cos(x) * (y * y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 0.000112:
		tmp = math.cos(x)
	elif y <= 1.35e+154:
		tmp = math.sinh(y) / y
	else:
		tmp = 0.16666666666666666 * (math.cos(x) * (y * y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 0.000112)
		tmp = cos(x);
	elseif (y <= 1.35e+154)
		tmp = Float64(sinh(y) / y);
	else
		tmp = Float64(0.16666666666666666 * Float64(cos(x) * Float64(y * y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 0.000112)
		tmp = cos(x);
	elseif (y <= 1.35e+154)
		tmp = sinh(y) / y;
	else
		tmp = 0.16666666666666666 * (cos(x) * (y * y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 0.000112], N[Cos[x], $MachinePrecision], If[LessEqual[y, 1.35e+154], N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision], N[(0.16666666666666666 * N[(N[Cos[x], $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.11999999999999998e-4

   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-/r/ 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in y around 0 68.3%

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

   if 1.11999999999999998e-4 < y < 1.35000000000000003e154

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-/r/ 99.9%

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

   3. Simplified 99.9%

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

      1. clear-num 99.9%

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

      2. inv-pow 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in x around 0 78.9%

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

   if 1.35000000000000003e154 < y

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]

   2. Taylor expanded in y around 0 100.0%

      \[\leadsto \cos x \cdot \color{blue}{\left(1 + 0.16666666666666666 \cdot {y}^{2}\right)} \]
   3. Step-by-step derivation

      1. unpow2 100.0%

         \[\leadsto \cos x \cdot \left(1 + 0.16666666666666666 \cdot \color{blue}{\left(y \cdot y\right)}\right) \]

   4. Simplified 100.0%

      \[\leadsto \cos x \cdot \color{blue}{\left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right)} \]

   5. Taylor expanded in y around inf 100.0%

      \[\leadsto \color{blue}{0.16666666666666666 \cdot \left(\cos x \cdot {y}^{2}\right)} \]
   6. Step-by-step derivation

      1. unpow2 100.0%

         \[\leadsto 0.16666666666666666 \cdot \left(\cos x \cdot \color{blue}{\left(y \cdot y\right)}\right) \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{0.16666666666666666 \cdot \left(\cos x \cdot \left(y \cdot y\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.000112:\\ \;\;\;\;\cos x\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{\sinh y}{y}\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot \left(\cos x \cdot \left(y \cdot y\right)\right)\\ \end{array} \]

Alternative 3: 84.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 0.038:\\ \;\;\;\;\cos x \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right)\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{\sinh y}{y}\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot \left(\cos x \cdot \left(y \cdot y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 0.038)
   (* (cos x) (+ 1.0 (* 0.16666666666666666 (* y y))))
   (if (<= y 1.35e+154)
     (/ (sinh y) y)
     (* 0.16666666666666666 (* (cos x) (* y y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 0.038) {
		tmp = cos(x) * (1.0 + (0.16666666666666666 * (y * y)));
	} else if (y <= 1.35e+154) {
		tmp = sinh(y) / y;
	} else {
		tmp = 0.16666666666666666 * (cos(x) * (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 <= 0.038d0) then
        tmp = cos(x) * (1.0d0 + (0.16666666666666666d0 * (y * y)))
    else if (y <= 1.35d+154) then
        tmp = sinh(y) / y
    else
        tmp = 0.16666666666666666d0 * (cos(x) * (y * y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 0.038) {
		tmp = Math.cos(x) * (1.0 + (0.16666666666666666 * (y * y)));
	} else if (y <= 1.35e+154) {
		tmp = Math.sinh(y) / y;
	} else {
		tmp = 0.16666666666666666 * (Math.cos(x) * (y * y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 0.038:
		tmp = math.cos(x) * (1.0 + (0.16666666666666666 * (y * y)))
	elif y <= 1.35e+154:
		tmp = math.sinh(y) / y
	else:
		tmp = 0.16666666666666666 * (math.cos(x) * (y * y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 0.038)
		tmp = Float64(cos(x) * Float64(1.0 + Float64(0.16666666666666666 * Float64(y * y))));
	elseif (y <= 1.35e+154)
		tmp = Float64(sinh(y) / y);
	else
		tmp = Float64(0.16666666666666666 * Float64(cos(x) * Float64(y * y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 0.038)
		tmp = cos(x) * (1.0 + (0.16666666666666666 * (y * y)));
	elseif (y <= 1.35e+154)
		tmp = sinh(y) / y;
	else
		tmp = 0.16666666666666666 * (cos(x) * (y * y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 0.038], N[(N[Cos[x], $MachinePrecision] * N[(1.0 + N[(0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.35e+154], N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision], N[(0.16666666666666666 * N[(N[Cos[x], $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 0.0379999999999999991

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]

   2. Taylor expanded in y around 0 85.0%

      \[\leadsto \cos x \cdot \color{blue}{\left(1 + 0.16666666666666666 \cdot {y}^{2}\right)} \]
   3. Step-by-step derivation

      1. unpow2 85.0%

         \[\leadsto \cos x \cdot \left(1 + 0.16666666666666666 \cdot \color{blue}{\left(y \cdot y\right)}\right) \]

   4. Simplified 85.0%

      \[\leadsto \cos x \cdot \color{blue}{\left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right)} \]

   if 0.0379999999999999991 < y < 1.35000000000000003e154

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-/r/ 99.9%

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

   3. Simplified 99.9%

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

      1. clear-num 99.9%

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

      2. inv-pow 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in x around 0 80.7%

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

   if 1.35000000000000003e154 < y

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]

   2. Taylor expanded in y around 0 100.0%

      \[\leadsto \cos x \cdot \color{blue}{\left(1 + 0.16666666666666666 \cdot {y}^{2}\right)} \]
   3. Step-by-step derivation

      1. unpow2 100.0%

         \[\leadsto \cos x \cdot \left(1 + 0.16666666666666666 \cdot \color{blue}{\left(y \cdot y\right)}\right) \]

   4. Simplified 100.0%

      \[\leadsto \cos x \cdot \color{blue}{\left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right)} \]

   5. Taylor expanded in y around inf 100.0%

      \[\leadsto \color{blue}{0.16666666666666666 \cdot \left(\cos x \cdot {y}^{2}\right)} \]
   6. Step-by-step derivation

      1. unpow2 100.0%

         \[\leadsto 0.16666666666666666 \cdot \left(\cos x \cdot \color{blue}{\left(y \cdot y\right)}\right) \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{0.16666666666666666 \cdot \left(\cos x \cdot \left(y \cdot y\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.038:\\ \;\;\;\;\cos x \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right)\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{\sinh y}{y}\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot \left(\cos x \cdot \left(y \cdot y\right)\right)\\ \end{array} \]

Alternative 4: 68.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 0.00015:\\ \;\;\;\;\cos x\\ \mathbf{else}:\\ \;\;\;\;\frac{\sinh y}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 0.00015) (cos x) (/ (sinh y) y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 0.00015) {
		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 <= 0.00015d0) 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 <= 0.00015) {
		tmp = Math.cos(x);
	} else {
		tmp = Math.sinh(y) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 0.00015:
		tmp = math.cos(x)
	else:
		tmp = math.sinh(y) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 0.00015)
		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 <= 0.00015)
		tmp = cos(x);
	else
		tmp = sinh(y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 0.00015], N[Cos[x], $MachinePrecision], N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.49999999999999987e-4

   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-/r/ 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in y around 0 68.3%

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

   if 1.49999999999999987e-4 < y

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-/r/ 100.0%

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

   3. Simplified 100.0%

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

      1. clear-num 100.0%

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

      2. inv-pow 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in x around 0 76.5%

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

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.00015:\\ \;\;\;\;\cos x\\ \mathbf{else}:\\ \;\;\;\;\frac{\sinh y}{y}\\ \end{array} \]

Alternative 5: 60.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 6200:\\ \;\;\;\;\cos x\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{+153}:\\ \;\;\;\;1 + \left(x \cdot x\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 6200.0)
   (cos x)
   (if (<= y 5.5e+153)
     (+ 1.0 (* (* x x) -0.5))
     (+ 1.0 (* 0.16666666666666666 (* y y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 6200.0) {
		tmp = cos(x);
	} else if (y <= 5.5e+153) {
		tmp = 1.0 + ((x * x) * -0.5);
	} else {
		tmp = 1.0 + (0.16666666666666666 * (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 <= 6200.0d0) then
        tmp = cos(x)
    else if (y <= 5.5d+153) then
        tmp = 1.0d0 + ((x * x) * (-0.5d0))
    else
        tmp = 1.0d0 + (0.16666666666666666d0 * (y * y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 6200.0) {
		tmp = Math.cos(x);
	} else if (y <= 5.5e+153) {
		tmp = 1.0 + ((x * x) * -0.5);
	} else {
		tmp = 1.0 + (0.16666666666666666 * (y * y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 6200.0:
		tmp = math.cos(x)
	elif y <= 5.5e+153:
		tmp = 1.0 + ((x * x) * -0.5)
	else:
		tmp = 1.0 + (0.16666666666666666 * (y * y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 6200.0)
		tmp = cos(x);
	elseif (y <= 5.5e+153)
		tmp = Float64(1.0 + Float64(Float64(x * x) * -0.5));
	else
		tmp = Float64(1.0 + Float64(0.16666666666666666 * Float64(y * y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 6200.0)
		tmp = cos(x);
	elseif (y <= 5.5e+153)
		tmp = 1.0 + ((x * x) * -0.5);
	else
		tmp = 1.0 + (0.16666666666666666 * (y * y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 6200.0], N[Cos[x], $MachinePrecision], If[LessEqual[y, 5.5e+153], N[(1.0 + N[(N[(x * x), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 6200

   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-/r/ 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in y around 0 67.3%

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

   if 6200 < y < 5.5000000000000003e153

   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-/r/ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 3.1%

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

   5. Taylor expanded in x around 0 9.1%

      \[\leadsto \color{blue}{1 + -0.5 \cdot {x}^{2}} \]
   6. Step-by-step derivation

      1. *-commutative 9.1%

         \[\leadsto 1 + \color{blue}{{x}^{2} \cdot -0.5} \]

      2. unpow2 9.1%

         \[\leadsto 1 + \color{blue}{\left(x \cdot x\right)} \cdot -0.5 \]

   7. Simplified 9.1%

      \[\leadsto \color{blue}{1 + \left(x \cdot x\right) \cdot -0.5} \]

   if 5.5000000000000003e153 < 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-/r/ 100.0%

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

   3. Simplified 100.0%

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

      1. clear-num 100.0%

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

      2. inv-pow 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in x around 0 74.3%

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

   7. Taylor expanded in y around 0 74.3%

      \[\leadsto \color{blue}{1 + 0.16666666666666666 \cdot {y}^{2}} \]
   8. Step-by-step derivation

      1. unpow2 100.0%

         \[\leadsto \cos x \cdot \left(1 + 0.16666666666666666 \cdot \color{blue}{\left(y \cdot y\right)}\right) \]

   9. Simplified 74.3%

      \[\leadsto \color{blue}{1 + 0.16666666666666666 \cdot \left(y \cdot y\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 61.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6200:\\ \;\;\;\;\cos x\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{+153}:\\ \;\;\;\;1 + \left(x \cdot x\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\\ \end{array} \]

Alternative 6: 47.4% accurate, 22.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.45 \cdot 10^{+192}:\\ \;\;\;\;1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(x \cdot x\right) \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 1.45e+192)
   (+ 1.0 (* 0.16666666666666666 (* y y)))
   (+ 1.0 (* (* x x) -0.5))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1.45e+192) {
		tmp = 1.0 + (0.16666666666666666 * (y * y));
	} else {
		tmp = 1.0 + ((x * x) * -0.5);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 1.45d+192) then
        tmp = 1.0d0 + (0.16666666666666666d0 * (y * y))
    else
        tmp = 1.0d0 + ((x * x) * (-0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 1.45e+192) {
		tmp = 1.0 + (0.16666666666666666 * (y * y));
	} else {
		tmp = 1.0 + ((x * x) * -0.5);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 1.45e+192:
		tmp = 1.0 + (0.16666666666666666 * (y * y))
	else:
		tmp = 1.0 + ((x * x) * -0.5)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1.45e+192)
		tmp = Float64(1.0 + Float64(0.16666666666666666 * Float64(y * y)));
	else
		tmp = Float64(1.0 + Float64(Float64(x * x) * -0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 1.45e+192)
		tmp = 1.0 + (0.16666666666666666 * (y * y));
	else
		tmp = 1.0 + ((x * x) * -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1.45e+192], N[(1.0 + N[(0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(x * x), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.4500000000000001e192

   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-/r/ 100.0%

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

   3. Simplified 100.0%

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

      1. clear-num 99.8%

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

      2. inv-pow 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in x around 0 74.4%

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

   7. Taylor expanded in y around 0 56.6%

      \[\leadsto \color{blue}{1 + 0.16666666666666666 \cdot {y}^{2}} \]
   8. Step-by-step derivation

      1. unpow2 79.0%

         \[\leadsto \cos x \cdot \left(1 + 0.16666666666666666 \cdot \color{blue}{\left(y \cdot y\right)}\right) \]

   9. Simplified 56.6%

      \[\leadsto \color{blue}{1 + 0.16666666666666666 \cdot \left(y \cdot y\right)} \]

   if 1.4500000000000001e192 < x

   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-/r/ 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in y around 0 39.3%

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

   5. Taylor expanded in x around 0 38.3%

      \[\leadsto \color{blue}{1 + -0.5 \cdot {x}^{2}} \]
   6. Step-by-step derivation

      1. *-commutative 38.3%

         \[\leadsto 1 + \color{blue}{{x}^{2} \cdot -0.5} \]

      2. unpow2 38.3%

         \[\leadsto 1 + \color{blue}{\left(x \cdot x\right)} \cdot -0.5 \]

   7. Simplified 38.3%

      \[\leadsto \color{blue}{1 + \left(x \cdot x\right) \cdot -0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 54.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.45 \cdot 10^{+192}:\\ \;\;\;\;1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(x \cdot x\right) \cdot -0.5\\ \end{array} \]

Alternative 7: 37.2% accurate, 29.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.95:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot \left(y \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.95) 1.0 (* 0.16666666666666666 (* y y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.95) {
		tmp = 1.0;
	} else {
		tmp = 0.16666666666666666 * (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 <= 1.95d0) then
        tmp = 1.0d0
    else
        tmp = 0.16666666666666666d0 * (y * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1.95) {
		tmp = 1.0;
	} else {
		tmp = 0.16666666666666666 * (y * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.95:
		tmp = 1.0
	else:
		tmp = 0.16666666666666666 * (y * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.95)
		tmp = 1.0;
	else
		tmp = Float64(0.16666666666666666 * Float64(y * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.95)
		tmp = 1.0;
	else
		tmp = 0.16666666666666666 * (y * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.95], 1.0, N[(0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.94999999999999996

   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-/r/ 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in y around 0 67.8%

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

   5. Taylor expanded in x around 0 43.8%

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

   if 1.94999999999999996 < y

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]

   2. Taylor expanded in y around 0 56.9%

      \[\leadsto \cos x \cdot \color{blue}{\left(1 + 0.16666666666666666 \cdot {y}^{2}\right)} \]
   3. Step-by-step derivation

      1. unpow2 56.9%

         \[\leadsto \cos x \cdot \left(1 + 0.16666666666666666 \cdot \color{blue}{\left(y \cdot y\right)}\right) \]

   4. Simplified 56.9%

      \[\leadsto \cos x \cdot \color{blue}{\left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right)} \]

   5. Taylor expanded in y around inf 56.7%

      \[\leadsto \color{blue}{0.16666666666666666 \cdot \left(\cos x \cdot {y}^{2}\right)} \]
   6. Step-by-step derivation

      1. unpow2 56.7%

         \[\leadsto 0.16666666666666666 \cdot \left(\cos x \cdot \color{blue}{\left(y \cdot y\right)}\right) \]

   7. Simplified 56.7%

      \[\leadsto \color{blue}{0.16666666666666666 \cdot \left(\cos x \cdot \left(y \cdot y\right)\right)} \]

   8. Taylor expanded in x around 0 42.1%

      \[\leadsto \color{blue}{0.16666666666666666 \cdot {y}^{2}} \]
   9. Step-by-step derivation

      1. unpow2 42.1%

         \[\leadsto 0.16666666666666666 \cdot \color{blue}{\left(y \cdot y\right)} \]

   10. Simplified 42.1%

       \[\leadsto \color{blue}{0.16666666666666666 \cdot \left(y \cdot y\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 43.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.95:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot \left(y \cdot y\right)\\ \end{array} \]

Alternative 8: 46.9% accurate, 29.3× speedup

Math

\[\begin{array}{l} \\ 1 + 0.16666666666666666 \cdot \left(y \cdot y\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (+ 1.0 (* 0.16666666666666666 (* y y))))

C

double code(double x, double y) {
	return 1.0 + (0.16666666666666666 * (y * y));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0 + (0.16666666666666666d0 * (y * y))
end function

Java

public static double code(double x, double y) {
	return 1.0 + (0.16666666666666666 * (y * y));
}

Python

def code(x, y):
	return 1.0 + (0.16666666666666666 * (y * y))

Julia

function code(x, y)
	return Float64(1.0 + Float64(0.16666666666666666 * Float64(y * y)))
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0 + (0.16666666666666666 * (y * y));
end

Wolfram

code[x_, y_] := N[(1.0 + N[(0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

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-/r/ 99.9%

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

3. Simplified 99.9%

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

   1. clear-num 99.8%

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

   2. inv-pow 99.8%

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

5. Applied egg-rr 99.8%

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

6. Taylor expanded in x around 0 70.1%

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

7. Taylor expanded in y around 0 52.5%

   \[\leadsto \color{blue}{1 + 0.16666666666666666 \cdot {y}^{2}} \]
8. Step-by-step derivation

   1. unpow2 77.6%

      \[\leadsto \cos x \cdot \left(1 + 0.16666666666666666 \cdot \color{blue}{\left(y \cdot y\right)}\right) \]

9. Simplified 52.5%

   \[\leadsto \color{blue}{1 + 0.16666666666666666 \cdot \left(y \cdot y\right)} \]

10. Final simplification 52.5%

    \[\leadsto 1 + 0.16666666666666666 \cdot \left(y \cdot y\right) \]

Alternative 9: 27.8% 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-/r/ 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in y around 0 51.5%

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

5. Taylor expanded in x around 0 33.3%

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

6. Final simplification 33.3%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023268 
(FPCore (x y)
  :name "Linear.Quaternion:$csin from linear-1.19.1.3"
  :precision binary64
  (* (cos x) (/ (sinh y) y)))