Linear.Quaternion:$ccosh from linear-1.19.1.3

Percentage Accurate: 88.8% → 99.9%

Time: 13.1s

Alternatives: 12

Speedup: 1.0×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 88.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.1%

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

   1. associate-*l/ 99.9%

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

3. Simplified 99.9%

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

4. Final simplification 99.9%

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

Alternative 2: 87.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sinh y \leq -1 \cdot 10^{+18}:\\ \;\;\;\;\sinh y\\ \mathbf{elif}\;\sinh y \leq 500:\\ \;\;\;\;\frac{\sin x}{x} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\sinh y \cdot \left(1 + \left(x \cdot x\right) \cdot -0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (sinh y) -1e+18)
   (sinh y)
   (if (<= (sinh y) 500.0)
     (* (/ (sin x) x) y)
     (* (sinh y) (+ 1.0 (* (* x x) -0.16666666666666666))))))

C

double code(double x, double y) {
	double tmp;
	if (sinh(y) <= -1e+18) {
		tmp = sinh(y);
	} else if (sinh(y) <= 500.0) {
		tmp = (sin(x) / x) * y;
	} else {
		tmp = sinh(y) * (1.0 + ((x * x) * -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 (sinh(y) <= (-1d+18)) then
        tmp = sinh(y)
    else if (sinh(y) <= 500.0d0) then
        tmp = (sin(x) / x) * y
    else
        tmp = sinh(y) * (1.0d0 + ((x * x) * (-0.16666666666666666d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (Math.sinh(y) <= -1e+18) {
		tmp = Math.sinh(y);
	} else if (Math.sinh(y) <= 500.0) {
		tmp = (Math.sin(x) / x) * y;
	} else {
		tmp = Math.sinh(y) * (1.0 + ((x * x) * -0.16666666666666666));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if math.sinh(y) <= -1e+18:
		tmp = math.sinh(y)
	elif math.sinh(y) <= 500.0:
		tmp = (math.sin(x) / x) * y
	else:
		tmp = math.sinh(y) * (1.0 + ((x * x) * -0.16666666666666666))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (sinh(y) <= -1e+18)
		tmp = sinh(y);
	elseif (sinh(y) <= 500.0)
		tmp = Float64(Float64(sin(x) / x) * y);
	else
		tmp = Float64(sinh(y) * Float64(1.0 + Float64(Float64(x * x) * -0.16666666666666666)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (sinh(y) <= -1e+18)
		tmp = sinh(y);
	elseif (sinh(y) <= 500.0)
		tmp = (sin(x) / x) * y;
	else
		tmp = sinh(y) * (1.0 + ((x * x) * -0.16666666666666666));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[Sinh[y], $MachinePrecision], -1e+18], N[Sinh[y], $MachinePrecision], If[LessEqual[N[Sinh[y], $MachinePrecision], 500.0], N[(N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision] * y), $MachinePrecision], N[(N[Sinh[y], $MachinePrecision] * N[(1.0 + N[(N[(x * x), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sinh.f64 y) < -1e18

   1. Initial program 100.0%

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

      1. associate-*l/ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 80.0%

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

   if -1e18 < (sinh.f64 y) < 500

   1. Initial program 82.6%

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

      1. associate-*l/ 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in y around 0 98.6%

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

   if 500 < (sinh.f64 y)

   1. Initial program 100.0%

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

      1. associate-*l/ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 74.0%

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

      1. *-commutative 74.0%

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

      2. unpow2 74.0%

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

   6. Simplified 74.0%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sinh y \leq -1 \cdot 10^{+18}:\\ \;\;\;\;\sinh y\\ \mathbf{elif}\;\sinh y \leq 500:\\ \;\;\;\;\frac{\sin x}{x} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\sinh y \cdot \left(1 + \left(x \cdot x\right) \cdot -0.16666666666666666\right)\\ \end{array} \]

Alternative 3: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.1%

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

   1. associate-*r/ 99.5%

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

3. Simplified 99.5%

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

4. Final simplification 99.5%

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

Alternative 4: 86.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.0014:\\ \;\;\;\;\sinh y\\ \mathbf{elif}\;y \leq 9 \cdot 10^{+17}:\\ \;\;\;\;\sin x \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -0.0014) (sinh y) (if (<= y 9e+17) (* (sin x) (/ y x)) (sinh y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -0.0014) {
		tmp = sinh(y);
	} else if (y <= 9e+17) {
		tmp = sin(x) * (y / x);
	} else {
		tmp = sinh(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 <= (-0.0014d0)) then
        tmp = sinh(y)
    else if (y <= 9d+17) then
        tmp = sin(x) * (y / x)
    else
        tmp = sinh(y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -0.0014) {
		tmp = Math.sinh(y);
	} else if (y <= 9e+17) {
		tmp = Math.sin(x) * (y / x);
	} else {
		tmp = Math.sinh(y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -0.0014:
		tmp = math.sinh(y)
	elif y <= 9e+17:
		tmp = math.sin(x) * (y / x)
	else:
		tmp = math.sinh(y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -0.0014)
		tmp = sinh(y);
	elseif (y <= 9e+17)
		tmp = Float64(sin(x) * Float64(y / x));
	else
		tmp = sinh(y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -0.0014)
		tmp = sinh(y);
	elseif (y <= 9e+17)
		tmp = sin(x) * (y / x);
	else
		tmp = sinh(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -0.0014], N[Sinh[y], $MachinePrecision], If[LessEqual[y, 9e+17], N[(N[Sin[x], $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision], N[Sinh[y], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.00139999999999999999 or 9e17 < y

   1. Initial program 100.0%

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

      1. associate-*l/ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 74.0%

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

   if -0.00139999999999999999 < y < 9e17

   1. Initial program 82.9%

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

      1. associate-*r/ 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in y around 0 97.0%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.0014:\\ \;\;\;\;\sinh y\\ \mathbf{elif}\;y \leq 9 \cdot 10^{+17}:\\ \;\;\;\;\sin x \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \]

Alternative 5: 86.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.046:\\ \;\;\;\;\sinh y\\ \mathbf{elif}\;y \leq 9 \cdot 10^{+17}:\\ \;\;\;\;\frac{\sin x}{x} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -0.046) (sinh y) (if (<= y 9e+17) (* (/ (sin x) x) y) (sinh y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -0.046) {
		tmp = sinh(y);
	} else if (y <= 9e+17) {
		tmp = (sin(x) / x) * y;
	} else {
		tmp = sinh(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 <= (-0.046d0)) then
        tmp = sinh(y)
    else if (y <= 9d+17) then
        tmp = (sin(x) / x) * y
    else
        tmp = sinh(y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -0.046) {
		tmp = Math.sinh(y);
	} else if (y <= 9e+17) {
		tmp = (Math.sin(x) / x) * y;
	} else {
		tmp = Math.sinh(y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -0.046:
		tmp = math.sinh(y)
	elif y <= 9e+17:
		tmp = (math.sin(x) / x) * y
	else:
		tmp = math.sinh(y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -0.046)
		tmp = sinh(y);
	elseif (y <= 9e+17)
		tmp = Float64(Float64(sin(x) / x) * y);
	else
		tmp = sinh(y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -0.046)
		tmp = sinh(y);
	elseif (y <= 9e+17)
		tmp = (sin(x) / x) * y;
	else
		tmp = sinh(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -0.046], N[Sinh[y], $MachinePrecision], If[LessEqual[y, 9e+17], N[(N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision] * y), $MachinePrecision], N[Sinh[y], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -0.045999999999999999 or 9e17 < y

   1. Initial program 100.0%

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

      1. associate-*l/ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 74.0%

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

   if -0.045999999999999999 < y < 9e17

   1. Initial program 82.9%

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

      1. associate-*l/ 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in y around 0 97.1%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.046:\\ \;\;\;\;\sinh y\\ \mathbf{elif}\;y \leq 9 \cdot 10^{+17}:\\ \;\;\;\;\frac{\sin x}{x} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \]

Alternative 6: 55.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.2 \cdot 10^{-63}:\\ \;\;\;\;x \cdot \frac{-1}{\frac{-x}{y}}\\ \mathbf{elif}\;y \leq 1.32 \cdot 10^{+154}:\\ \;\;\;\;y \cdot \left(1 + \left(x \cdot x\right) \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{y \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 3.2e-63)
   (* x (/ -1.0 (/ (- x) y)))
   (if (<= y 1.32e+154)
     (* y (+ 1.0 (* (* x x) -0.16666666666666666)))
     (sqrt (* y y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 3.2e-63) {
		tmp = x * (-1.0 / (-x / y));
	} else if (y <= 1.32e+154) {
		tmp = y * (1.0 + ((x * x) * -0.16666666666666666));
	} else {
		tmp = sqrt((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 <= 3.2d-63) then
        tmp = x * ((-1.0d0) / (-x / y))
    else if (y <= 1.32d+154) then
        tmp = y * (1.0d0 + ((x * x) * (-0.16666666666666666d0)))
    else
        tmp = sqrt((y * y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 3.2e-63) {
		tmp = x * (-1.0 / (-x / y));
	} else if (y <= 1.32e+154) {
		tmp = y * (1.0 + ((x * x) * -0.16666666666666666));
	} else {
		tmp = Math.sqrt((y * y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 3.2e-63:
		tmp = x * (-1.0 / (-x / y))
	elif y <= 1.32e+154:
		tmp = y * (1.0 + ((x * x) * -0.16666666666666666))
	else:
		tmp = math.sqrt((y * y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 3.2e-63)
		tmp = Float64(x * Float64(-1.0 / Float64(Float64(-x) / y)));
	elseif (y <= 1.32e+154)
		tmp = Float64(y * Float64(1.0 + Float64(Float64(x * x) * -0.16666666666666666)));
	else
		tmp = sqrt(Float64(y * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 3.2e-63)
		tmp = x * (-1.0 / (-x / y));
	elseif (y <= 1.32e+154)
		tmp = y * (1.0 + ((x * x) * -0.16666666666666666));
	else
		tmp = sqrt((y * y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 3.2e-63], N[(x * N[(-1.0 / N[((-x) / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.32e+154], N[(y * N[(1.0 + N[(N[(x * x), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(y * y), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 3.19999999999999989e-63

   1. Initial program 88.1%

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

      1. associate-*r/ 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in y around 0 50.6%

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

      1. *-commutative 50.6%

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

      2. associate-/l* 71.9%

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

   6. Simplified 71.9%

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

      1. frac-2neg 71.9%

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

      2. div-inv 72.8%

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

      3. distribute-neg-frac 72.8%

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

   8. Applied egg-rr 72.8%

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

   9. Taylor expanded in x around 0 58.1%

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

   if 3.19999999999999989e-63 < y < 1.31999999999999998e154

   1. Initial program 99.9%

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

      1. associate-*l/ 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around 0 67.3%

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

      1. *-commutative 67.3%

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

      2. unpow2 67.3%

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

   6. Simplified 67.3%

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

   7. Taylor expanded in y around 0 48.6%

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

   if 1.31999999999999998e154 < y

   1. Initial program 100.0%

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

      1. associate-*r/ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 5.3%

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

      1. *-commutative 5.3%

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

      2. associate-/l* 43.8%

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

   6. Simplified 43.8%

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

   7. Taylor expanded in x around 0 43.6%

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

      1. associate-/r/ 5.2%

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

      2. *-inverses 5.2%

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

      3. *-un-lft-identity 5.2%

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

      4. add-sqr-sqrt 5.2%

         \[\leadsto \color{blue}{\sqrt{y} \cdot \sqrt{y}} \]

      5. sqrt-unprod 72.4%

         \[\leadsto \color{blue}{\sqrt{y \cdot y}} \]

   9. Applied egg-rr 72.4%

      \[\leadsto \color{blue}{\sqrt{y \cdot y}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 58.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.2 \cdot 10^{-63}:\\ \;\;\;\;x \cdot \frac{-1}{\frac{-x}{y}}\\ \mathbf{elif}\;y \leq 1.32 \cdot 10^{+154}:\\ \;\;\;\;y \cdot \left(1 + \left(x \cdot x\right) \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{y \cdot y}\\ \end{array} \]

Alternative 7: 74.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.4 \cdot 10^{-70}:\\ \;\;\;\;\sinh y\\ \mathbf{elif}\;y \leq 2.1 \cdot 10^{-63}:\\ \;\;\;\;\frac{x}{\frac{x}{y}}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -5.4e-70) (sinh y) (if (<= y 2.1e-63) (/ x (/ x y)) (sinh y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -5.4e-70) {
		tmp = sinh(y);
	} else if (y <= 2.1e-63) {
		tmp = x / (x / y);
	} else {
		tmp = sinh(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 <= (-5.4d-70)) then
        tmp = sinh(y)
    else if (y <= 2.1d-63) then
        tmp = x / (x / y)
    else
        tmp = sinh(y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -5.4e-70) {
		tmp = Math.sinh(y);
	} else if (y <= 2.1e-63) {
		tmp = x / (x / y);
	} else {
		tmp = Math.sinh(y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -5.4e-70:
		tmp = math.sinh(y)
	elif y <= 2.1e-63:
		tmp = x / (x / y)
	else:
		tmp = math.sinh(y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -5.4e-70)
		tmp = sinh(y);
	elseif (y <= 2.1e-63)
		tmp = Float64(x / Float64(x / y));
	else
		tmp = sinh(y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -5.4e-70)
		tmp = sinh(y);
	elseif (y <= 2.1e-63)
		tmp = x / (x / y);
	else
		tmp = sinh(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -5.4e-70], N[Sinh[y], $MachinePrecision], If[LessEqual[y, 2.1e-63], N[(x / N[(x / y), $MachinePrecision]), $MachinePrecision], N[Sinh[y], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -5.4000000000000003e-70 or 2.1e-63 < y

   1. Initial program 99.2%

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

      1. associate-*l/ 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 70.2%

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

   if -5.4000000000000003e-70 < y < 2.1e-63

   1. Initial program 78.4%

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

      1. associate-*r/ 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in y around 0 78.4%

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

      1. *-commutative 78.4%

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

      2. associate-/l* 99.3%

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

   6. Simplified 99.3%

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

   7. Taylor expanded in x around 0 77.7%

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

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.4 \cdot 10^{-70}:\\ \;\;\;\;\sinh y\\ \mathbf{elif}\;y \leq 2.1 \cdot 10^{-63}:\\ \;\;\;\;\frac{x}{\frac{x}{y}}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \]

Alternative 8: 50.2% accurate, 15.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 8.8 \cdot 10^{+142}:\\ \;\;\;\;x \cdot \frac{y}{x}\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{+170}:\\ \;\;\;\;y \cdot \left(1 + \left(x \cdot x\right) \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{x}{y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 8.8e+142)
   (* x (/ y x))
   (if (<= x 2.6e+170)
     (* y (+ 1.0 (* (* x x) -0.16666666666666666)))
     (/ x (/ x y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 8.8e+142) {
		tmp = x * (y / x);
	} else if (x <= 2.6e+170) {
		tmp = y * (1.0 + ((x * x) * -0.16666666666666666));
	} else {
		tmp = x / (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 (x <= 8.8d+142) then
        tmp = x * (y / x)
    else if (x <= 2.6d+170) then
        tmp = y * (1.0d0 + ((x * x) * (-0.16666666666666666d0)))
    else
        tmp = x / (x / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 8.8e+142) {
		tmp = x * (y / x);
	} else if (x <= 2.6e+170) {
		tmp = y * (1.0 + ((x * x) * -0.16666666666666666));
	} else {
		tmp = x / (x / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 8.8e+142:
		tmp = x * (y / x)
	elif x <= 2.6e+170:
		tmp = y * (1.0 + ((x * x) * -0.16666666666666666))
	else:
		tmp = x / (x / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 8.8e+142)
		tmp = Float64(x * Float64(y / x));
	elseif (x <= 2.6e+170)
		tmp = Float64(y * Float64(1.0 + Float64(Float64(x * x) * -0.16666666666666666)));
	else
		tmp = Float64(x / Float64(x / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 8.8e+142)
		tmp = x * (y / x);
	elseif (x <= 2.6e+170)
		tmp = y * (1.0 + ((x * x) * -0.16666666666666666));
	else
		tmp = x / (x / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 8.8e+142], N[(x * N[(y / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.6e+170], N[(y * N[(1.0 + N[(N[(x * x), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(x / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 8.79999999999999947e142

   1. Initial program 89.5%

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

      1. associate-*r/ 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in y around 0 41.7%

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

      1. *-commutative 41.7%

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

      2. associate-/l* 65.8%

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

   6. Simplified 65.8%

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

   7. Taylor expanded in x around 0 51.7%

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

      1. clear-num 51.6%

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

      2. associate-/r/ 53.0%

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

      3. clear-num 52.6%

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

   9. Applied egg-rr 52.6%

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

   if 8.79999999999999947e142 < x < 2.5999999999999998e170

   1. Initial program 100.0%

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

      1. associate-*l/ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 66.7%

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

      1. *-commutative 66.7%

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

      2. unpow2 66.7%

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

   6. Simplified 66.7%

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

   7. Taylor expanded in y around 0 66.7%

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

   if 2.5999999999999998e170 < x

   1. Initial program 100.0%

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

      1. associate-*r/ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 60.0%

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

      1. *-commutative 60.0%

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

      2. associate-/l* 58.6%

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

   6. Simplified 58.6%

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

   7. Taylor expanded in x around 0 53.0%

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

4. Final simplification 52.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 8.8 \cdot 10^{+142}:\\ \;\;\;\;x \cdot \frac{y}{x}\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{+170}:\\ \;\;\;\;y \cdot \left(1 + \left(x \cdot x\right) \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{x}{y}}\\ \end{array} \]

Alternative 9: 50.2% accurate, 15.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 8.8 \cdot 10^{+142}:\\ \;\;\;\;x \cdot \frac{y}{x}\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{+170}:\\ \;\;\;\;y + x \cdot \left(y \cdot \left(x \cdot -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{x}{y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 8.8e+142)
   (* x (/ y x))
   (if (<= x 2.6e+170)
     (+ y (* x (* y (* x -0.16666666666666666))))
     (/ x (/ x y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 8.8e+142) {
		tmp = x * (y / x);
	} else if (x <= 2.6e+170) {
		tmp = y + (x * (y * (x * -0.16666666666666666)));
	} else {
		tmp = x / (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 (x <= 8.8d+142) then
        tmp = x * (y / x)
    else if (x <= 2.6d+170) then
        tmp = y + (x * (y * (x * (-0.16666666666666666d0))))
    else
        tmp = x / (x / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 8.8e+142) {
		tmp = x * (y / x);
	} else if (x <= 2.6e+170) {
		tmp = y + (x * (y * (x * -0.16666666666666666)));
	} else {
		tmp = x / (x / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 8.8e+142:
		tmp = x * (y / x)
	elif x <= 2.6e+170:
		tmp = y + (x * (y * (x * -0.16666666666666666)))
	else:
		tmp = x / (x / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 8.8e+142)
		tmp = Float64(x * Float64(y / x));
	elseif (x <= 2.6e+170)
		tmp = Float64(y + Float64(x * Float64(y * Float64(x * -0.16666666666666666))));
	else
		tmp = Float64(x / Float64(x / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 8.8e+142)
		tmp = x * (y / x);
	elseif (x <= 2.6e+170)
		tmp = y + (x * (y * (x * -0.16666666666666666)));
	else
		tmp = x / (x / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 8.8e+142], N[(x * N[(y / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.6e+170], N[(y + N[(x * N[(y * N[(x * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(x / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 8.79999999999999947e142

   1. Initial program 89.5%

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

      1. associate-*r/ 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in y around 0 41.7%

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

      1. *-commutative 41.7%

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

      2. associate-/l* 65.8%

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

   6. Simplified 65.8%

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

   7. Taylor expanded in x around 0 51.7%

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

      1. clear-num 51.6%

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

      2. associate-/r/ 53.0%

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

      3. clear-num 52.6%

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

   9. Applied egg-rr 52.6%

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

   if 8.79999999999999947e142 < x < 2.5999999999999998e170

   1. Initial program 100.0%

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

      1. associate-*l/ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 3.1%

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

   5. Taylor expanded in x around 0 66.7%

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

      1. unpow2 66.7%

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

   7. Simplified 66.7%

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

   8. Taylor expanded in y around 0 66.7%

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

      1. *-commutative 66.7%

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

      2. *-commutative 66.7%

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

      3. unpow2 66.7%

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

      4. associate-*l* 66.7%

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

      5. *-commutative 66.7%

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

      6. associate-*l* 66.7%

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

      7. associate-*l* 66.7%

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

   10. Simplified 66.7%

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

   if 2.5999999999999998e170 < x

   1. Initial program 100.0%

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

      1. associate-*r/ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 60.0%

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

      1. *-commutative 60.0%

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

      2. associate-/l* 58.6%

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

   6. Simplified 58.6%

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

   7. Taylor expanded in x around 0 53.0%

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

4. Final simplification 52.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 8.8 \cdot 10^{+142}:\\ \;\;\;\;x \cdot \frac{y}{x}\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{+170}:\\ \;\;\;\;y + x \cdot \left(y \cdot \left(x \cdot -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{x}{y}}\\ \end{array} \]

Alternative 10: 50.8% accurate, 25.6× speedup

Math

\[\begin{array}{l} \\ x \cdot \frac{-1}{\frac{-x}{y}} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* x (/ -1.0 (/ (- x) y))))

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return x * (-1.0 / (-x / y))

Julia

function code(x, y)
	return Float64(x * Float64(-1.0 / Float64(Float64(-x) / y)))
end

MATLAB

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

Wolfram

code[x_, y_] := N[(x * N[(-1.0 / N[((-x) / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 91.1%

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

   1. associate-*r/ 99.5%

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

3. Simplified 99.5%

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

4. Taylor expanded in y around 0 43.8%

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

   1. *-commutative 43.8%

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

   2. associate-/l* 64.0%

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

6. Simplified 64.0%

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

   1. frac-2neg 64.0%

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

   2. div-inv 65.1%

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

   3. distribute-neg-frac 65.1%

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

8. Applied egg-rr 65.1%

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

9. Taylor expanded in x around 0 52.4%

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

10. Final simplification 52.4%

    \[\leadsto x \cdot \frac{-1}{\frac{-x}{y}} \]

Alternative 11: 50.2% accurate, 41.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return x * (y / x)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.1%

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

   1. associate-*r/ 99.5%

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

3. Simplified 99.5%

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

4. Taylor expanded in y around 0 43.8%

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

   1. *-commutative 43.8%

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

   2. associate-/l* 64.0%

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

6. Simplified 64.0%

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

7. Taylor expanded in x around 0 51.3%

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

   1. clear-num 51.3%

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

   2. associate-/r/ 52.4%

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

   3. clear-num 51.9%

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

9. Applied egg-rr 51.9%

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

10. Final simplification 51.9%

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

Alternative 12: 28.4% accurate, 205.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 y)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return y

Julia

function code(x, y)
	return y
end

MATLAB

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

Wolfram

code[x_, y_] := y

Derivation

1. Initial program 91.1%

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

   1. associate-*l/ 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in y around 0 52.6%

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

5. Taylor expanded in x around 0 26.8%

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

6. Final simplification 26.8%

   \[\leadsto y \]

Developer target: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023279 
(FPCore (x y)
  :name "Linear.Quaternion:$ccosh from linear-1.19.1.3"
  :precision binary64

  :herbie-target
  (* (sin x) (/ (sinh y) x))

  (/ (* (sin x) (sinh y)) x))