Linear.Quaternion:$ccosh from linear-1.19.1.3

Percentage Accurate: 89.0% → 99.9%

Time: 6.8s

Alternatives: 10

Speedup: 1.0×

• 49.8% 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: 89.0% 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 88.3%

   \[\frac{\sin x \cdot \sinh y}{x} \]
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: 87.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sinh y \leq -4 \cdot 10^{-7}:\\ \;\;\;\;\sinh y\\ \mathbf{elif}\;\sinh y \leq 0.02:\\ \;\;\;\;\frac{\sin x}{x} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\sinh y \cdot \left(1 + \left(x \cdot x\right) \cdot -0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (sinh y) -4e-7)
   (sinh y)
   (if (<= (sinh y) 0.02)
     (* (/ (sin x) x) y)
     (* (sinh y) (+ 1.0 (* (* x x) -0.16666666666666666))))))

C

double code(double x, double y) {
	double tmp;
	if (sinh(y) <= -4e-7) {
		tmp = sinh(y);
	} else if (sinh(y) <= 0.02) {
		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) <= (-4d-7)) then
        tmp = sinh(y)
    else if (sinh(y) <= 0.02d0) 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) <= -4e-7) {
		tmp = Math.sinh(y);
	} else if (Math.sinh(y) <= 0.02) {
		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) <= -4e-7:
		tmp = math.sinh(y)
	elif math.sinh(y) <= 0.02:
		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) <= -4e-7)
		tmp = sinh(y);
	elseif (sinh(y) <= 0.02)
		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) <= -4e-7)
		tmp = sinh(y);
	elseif (sinh(y) <= 0.02)
		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], -4e-7], N[Sinh[y], $MachinePrecision], If[LessEqual[N[Sinh[y], $MachinePrecision], 0.02], 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) < -3.9999999999999998e-7

   1. Initial program 99.6%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 83.8%

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

   if -3.9999999999999998e-7 < (sinh.f64 y) < 0.0200000000000000004

   1. Initial program 78.4%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 98.9%

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

   if 0.0200000000000000004 < (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 80.0%

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

      1. *-commutative 80.0%

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

      2. unpow2 80.0%

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

   6. Simplified 80.0%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sinh y \leq -4 \cdot 10^{-7}:\\ \;\;\;\;\sinh y\\ \mathbf{elif}\;\sinh y \leq 0.02:\\ \;\;\;\;\frac{\sin x}{x} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\sinh y \cdot \left(1 + \left(x \cdot x\right) \cdot -0.16666666666666666\right)\\ \end{array} \]

Alternative 3: 87.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sinh y \leq -4 \cdot 10^{-7}:\\ \;\;\;\;\sinh y\\ \mathbf{elif}\;\sinh y \leq 0.02:\\ \;\;\;\;\sin x \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (sinh y) -4e-7)
   (sinh y)
   (if (<= (sinh y) 0.02) (* (sin x) (/ y x)) (sinh y))))

C

double code(double x, double y) {
	double tmp;
	if (sinh(y) <= -4e-7) {
		tmp = sinh(y);
	} else if (sinh(y) <= 0.02) {
		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) <= (-4d-7)) then
        tmp = sinh(y)
    else if (sinh(y) <= 0.02d0) 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) <= -4e-7) {
		tmp = Math.sinh(y);
	} else if (Math.sinh(y) <= 0.02) {
		tmp = Math.sin(x) * (y / x);
	} else {
		tmp = Math.sinh(y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if math.sinh(y) <= -4e-7:
		tmp = math.sinh(y)
	elif math.sinh(y) <= 0.02:
		tmp = math.sin(x) * (y / x)
	else:
		tmp = math.sinh(y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (sinh(y) <= -4e-7)
		tmp = sinh(y);
	elseif (sinh(y) <= 0.02)
		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) <= -4e-7)
		tmp = sinh(y);
	elseif (sinh(y) <= 0.02)
		tmp = sin(x) * (y / x);
	else
		tmp = sinh(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[Sinh[y], $MachinePrecision], -4e-7], N[Sinh[y], $MachinePrecision], If[LessEqual[N[Sinh[y], $MachinePrecision], 0.02], 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) < -3.9999999999999998e-7 or 0.0200000000000000004 < (sinh.f64 y)

   1. Initial program 99.7%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 78.8%

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

   if -3.9999999999999998e-7 < (sinh.f64 y) < 0.0200000000000000004

   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 77.4%

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

      1. associate-/l* 98.8%

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

      2. associate-/r/ 98.8%

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

   6. Simplified 98.8%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sinh y \leq -4 \cdot 10^{-7}:\\ \;\;\;\;\sinh y\\ \mathbf{elif}\;\sinh y \leq 0.02:\\ \;\;\;\;\sin x \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \]

Alternative 4: 87.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sinh y \leq -4 \cdot 10^{-7}:\\ \;\;\;\;\sinh y\\ \mathbf{elif}\;\sinh y \leq 0.02:\\ \;\;\;\;\frac{\sin x}{x} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (sinh y) -4e-7)
   (sinh y)
   (if (<= (sinh y) 0.02) (* (/ (sin x) x) y) (sinh y))))

C

double code(double x, double y) {
	double tmp;
	if (sinh(y) <= -4e-7) {
		tmp = sinh(y);
	} else if (sinh(y) <= 0.02) {
		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) <= (-4d-7)) then
        tmp = sinh(y)
    else if (sinh(y) <= 0.02d0) 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) <= -4e-7) {
		tmp = Math.sinh(y);
	} else if (Math.sinh(y) <= 0.02) {
		tmp = (Math.sin(x) / x) * y;
	} else {
		tmp = Math.sinh(y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if math.sinh(y) <= -4e-7:
		tmp = math.sinh(y)
	elif math.sinh(y) <= 0.02:
		tmp = (math.sin(x) / x) * y
	else:
		tmp = math.sinh(y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (sinh(y) <= -4e-7)
		tmp = sinh(y);
	elseif (sinh(y) <= 0.02)
		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) <= -4e-7)
		tmp = sinh(y);
	elseif (sinh(y) <= 0.02)
		tmp = (sin(x) / x) * y;
	else
		tmp = sinh(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[Sinh[y], $MachinePrecision], -4e-7], N[Sinh[y], $MachinePrecision], If[LessEqual[N[Sinh[y], $MachinePrecision], 0.02], 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) < -3.9999999999999998e-7 or 0.0200000000000000004 < (sinh.f64 y)

   1. Initial program 99.7%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 78.8%

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

   if -3.9999999999999998e-7 < (sinh.f64 y) < 0.0200000000000000004

   1. Initial program 78.4%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 98.9%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sinh y \leq -4 \cdot 10^{-7}:\\ \;\;\;\;\sinh y\\ \mathbf{elif}\;\sinh y \leq 0.02:\\ \;\;\;\;\frac{\sin x}{x} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \]

Alternative 5: 74.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sinh y \leq -4 \cdot 10^{-7}:\\ \;\;\;\;\sinh y\\ \mathbf{elif}\;\sinh y \leq 2 \cdot 10^{-50}:\\ \;\;\;\;\frac{x}{\frac{x}{y}}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (sinh y) -4e-7)
   (sinh y)
   (if (<= (sinh y) 2e-50) (/ x (/ x y)) (sinh y))))

C

double code(double x, double y) {
	double tmp;
	if (sinh(y) <= -4e-7) {
		tmp = sinh(y);
	} else if (sinh(y) <= 2e-50) {
		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 (sinh(y) <= (-4d-7)) then
        tmp = sinh(y)
    else if (sinh(y) <= 2d-50) 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 (Math.sinh(y) <= -4e-7) {
		tmp = Math.sinh(y);
	} else if (Math.sinh(y) <= 2e-50) {
		tmp = x / (x / y);
	} else {
		tmp = Math.sinh(y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if math.sinh(y) <= -4e-7:
		tmp = math.sinh(y)
	elif math.sinh(y) <= 2e-50:
		tmp = x / (x / y)
	else:
		tmp = math.sinh(y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (sinh(y) <= -4e-7)
		tmp = sinh(y);
	elseif (sinh(y) <= 2e-50)
		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 (sinh(y) <= -4e-7)
		tmp = sinh(y);
	elseif (sinh(y) <= 2e-50)
		tmp = x / (x / y);
	else
		tmp = sinh(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[Sinh[y], $MachinePrecision], -4e-7], N[Sinh[y], $MachinePrecision], If[LessEqual[N[Sinh[y], $MachinePrecision], 2e-50], N[(x / N[(x / y), $MachinePrecision]), $MachinePrecision], N[Sinh[y], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (sinh.f64 y) < -3.9999999999999998e-7 or 2.00000000000000002e-50 < (sinh.f64 y)

   1. Initial program 99.8%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 76.9%

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

   if -3.9999999999999998e-7 < (sinh.f64 y) < 2.00000000000000002e-50

   1. Initial program 77.3%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 77.2%

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

      1. associate-/l* 99.7%

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

      2. associate-/r/ 99.7%

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

   6. Simplified 99.7%

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

   7. Taylor expanded in x around 0 70.5%

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

      1. *-commutative 70.5%

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

      2. clear-num 72.5%

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

      3. un-div-inv 72.6%

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

   9. Applied egg-rr 72.6%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sinh y \leq -4 \cdot 10^{-7}:\\ \;\;\;\;\sinh y\\ \mathbf{elif}\;\sinh y \leq 2 \cdot 10^{-50}:\\ \;\;\;\;\frac{x}{\frac{x}{y}}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \]

Alternative 6: 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 88.3%

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

   1. associate-*r/ 99.9%

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

3. Simplified 99.9%

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

4. Final simplification 99.9%

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

Alternative 7: 53.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.9 \cdot 10^{+188}:\\ \;\;\;\;x \cdot \frac{-1}{\frac{-x}{y}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{y \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 4.9e+188) (* x (/ -1.0 (/ (- x) y))) (sqrt (* y y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 4.9e+188) {
		tmp = x * (-1.0 / (-x / y));
	} 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 <= 4.9d+188) then
        tmp = x * ((-1.0d0) / (-x / y))
    else
        tmp = sqrt((y * y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 4.9e+188) {
		tmp = x * (-1.0 / (-x / y));
	} else {
		tmp = Math.sqrt((y * y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 4.9e+188:
		tmp = x * (-1.0 / (-x / y))
	else:
		tmp = math.sqrt((y * y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 4.9e+188)
		tmp = Float64(x * Float64(-1.0 / Float64(Float64(-x) / y)));
	else
		tmp = sqrt(Float64(y * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 4.9e+188)
		tmp = x * (-1.0 / (-x / y));
	else
		tmp = sqrt((y * y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 4.9e+188], N[(x * N[(-1.0 / N[((-x) / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(y * y), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.9e188

   1. Initial program 86.9%

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

      1. associate-*r/ 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in y around 0 48.5%

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

      1. associate-/l* 61.4%

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

      2. associate-/r/ 73.7%

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

   6. Simplified 73.7%

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

   7. Taylor expanded in x around 0 56.0%

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

      1. *-commutative 56.0%

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

      2. clear-num 57.1%

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

      3. un-div-inv 55.5%

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

   9. Applied egg-rr 55.5%

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

       1. frac-2neg 55.5%

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

       2. div-inv 57.1%

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

       3. distribute-neg-frac 57.1%

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

   11. Applied egg-rr 57.1%

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

   if 4.9e188 < 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 6.2%

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

      1. associate-/l* 6.2%

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

      2. associate-/r/ 48.7%

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

   6. Simplified 48.7%

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

   7. Taylor expanded in x around 0 48.5%

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

      1. add-sqr-sqrt 48.5%

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

      2. sqrt-unprod 76.9%

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

      3. pow2 76.9%

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

      4. *-commutative 76.9%

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

   9. Applied egg-rr 76.9%

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

       1. *-commutative 76.9%

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

       2. associate-/r/ 76.9%

          \[\leadsto \sqrt{{\color{blue}{\left(\frac{y}{\frac{x}{x}}\right)}}^{2}} \]

       3. *-inverses 76.9%

          \[\leadsto \sqrt{{\left(\frac{y}{\color{blue}{1}}\right)}^{2}} \]

       4. /-rgt-identity 76.9%

          \[\leadsto \sqrt{{\color{blue}{y}}^{2}} \]

       5. unpow2 76.9%

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

   11. Simplified 76.9%

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

4. Final simplification 59.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.9 \cdot 10^{+188}:\\ \;\;\;\;x \cdot \frac{-1}{\frac{-x}{y}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{y \cdot y}\\ \end{array} \]

Alternative 8: 51.0% 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 88.3%

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

   1. associate-*r/ 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in y around 0 44.2%

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

   1. associate-/l* 55.8%

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

   2. associate-/r/ 71.2%

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

6. Simplified 71.2%

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

7. Taylor expanded in x around 0 55.2%

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

   1. *-commutative 55.2%

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

   2. clear-num 56.2%

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

   3. un-div-inv 54.8%

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

9. Applied egg-rr 54.8%

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

    1. frac-2neg 54.8%

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

    2. div-inv 56.2%

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

    3. distribute-neg-frac 56.2%

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

11. Applied egg-rr 56.2%

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

12. Final simplification 56.2%

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

Alternative 9: 50.6% 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 88.3%

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

   1. associate-*r/ 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in y around 0 44.2%

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

   1. associate-/l* 55.8%

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

   2. associate-/r/ 71.2%

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

6. Simplified 71.2%

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

7. Taylor expanded in x around 0 55.2%

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

8. Final simplification 55.2%

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

Alternative 10: 28.2% 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 88.3%

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

   1. associate-*r/ 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in y around 0 44.2%

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

   1. associate-/l* 55.8%

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

   2. associate-/r/ 71.2%

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

6. Simplified 71.2%

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

7. Taylor expanded in x around 0 27.1%

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

8. Final simplification 27.1%

   \[\leadsto y \]

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 2023238 
(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))