Linear.Quaternion:$ccosh from linear-1.19.1.3

Percentage Accurate: 88.1% → 99.9%

Time: 8.0s

Alternatives: 10

Speedup: 1.0×

• 49.0% 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: 88.1% 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 86.0%

   \[\frac{\sin x \cdot \sinh y}{x} \]
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: 87.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sinh y \leq -2 \cdot 10^{-7}:\\ \;\;\;\;\sinh y\\ \mathbf{elif}\;\sinh y \leq 0.005:\\ \;\;\;\;\sin x \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (sinh y) -2e-7)
   (sinh y)
   (if (<= (sinh y) 0.005) (* (sin x) (/ y x)) (sinh y))))

C

double code(double x, double y) {
	double tmp;
	if (sinh(y) <= -2e-7) {
		tmp = sinh(y);
	} else if (sinh(y) <= 0.005) {
		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 (sinh(y) <= (-2d-7)) then
        tmp = sinh(y)
    else if (sinh(y) <= 0.005d0) 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 (Math.sinh(y) <= -2e-7) {
		tmp = Math.sinh(y);
	} else if (Math.sinh(y) <= 0.005) {
		tmp = Math.sin(x) * (y / x);
	} else {
		tmp = Math.sinh(y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if math.sinh(y) <= -2e-7:
		tmp = math.sinh(y)
	elif math.sinh(y) <= 0.005:
		tmp = math.sin(x) * (y / x)
	else:
		tmp = math.sinh(y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (sinh(y) <= -2e-7)
		tmp = sinh(y);
	elseif (sinh(y) <= 0.005)
		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 (sinh(y) <= -2e-7)
		tmp = sinh(y);
	elseif (sinh(y) <= 0.005)
		tmp = sin(x) * (y / x);
	else
		tmp = sinh(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[Sinh[y], $MachinePrecision], -2e-7], N[Sinh[y], $MachinePrecision], If[LessEqual[N[Sinh[y], $MachinePrecision], 0.005], N[(N[Sin[x], $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision], N[Sinh[y], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (sinh.f64 y) < -1.9999999999999999e-7 or 0.0050000000000000001 < (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 76.7%

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

   if -1.9999999999999999e-7 < (sinh.f64 y) < 0.0050000000000000001

   1. Initial program 70.9%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 70.1%

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

      1. associate-/l* 99.1%

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

      2. associate-/r/ 98.9%

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

   6. Simplified 98.9%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sinh y \leq -2 \cdot 10^{-7}:\\ \;\;\;\;\sinh y\\ \mathbf{elif}\;\sinh y \leq 0.005:\\ \;\;\;\;\sin x \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \]

Alternative 3: 87.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sinh y \leq -2 \cdot 10^{-7}:\\ \;\;\;\;\sinh y\\ \mathbf{elif}\;\sinh y \leq 0.005:\\ \;\;\;\;\frac{\sin x}{x} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (sinh y) -2e-7)
   (sinh y)
   (if (<= (sinh y) 0.005) (* (/ (sin x) x) y) (sinh y))))

C

double code(double x, double y) {
	double tmp;
	if (sinh(y) <= -2e-7) {
		tmp = sinh(y);
	} else if (sinh(y) <= 0.005) {
		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 (sinh(y) <= (-2d-7)) then
        tmp = sinh(y)
    else if (sinh(y) <= 0.005d0) 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 (Math.sinh(y) <= -2e-7) {
		tmp = Math.sinh(y);
	} else if (Math.sinh(y) <= 0.005) {
		tmp = (Math.sin(x) / x) * y;
	} else {
		tmp = Math.sinh(y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if math.sinh(y) <= -2e-7:
		tmp = math.sinh(y)
	elif math.sinh(y) <= 0.005:
		tmp = (math.sin(x) / x) * y
	else:
		tmp = math.sinh(y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (sinh(y) <= -2e-7)
		tmp = sinh(y);
	elseif (sinh(y) <= 0.005)
		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 (sinh(y) <= -2e-7)
		tmp = sinh(y);
	elseif (sinh(y) <= 0.005)
		tmp = (sin(x) / x) * y;
	else
		tmp = sinh(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[Sinh[y], $MachinePrecision], -2e-7], N[Sinh[y], $MachinePrecision], If[LessEqual[N[Sinh[y], $MachinePrecision], 0.005], N[(N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision] * y), $MachinePrecision], N[Sinh[y], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (sinh.f64 y) < -1.9999999999999999e-7 or 0.0050000000000000001 < (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 76.7%

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

   if -1.9999999999999999e-7 < (sinh.f64 y) < 0.0050000000000000001

   1. Initial program 70.9%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 99.1%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sinh y \leq -2 \cdot 10^{-7}:\\ \;\;\;\;\sinh y\\ \mathbf{elif}\;\sinh y \leq 0.005:\\ \;\;\;\;\frac{\sin x}{x} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \]

Alternative 4: 87.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sinh y \leq -2 \cdot 10^{-7}:\\ \;\;\;\;\sinh y\\ \mathbf{elif}\;\sinh y \leq 0.005:\\ \;\;\;\;\frac{y}{\frac{x}{\sin x}}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (sinh y) -2e-7)
   (sinh y)
   (if (<= (sinh y) 0.005) (/ y (/ x (sin x))) (sinh y))))

C

double code(double x, double y) {
	double tmp;
	if (sinh(y) <= -2e-7) {
		tmp = sinh(y);
	} else if (sinh(y) <= 0.005) {
		tmp = y / (x / sin(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 (sinh(y) <= (-2d-7)) then
        tmp = sinh(y)
    else if (sinh(y) <= 0.005d0) then
        tmp = y / (x / sin(x))
    else
        tmp = sinh(y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (Math.sinh(y) <= -2e-7) {
		tmp = Math.sinh(y);
	} else if (Math.sinh(y) <= 0.005) {
		tmp = y / (x / Math.sin(x));
	} else {
		tmp = Math.sinh(y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if math.sinh(y) <= -2e-7:
		tmp = math.sinh(y)
	elif math.sinh(y) <= 0.005:
		tmp = y / (x / math.sin(x))
	else:
		tmp = math.sinh(y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (sinh(y) <= -2e-7)
		tmp = sinh(y);
	elseif (sinh(y) <= 0.005)
		tmp = Float64(y / Float64(x / sin(x)));
	else
		tmp = sinh(y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (sinh(y) <= -2e-7)
		tmp = sinh(y);
	elseif (sinh(y) <= 0.005)
		tmp = y / (x / sin(x));
	else
		tmp = sinh(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[Sinh[y], $MachinePrecision], -2e-7], N[Sinh[y], $MachinePrecision], If[LessEqual[N[Sinh[y], $MachinePrecision], 0.005], N[(y / N[(x / N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sinh[y], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (sinh.f64 y) < -1.9999999999999999e-7 or 0.0050000000000000001 < (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 76.7%

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

   if -1.9999999999999999e-7 < (sinh.f64 y) < 0.0050000000000000001

   1. Initial program 70.9%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 70.1%

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

      1. associate-/l* 99.1%

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

   6. Simplified 99.1%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sinh y \leq -2 \cdot 10^{-7}:\\ \;\;\;\;\sinh y\\ \mathbf{elif}\;\sinh y \leq 0.005:\\ \;\;\;\;\frac{y}{\frac{x}{\sin x}}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \]

Alternative 5: 86.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sinh y \leq -2 \cdot 10^{-7}:\\ \;\;\;\;\left(1 + -0.16666666666666666 \cdot \left(x \cdot x\right)\right) \cdot \sinh y\\ \mathbf{elif}\;\sinh y \leq 0.005:\\ \;\;\;\;\frac{y}{\frac{x}{\sin x}}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (sinh y) -2e-7)
   (* (+ 1.0 (* -0.16666666666666666 (* x x))) (sinh y))
   (if (<= (sinh y) 0.005) (/ y (/ x (sin x))) (sinh y))))

C

double code(double x, double y) {
	double tmp;
	if (sinh(y) <= -2e-7) {
		tmp = (1.0 + (-0.16666666666666666 * (x * x))) * sinh(y);
	} else if (sinh(y) <= 0.005) {
		tmp = y / (x / sin(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 (sinh(y) <= (-2d-7)) then
        tmp = (1.0d0 + ((-0.16666666666666666d0) * (x * x))) * sinh(y)
    else if (sinh(y) <= 0.005d0) then
        tmp = y / (x / sin(x))
    else
        tmp = sinh(y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (Math.sinh(y) <= -2e-7) {
		tmp = (1.0 + (-0.16666666666666666 * (x * x))) * Math.sinh(y);
	} else if (Math.sinh(y) <= 0.005) {
		tmp = y / (x / Math.sin(x));
	} else {
		tmp = Math.sinh(y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if math.sinh(y) <= -2e-7:
		tmp = (1.0 + (-0.16666666666666666 * (x * x))) * math.sinh(y)
	elif math.sinh(y) <= 0.005:
		tmp = y / (x / math.sin(x))
	else:
		tmp = math.sinh(y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (sinh(y) <= -2e-7)
		tmp = Float64(Float64(1.0 + Float64(-0.16666666666666666 * Float64(x * x))) * sinh(y));
	elseif (sinh(y) <= 0.005)
		tmp = Float64(y / Float64(x / sin(x)));
	else
		tmp = sinh(y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (sinh(y) <= -2e-7)
		tmp = (1.0 + (-0.16666666666666666 * (x * x))) * sinh(y);
	elseif (sinh(y) <= 0.005)
		tmp = y / (x / sin(x));
	else
		tmp = sinh(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[Sinh[y], $MachinePrecision], -2e-7], N[(N[(1.0 + N[(-0.16666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sinh[y], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sinh[y], $MachinePrecision], 0.005], N[(y / N[(x / N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sinh[y], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sinh.f64 y) < -1.9999999999999999e-7

   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 89.4%

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

      1. unpow2 26.5%

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

   6. Simplified 89.4%

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

   if -1.9999999999999999e-7 < (sinh.f64 y) < 0.0050000000000000001

   1. Initial program 70.9%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 70.1%

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

      1. associate-/l* 99.1%

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

   6. Simplified 99.1%

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

   if 0.0050000000000000001 < (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 82.1%

      \[\leadsto \color{blue}{1} \cdot \sinh y \]
3. Recombined 3 regimes into one program.

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sinh y \leq -2 \cdot 10^{-7}:\\ \;\;\;\;\left(1 + -0.16666666666666666 \cdot \left(x \cdot x\right)\right) \cdot \sinh y\\ \mathbf{elif}\;\sinh y \leq 0.005:\\ \;\;\;\;\frac{y}{\frac{x}{\sin x}}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \]

Alternative 6: 63.6% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 (sinh y))

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x, y)
	return sinh(y)
end

MATLAB

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

Wolfram

code[x_, y_] := N[Sinh[y], $MachinePrecision]

Derivation

1. Initial program 86.0%

   \[\frac{\sin x \cdot \sinh y}{x} \]
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 67.9%

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

5. Final simplification 67.9%

   \[\leadsto \sinh y \]

Alternative 7: 35.0% accurate, 18.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1200000:\\ \;\;\;\;y \cdot \left(-0.16666666666666666 \cdot \left(x \cdot x\right)\right)\\ \mathbf{elif}\;y \leq 4.9 \cdot 10^{-8}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1200000.0)
   (* y (* -0.16666666666666666 (* x x)))
   (if (<= y 4.9e-8) y (* (* x y) (/ 1.0 x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1200000.0) {
		tmp = y * (-0.16666666666666666 * (x * x));
	} else if (y <= 4.9e-8) {
		tmp = y;
	} else {
		tmp = (x * y) * (1.0 / x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1200000.0d0)) then
        tmp = y * ((-0.16666666666666666d0) * (x * x))
    else if (y <= 4.9d-8) then
        tmp = y
    else
        tmp = (x * y) * (1.0d0 / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1200000.0) {
		tmp = y * (-0.16666666666666666 * (x * x));
	} else if (y <= 4.9e-8) {
		tmp = y;
	} else {
		tmp = (x * y) * (1.0 / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1200000.0:
		tmp = y * (-0.16666666666666666 * (x * x))
	elif y <= 4.9e-8:
		tmp = y
	else:
		tmp = (x * y) * (1.0 / x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1200000.0)
		tmp = Float64(y * Float64(-0.16666666666666666 * Float64(x * x)));
	elseif (y <= 4.9e-8)
		tmp = y;
	else
		tmp = Float64(Float64(x * y) * Float64(1.0 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1200000.0)
		tmp = y * (-0.16666666666666666 * (x * x));
	elseif (y <= 4.9e-8)
		tmp = y;
	else
		tmp = (x * y) * (1.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1200000.0], N[(y * N[(-0.16666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.9e-8], y, N[(N[(x * y), $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.2e6

   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 4.9%

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

   5. Taylor expanded in x around 0 25.9%

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

      1. unpow2 25.9%

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

   7. Simplified 25.9%

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

   8. Taylor expanded in x around inf 23.1%

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

      1. unpow2 23.1%

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

   10. Simplified 23.1%

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

   11. Taylor expanded in y around 0 23.1%

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

       1. associate-*r* 23.1%

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

       2. *-commutative 23.1%

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

       3. associate-*l* 23.1%

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

       4. unpow2 23.1%

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

   13. Simplified 23.1%

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

   if -1.2e6 < y < 4.9000000000000002e-8

   1. Initial program 70.6%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 99.0%

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

   5. Taylor expanded in x around 0 58.6%

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

   if 4.9000000000000002e-8 < y

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 29.8%

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

      1. associate-/l* 7.2%

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

      2. div-inv 7.3%

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

      3. *-commutative 7.3%

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

   6. Applied egg-rr 7.3%

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

   7. Taylor expanded in x around 0 27.5%

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

4. Final simplification 41.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1200000:\\ \;\;\;\;y \cdot \left(-0.16666666666666666 \cdot \left(x \cdot x\right)\right)\\ \mathbf{elif}\;y \leq 4.9 \cdot 10^{-8}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{1}{x}\\ \end{array} \]

Alternative 8: 35.7% accurate, 18.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5.2 \cdot 10^{+111}:\\ \;\;\;\;y \cdot \left(1 + -0.16666666666666666 \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 5.2e+111)
   (* y (+ 1.0 (* -0.16666666666666666 (* x x))))
   (* (* x y) (/ 1.0 x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 5.2e+111) {
		tmp = y * (1.0 + (-0.16666666666666666 * (x * x)));
	} else {
		tmp = (x * y) * (1.0 / x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 5.2d+111) then
        tmp = y * (1.0d0 + ((-0.16666666666666666d0) * (x * x)))
    else
        tmp = (x * y) * (1.0d0 / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 5.2e+111) {
		tmp = y * (1.0 + (-0.16666666666666666 * (x * x)));
	} else {
		tmp = (x * y) * (1.0 / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 5.2e+111:
		tmp = y * (1.0 + (-0.16666666666666666 * (x * x)))
	else:
		tmp = (x * y) * (1.0 / x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 5.2e+111)
		tmp = Float64(y * Float64(1.0 + Float64(-0.16666666666666666 * Float64(x * x))));
	else
		tmp = Float64(Float64(x * y) * Float64(1.0 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 5.2e+111)
		tmp = y * (1.0 + (-0.16666666666666666 * (x * x)));
	else
		tmp = (x * y) * (1.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 5.2e+111], N[(y * N[(1.0 + N[(-0.16666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * y), $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5.1999999999999997e111

   1. Initial program 82.9%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 60.3%

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

   5. Taylor expanded in x around 0 43.6%

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

      1. unpow2 43.6%

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

   7. Simplified 43.6%

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

   if 5.1999999999999997e111 < y

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 37.5%

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

      1. associate-/l* 5.2%

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

      2. div-inv 5.2%

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

      3. *-commutative 5.2%

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

   6. Applied egg-rr 5.2%

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

   7. Taylor expanded in x around 0 36.1%

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

4. Final simplification 42.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5.2 \cdot 10^{+111}:\\ \;\;\;\;y \cdot \left(1 + -0.16666666666666666 \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{1}{x}\\ \end{array} \]

Alternative 9: 32.5% accurate, 22.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 6.5 \cdot 10^{+26}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(y \cdot \left(x \cdot x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 6.5e+26) y (* -0.16666666666666666 (* y (* x x)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 6.5e+26) {
		tmp = y;
	} else {
		tmp = -0.16666666666666666 * (y * (x * x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 6.5d+26) then
        tmp = y
    else
        tmp = (-0.16666666666666666d0) * (y * (x * x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 6.5e+26) {
		tmp = y;
	} else {
		tmp = -0.16666666666666666 * (y * (x * x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 6.5e+26:
		tmp = y
	else:
		tmp = -0.16666666666666666 * (y * (x * x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 6.5e+26)
		tmp = y;
	else
		tmp = Float64(-0.16666666666666666 * Float64(y * Float64(x * x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 6.5e+26)
		tmp = y;
	else
		tmp = -0.16666666666666666 * (y * (x * x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 6.5e+26], y, N[(-0.16666666666666666 * N[(y * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 6.50000000000000022e26

   1. Initial program 82.2%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 y around 0 49.3%

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

   5. Taylor expanded in x around 0 37.8%

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

   if 6.50000000000000022e26 < x

   1. Initial program 99.9%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 54.3%

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

   5. Taylor expanded in x around 0 21.3%

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

      1. unpow2 21.3%

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

   7. Simplified 21.3%

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

   8. Taylor expanded in x around inf 21.3%

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

      1. unpow2 21.3%

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

   10. Simplified 21.3%

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

4. Final simplification 34.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6.5 \cdot 10^{+26}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(y \cdot \left(x \cdot x\right)\right)\\ \end{array} \]

Alternative 10: 28.5% 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 86.0%

   \[\frac{\sin x \cdot \sinh y}{x} \]
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 50.4%

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

5. Taylor expanded in x around 0 30.8%

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

6. Final simplification 30.8%

   \[\leadsto y \]

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 2023268 
(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))