Linear.Quaternion:$ccos from linear-1.19.1.3

Percentage Accurate: 100.0% → 100.0%

Time: 9.8s

Alternatives: 13

Speedup: 1.0×

• 50.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

2. Final simplification 100.0%

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

Alternative 2: 71.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 0.3:\\ \;\;\;\;\sin x\\ \mathbf{elif}\;y \leq 3.3 \cdot 10^{+154}:\\ \;\;\;\;x \cdot \frac{\sinh y}{y}\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \left(y \cdot \left(y \cdot 0.16666666666666666\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 0.3)
   (sin x)
   (if (<= y 3.3e+154)
     (* x (/ (sinh y) y))
     (* (sin x) (* y (* y 0.16666666666666666))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 0.3) {
		tmp = sin(x);
	} else if (y <= 3.3e+154) {
		tmp = x * (sinh(y) / y);
	} else {
		tmp = sin(x) * (y * (y * 0.16666666666666666));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 0.3d0) then
        tmp = sin(x)
    else if (y <= 3.3d+154) then
        tmp = x * (sinh(y) / y)
    else
        tmp = sin(x) * (y * (y * 0.16666666666666666d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 0.3) {
		tmp = Math.sin(x);
	} else if (y <= 3.3e+154) {
		tmp = x * (Math.sinh(y) / y);
	} else {
		tmp = Math.sin(x) * (y * (y * 0.16666666666666666));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 0.3:
		tmp = math.sin(x)
	elif y <= 3.3e+154:
		tmp = x * (math.sinh(y) / y)
	else:
		tmp = math.sin(x) * (y * (y * 0.16666666666666666))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 0.3)
		tmp = sin(x);
	elseif (y <= 3.3e+154)
		tmp = Float64(x * Float64(sinh(y) / y));
	else
		tmp = Float64(sin(x) * Float64(y * Float64(y * 0.16666666666666666)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 0.3)
		tmp = sin(x);
	elseif (y <= 3.3e+154)
		tmp = x * (sinh(y) / y);
	else
		tmp = sin(x) * (y * (y * 0.16666666666666666));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 0.3], N[Sin[x], $MachinePrecision], If[LessEqual[y, 3.3e+154], N[(x * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(N[Sin[x], $MachinePrecision] * N[(y * N[(y * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 0.299999999999999989

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 67.9%

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

   if 0.299999999999999989 < y < 3.3e154

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-/r/ 90.6%

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

   3. Applied egg-rr 90.6%

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

   4. Taylor expanded in x around 0 56.3%

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

      1. associate-/r/ 65.6%

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

   6. Applied egg-rr 65.6%

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

   if 3.3e154 < y

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-/r/ 71.9%

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

   3. Applied egg-rr 71.9%

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

   4. Taylor expanded in y around 0 100.0%

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

      1. associate-*r* 100.0%

         \[\leadsto \sin x + \color{blue}{\left(0.16666666666666666 \cdot {y}^{2}\right) \cdot \sin x} \]

      2. distribute-rgt1-in 100.0%

         \[\leadsto \color{blue}{\left(0.16666666666666666 \cdot {y}^{2} + 1\right) \cdot \sin x} \]

      3. unpow2 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in y around inf 100.0%

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

      1. associate-*r* 100.0%

         \[\leadsto \color{blue}{\left(0.16666666666666666 \cdot {y}^{2}\right) \cdot \sin x} \]

      2. unpow2 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*r* 100.0%

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

      5. *-commutative 100.0%

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

   9. Simplified 100.0%

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

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.3:\\ \;\;\;\;\sin x\\ \mathbf{elif}\;y \leq 3.3 \cdot 10^{+154}:\\ \;\;\;\;x \cdot \frac{\sinh y}{y}\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \left(y \cdot \left(y \cdot 0.16666666666666666\right)\right)\\ \end{array} \]

Alternative 3: 84.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 0.31:\\ \;\;\;\;\sin x \cdot \left(0.16666666666666666 \cdot \left(y \cdot y\right) + 1\right)\\ \mathbf{elif}\;y \leq 3.3 \cdot 10^{+154}:\\ \;\;\;\;x \cdot \frac{\sinh y}{y}\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \left(y \cdot \left(y \cdot 0.16666666666666666\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 0.31)
   (* (sin x) (+ (* 0.16666666666666666 (* y y)) 1.0))
   (if (<= y 3.3e+154)
     (* x (/ (sinh y) y))
     (* (sin x) (* y (* y 0.16666666666666666))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 0.31) {
		tmp = sin(x) * ((0.16666666666666666 * (y * y)) + 1.0);
	} else if (y <= 3.3e+154) {
		tmp = x * (sinh(y) / y);
	} else {
		tmp = sin(x) * (y * (y * 0.16666666666666666));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 0.31d0) then
        tmp = sin(x) * ((0.16666666666666666d0 * (y * y)) + 1.0d0)
    else if (y <= 3.3d+154) then
        tmp = x * (sinh(y) / y)
    else
        tmp = sin(x) * (y * (y * 0.16666666666666666d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 0.31) {
		tmp = Math.sin(x) * ((0.16666666666666666 * (y * y)) + 1.0);
	} else if (y <= 3.3e+154) {
		tmp = x * (Math.sinh(y) / y);
	} else {
		tmp = Math.sin(x) * (y * (y * 0.16666666666666666));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 0.31:
		tmp = math.sin(x) * ((0.16666666666666666 * (y * y)) + 1.0)
	elif y <= 3.3e+154:
		tmp = x * (math.sinh(y) / y)
	else:
		tmp = math.sin(x) * (y * (y * 0.16666666666666666))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 0.31)
		tmp = Float64(sin(x) * Float64(Float64(0.16666666666666666 * Float64(y * y)) + 1.0));
	elseif (y <= 3.3e+154)
		tmp = Float64(x * Float64(sinh(y) / y));
	else
		tmp = Float64(sin(x) * Float64(y * Float64(y * 0.16666666666666666)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 0.31)
		tmp = sin(x) * ((0.16666666666666666 * (y * y)) + 1.0);
	elseif (y <= 3.3e+154)
		tmp = x * (sinh(y) / y);
	else
		tmp = sin(x) * (y * (y * 0.16666666666666666));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 0.31], N[(N[Sin[x], $MachinePrecision] * N[(N[(0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.3e+154], N[(x * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(N[Sin[x], $MachinePrecision] * N[(y * N[(y * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 0.309999999999999998

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-/r/ 91.5%

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

   3. Applied egg-rr 91.5%

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

   4. Taylor expanded in y around 0 86.0%

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

      1. associate-*r* 86.0%

         \[\leadsto \sin x + \color{blue}{\left(0.16666666666666666 \cdot {y}^{2}\right) \cdot \sin x} \]

      2. distribute-rgt1-in 86.0%

         \[\leadsto \color{blue}{\left(0.16666666666666666 \cdot {y}^{2} + 1\right) \cdot \sin x} \]

      3. unpow2 86.0%

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

   6. Simplified 86.0%

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

   if 0.309999999999999998 < y < 3.3e154

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-/r/ 90.6%

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

   3. Applied egg-rr 90.6%

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

   4. Taylor expanded in x around 0 56.3%

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

      1. associate-/r/ 65.6%

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

   6. Applied egg-rr 65.6%

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

   if 3.3e154 < y

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-/r/ 71.9%

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

   3. Applied egg-rr 71.9%

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

   4. Taylor expanded in y around 0 100.0%

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

      1. associate-*r* 100.0%

         \[\leadsto \sin x + \color{blue}{\left(0.16666666666666666 \cdot {y}^{2}\right) \cdot \sin x} \]

      2. distribute-rgt1-in 100.0%

         \[\leadsto \color{blue}{\left(0.16666666666666666 \cdot {y}^{2} + 1\right) \cdot \sin x} \]

      3. unpow2 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in y around inf 100.0%

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

      1. associate-*r* 100.0%

         \[\leadsto \color{blue}{\left(0.16666666666666666 \cdot {y}^{2}\right) \cdot \sin x} \]

      2. unpow2 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*r* 100.0%

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

      5. *-commutative 100.0%

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

   9. Simplified 100.0%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.31:\\ \;\;\;\;\sin x \cdot \left(0.16666666666666666 \cdot \left(y \cdot y\right) + 1\right)\\ \mathbf{elif}\;y \leq 3.3 \cdot 10^{+154}:\\ \;\;\;\;x \cdot \frac{\sinh y}{y}\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \left(y \cdot \left(y \cdot 0.16666666666666666\right)\right)\\ \end{array} \]

Alternative 4: 67.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 0.3:\\ \;\;\;\;\sin x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\sinh y}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 0.3) (sin x) (* x (/ (sinh y) y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 0.3) {
		tmp = sin(x);
	} else {
		tmp = x * (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 <= 0.3d0) then
        tmp = sin(x)
    else
        tmp = x * (sinh(y) / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 0.3) {
		tmp = Math.sin(x);
	} else {
		tmp = x * (Math.sinh(y) / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 0.3:
		tmp = math.sin(x)
	else:
		tmp = x * (math.sinh(y) / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 0.3)
		tmp = sin(x);
	else
		tmp = Float64(x * Float64(sinh(y) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 0.3)
		tmp = sin(x);
	else
		tmp = x * (sinh(y) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 0.3], N[Sin[x], $MachinePrecision], N[(x * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 0.299999999999999989

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 67.9%

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

   if 0.299999999999999989 < y

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-/r/ 81.3%

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

   3. Applied egg-rr 81.3%

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

   4. Taylor expanded in x around 0 48.4%

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

      1. associate-/r/ 67.2%

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

   6. Applied egg-rr 67.2%

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

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.3:\\ \;\;\;\;\sin x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\sinh y}{y}\\ \end{array} \]

Alternative 5: 60.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 8.8 \cdot 10^{+67}:\\ \;\;\;\;\sin x\\ \mathbf{else}:\\ \;\;\;\;x + 0.16666666666666666 \cdot \left(x \cdot \left(y \cdot y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 8.8e+67) (sin x) (+ x (* 0.16666666666666666 (* x (* y y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 8.8e+67) {
		tmp = sin(x);
	} else {
		tmp = x + (0.16666666666666666 * (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 <= 8.8d+67) then
        tmp = sin(x)
    else
        tmp = x + (0.16666666666666666d0 * (x * (y * y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 8.8e+67) {
		tmp = Math.sin(x);
	} else {
		tmp = x + (0.16666666666666666 * (x * (y * y)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 8.8e+67:
		tmp = math.sin(x)
	else:
		tmp = x + (0.16666666666666666 * (x * (y * y)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 8.8e+67)
		tmp = sin(x);
	else
		tmp = Float64(x + Float64(0.16666666666666666 * Float64(x * Float64(y * y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 8.8e+67)
		tmp = sin(x);
	else
		tmp = x + (0.16666666666666666 * (x * (y * y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 8.8e+67], N[Sin[x], $MachinePrecision], N[(x + N[(0.16666666666666666 * N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 8.8e67

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 64.4%

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

   if 8.8e67 < y

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-/r/ 77.4%

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

   3. Applied egg-rr 77.4%

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

   4. Taylor expanded in x around 0 47.2%

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

      1. associate-/r/ 69.8%

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

   6. Applied egg-rr 69.8%

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

   7. Taylor expanded in y around 0 53.6%

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

      1. unpow2 53.6%

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

   9. Simplified 53.6%

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

4. Final simplification 62.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 8.8 \cdot 10^{+67}:\\ \;\;\;\;\sin x\\ \mathbf{else}:\\ \;\;\;\;x + 0.16666666666666666 \cdot \left(x \cdot \left(y \cdot y\right)\right)\\ \end{array} \]

Alternative 6: 36.6% accurate, 22.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 0.2:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot \left(x \cdot \left(y \cdot y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 0.2) x (* 0.16666666666666666 (* x (* y y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 0.2) {
		tmp = x;
	} else {
		tmp = 0.16666666666666666 * (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 <= 0.2d0) then
        tmp = x
    else
        tmp = 0.16666666666666666d0 * (x * (y * y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 0.2) {
		tmp = x;
	} else {
		tmp = 0.16666666666666666 * (x * (y * y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 0.2:
		tmp = x
	else:
		tmp = 0.16666666666666666 * (x * (y * y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 0.2)
		tmp = x;
	else
		tmp = Float64(0.16666666666666666 * Float64(x * Float64(y * y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 0.2)
		tmp = x;
	else
		tmp = 0.16666666666666666 * (x * (y * y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 0.2], x, N[(0.16666666666666666 * N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 0.20000000000000001

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-/r/ 91.5%

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

   3. Applied egg-rr 91.5%

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

   4. Taylor expanded in x around 0 50.9%

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

   5. Taylor expanded in y around 0 35.2%

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

   if 0.20000000000000001 < y

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-/r/ 81.5%

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

   3. Applied egg-rr 81.5%

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

   4. Taylor expanded in y around 0 52.4%

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

      1. associate-*r* 52.4%

         \[\leadsto \sin x + \color{blue}{\left(0.16666666666666666 \cdot {y}^{2}\right) \cdot \sin x} \]

      2. distribute-rgt1-in 52.4%

         \[\leadsto \color{blue}{\left(0.16666666666666666 \cdot {y}^{2} + 1\right) \cdot \sin x} \]

      3. unpow2 52.4%

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

   6. Simplified 52.4%

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

   7. Taylor expanded in y around inf 52.0%

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

      1. associate-*r* 52.0%

         \[\leadsto \color{blue}{\left(0.16666666666666666 \cdot {y}^{2}\right) \cdot \sin x} \]

      2. unpow2 52.0%

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

      3. *-commutative 52.0%

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

      4. associate-*r* 52.0%

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

      5. *-commutative 52.0%

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

   9. Simplified 52.0%

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

   10. Taylor expanded in x around 0 44.2%

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

       1. unpow2 44.2%

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

   12. Simplified 44.2%

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

4. Final simplification 37.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.2:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot \left(x \cdot \left(y \cdot y\right)\right)\\ \end{array} \]

Alternative 7: 47.6% accurate, 22.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. *-commutative 100.0%

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

   2. associate-/r/ 89.0%

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

3. Applied egg-rr 89.0%

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

4. Taylor expanded in x around 0 50.1%

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

   1. associate-/r/ 61.1%

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

6. Applied egg-rr 61.1%

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

7. Taylor expanded in y around 0 48.1%

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

   1. unpow2 48.1%

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

9. Simplified 48.1%

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

10. Final simplification 48.1%

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

Alternative 8: 4.1% accurate, 205.0× speedup

Math

\[\begin{array}{l} \\ -8 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 -8.0)

C

double code(double x, double y) {
	return -8.0;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = -8.0d0
end function

Java

public static double code(double x, double y) {
	return -8.0;
}

Python

def code(x, y):
	return -8.0

Julia

function code(x, y)
	return -8.0
end

MATLAB

function tmp = code(x, y)
	tmp = -8.0;
end

Wolfram

code[x_, y_] := -8.0

Derivation

1. Initial program 100.0%

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

   1. *-commutative 100.0%

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

   2. associate-/r/ 89.0%

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

3. Applied egg-rr 89.0%

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

5. Applied egg-rr 4.1%

   \[\leadsto \color{blue}{-8} \]

6. Final simplification 4.1%

   \[\leadsto -8 \]

Alternative 9: 4.7% 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%

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

   1. *-commutative 100.0%

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

   2. associate-/r/ 89.0%

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

3. Applied egg-rr 89.0%

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

5. Applied egg-rr 4.7%

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

6. Final simplification 4.7%

   \[\leadsto -1 \]

Alternative 10: 4.6% accurate, 205.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 0.5)

C

double code(double x, double y) {
	return 0.5;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 0.5d0
end function

Java

public static double code(double x, double y) {
	return 0.5;
}

Python

def code(x, y):
	return 0.5

Julia

function code(x, y)
	return 0.5
end

MATLAB

function tmp = code(x, y)
	tmp = 0.5;
end

Wolfram

code[x_, y_] := 0.5

Derivation

1. Initial program 100.0%

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

   1. *-commutative 100.0%

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

   2. associate-/r/ 89.0%

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

3. Applied egg-rr 89.0%

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

5. Applied egg-rr 4.8%

   \[\leadsto \color{blue}{0.5} \]

6. Final simplification 4.8%

   \[\leadsto 0.5 \]

Alternative 11: 4.6% accurate, 205.0× speedup

Math

\[\begin{array}{l} \\ 0.75 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 0.75)

C

double code(double x, double y) {
	return 0.75;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 0.75d0
end function

Java

public static double code(double x, double y) {
	return 0.75;
}

Python

def code(x, y):
	return 0.75

Julia

function code(x, y)
	return 0.75
end

MATLAB

function tmp = code(x, y)
	tmp = 0.75;
end

Wolfram

code[x_, y_] := 0.75

Derivation

1. Initial program 100.0%

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

   1. *-commutative 100.0%

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

   2. associate-/r/ 89.0%

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

3. Applied egg-rr 89.0%

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

5. Applied egg-rr 5.0%

   \[\leadsto \color{blue}{0.75} \]

6. Final simplification 5.0%

   \[\leadsto 0.75 \]

Alternative 12: 4.7% 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%

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

   1. *-commutative 100.0%

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

   2. associate-/r/ 89.0%

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

3. Applied egg-rr 89.0%

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

5. Applied egg-rr 5.1%

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

6. Final simplification 5.1%

   \[\leadsto 1 \]

Alternative 13: 26.3% accurate, 205.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 x)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return x

Julia

function code(x, y)
	return x
end

MATLAB

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

Wolfram

code[x_, y_] := x

Derivation

1. Initial program 100.0%

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

   1. *-commutative 100.0%

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

   2. associate-/r/ 89.0%

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

3. Applied egg-rr 89.0%

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

4. Taylor expanded in x around 0 50.1%

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

5. Taylor expanded in y around 0 26.9%

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

6. Final simplification 26.9%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023297 
(FPCore (x y)
  :name "Linear.Quaternion:$ccos from linear-1.19.1.3"
  :precision binary64
  (* (sin x) (/ (sinh y) y)))