Linear.Quaternion:$ccosh from linear-1.19.1.3

Percentage Accurate: 88.3% → 99.9%

Time: 10.0s

Alternatives: 10

Speedup: 1.0×

• 49.3% 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: 88.3% 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 90.1%

   \[\frac{\sin x \cdot \sinh y}{x} \]
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. Final simplification 100.0%

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

Alternative 2: 87.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sinh y \leq -0.0001:\\ \;\;\;\;\sinh y\\ \mathbf{elif}\;\sinh y \leq 0.002:\\ \;\;\;\;\frac{y}{\frac{x}{\sin x}}\\ \mathbf{else}:\\ \;\;\;\;\sinh y \cdot \left(1 + -0.16666666666666666 \cdot \left(x \cdot x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (sinh y) -0.0001)
   (sinh y)
   (if (<= (sinh y) 0.002)
     (/ y (/ x (sin x)))
     (* (sinh y) (+ 1.0 (* -0.16666666666666666 (* x x)))))))

C

double code(double x, double y) {
	double tmp;
	if (sinh(y) <= -0.0001) {
		tmp = sinh(y);
	} else if (sinh(y) <= 0.002) {
		tmp = y / (x / sin(x));
	} else {
		tmp = sinh(y) * (1.0 + (-0.16666666666666666 * (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 (sinh(y) <= (-0.0001d0)) then
        tmp = sinh(y)
    else if (sinh(y) <= 0.002d0) then
        tmp = y / (x / sin(x))
    else
        tmp = sinh(y) * (1.0d0 + ((-0.16666666666666666d0) * (x * x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (Math.sinh(y) <= -0.0001) {
		tmp = Math.sinh(y);
	} else if (Math.sinh(y) <= 0.002) {
		tmp = y / (x / Math.sin(x));
	} else {
		tmp = Math.sinh(y) * (1.0 + (-0.16666666666666666 * (x * x)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if math.sinh(y) <= -0.0001:
		tmp = math.sinh(y)
	elif math.sinh(y) <= 0.002:
		tmp = y / (x / math.sin(x))
	else:
		tmp = math.sinh(y) * (1.0 + (-0.16666666666666666 * (x * x)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (sinh(y) <= -0.0001)
		tmp = sinh(y);
	elseif (sinh(y) <= 0.002)
		tmp = Float64(y / Float64(x / sin(x)));
	else
		tmp = Float64(sinh(y) * Float64(1.0 + Float64(-0.16666666666666666 * Float64(x * x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (sinh(y) <= -0.0001)
		tmp = sinh(y);
	elseif (sinh(y) <= 0.002)
		tmp = y / (x / sin(x));
	else
		tmp = sinh(y) * (1.0 + (-0.16666666666666666 * (x * x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[Sinh[y], $MachinePrecision], -0.0001], N[Sinh[y], $MachinePrecision], If[LessEqual[N[Sinh[y], $MachinePrecision], 0.002], N[(y / N[(x / N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sinh[y], $MachinePrecision] * N[(1.0 + N[(-0.16666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sinh.f64 y) < -1.00000000000000005e-4

   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 76.9%

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

   if -1.00000000000000005e-4 < (sinh.f64 y) < 2e-3

   1. Initial program 80.9%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 80.3%

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

      1. associate-/l* 99.4%

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

   6. Simplified 99.4%

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

   if 2e-3 < (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 77.6%

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

      1. unpow2 77.6%

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

   6. Simplified 77.6%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sinh y \leq -0.0001:\\ \;\;\;\;\sinh y\\ \mathbf{elif}\;\sinh y \leq 0.002:\\ \;\;\;\;\frac{y}{\frac{x}{\sin x}}\\ \mathbf{else}:\\ \;\;\;\;\sinh y \cdot \left(1 + -0.16666666666666666 \cdot \left(x \cdot x\right)\right)\\ \end{array} \]

Alternative 3: 87.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sinh y \leq -0.0001:\\ \;\;\;\;\sinh y\\ \mathbf{elif}\;\sinh y \leq 0.002:\\ \;\;\;\;\sin x \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (sinh y) -0.0001)
   (sinh y)
   (if (<= (sinh y) 0.002) (* (sin x) (/ y x)) (sinh y))))

C

double code(double x, double y) {
	double tmp;
	if (sinh(y) <= -0.0001) {
		tmp = sinh(y);
	} else if (sinh(y) <= 0.002) {
		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) <= (-0.0001d0)) then
        tmp = sinh(y)
    else if (sinh(y) <= 0.002d0) 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) <= -0.0001) {
		tmp = Math.sinh(y);
	} else if (Math.sinh(y) <= 0.002) {
		tmp = Math.sin(x) * (y / x);
	} else {
		tmp = Math.sinh(y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if math.sinh(y) <= -0.0001:
		tmp = math.sinh(y)
	elif math.sinh(y) <= 0.002:
		tmp = math.sin(x) * (y / x)
	else:
		tmp = math.sinh(y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (sinh(y) <= -0.0001)
		tmp = sinh(y);
	elseif (sinh(y) <= 0.002)
		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) <= -0.0001)
		tmp = sinh(y);
	elseif (sinh(y) <= 0.002)
		tmp = sin(x) * (y / x);
	else
		tmp = sinh(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[Sinh[y], $MachinePrecision], -0.0001], N[Sinh[y], $MachinePrecision], If[LessEqual[N[Sinh[y], $MachinePrecision], 0.002], 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) < -1.00000000000000005e-4 or 2e-3 < (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 76.4%

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

   if -1.00000000000000005e-4 < (sinh.f64 y) < 2e-3

   1. Initial program 80.9%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 80.3%

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

      1. associate-/l* 99.4%

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

      2. associate-/r/ 99.3%

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

   6. Simplified 99.3%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sinh y \leq -0.0001:\\ \;\;\;\;\sinh y\\ \mathbf{elif}\;\sinh y \leq 0.002:\\ \;\;\;\;\sin x \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \]

Alternative 4: 87.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sinh y \leq -0.0001:\\ \;\;\;\;\sinh y\\ \mathbf{elif}\;\sinh y \leq 0.002:\\ \;\;\;\;\frac{y}{\frac{x}{\sin x}}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (sinh y) -0.0001)
   (sinh y)
   (if (<= (sinh y) 0.002) (/ y (/ x (sin x))) (sinh y))))

C

double code(double x, double y) {
	double tmp;
	if (sinh(y) <= -0.0001) {
		tmp = sinh(y);
	} else if (sinh(y) <= 0.002) {
		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) <= (-0.0001d0)) then
        tmp = sinh(y)
    else if (sinh(y) <= 0.002d0) 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) <= -0.0001) {
		tmp = Math.sinh(y);
	} else if (Math.sinh(y) <= 0.002) {
		tmp = y / (x / Math.sin(x));
	} else {
		tmp = Math.sinh(y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if math.sinh(y) <= -0.0001:
		tmp = math.sinh(y)
	elif math.sinh(y) <= 0.002:
		tmp = y / (x / math.sin(x))
	else:
		tmp = math.sinh(y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (sinh(y) <= -0.0001)
		tmp = sinh(y);
	elseif (sinh(y) <= 0.002)
		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) <= -0.0001)
		tmp = sinh(y);
	elseif (sinh(y) <= 0.002)
		tmp = y / (x / sin(x));
	else
		tmp = sinh(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[Sinh[y], $MachinePrecision], -0.0001], N[Sinh[y], $MachinePrecision], If[LessEqual[N[Sinh[y], $MachinePrecision], 0.002], 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) < -1.00000000000000005e-4 or 2e-3 < (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 76.4%

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

   if -1.00000000000000005e-4 < (sinh.f64 y) < 2e-3

   1. Initial program 80.9%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 80.3%

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

      1. associate-/l* 99.4%

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

   6. Simplified 99.4%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sinh y \leq -0.0001:\\ \;\;\;\;\sinh y\\ \mathbf{elif}\;\sinh y \leq 0.002:\\ \;\;\;\;\frac{y}{\frac{x}{\sin x}}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \]

Alternative 5: 75.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sinh y \leq -1 \cdot 10^{-11}:\\ \;\;\;\;\sinh y\\ \mathbf{elif}\;\sinh y \leq 2 \cdot 10^{-43}:\\ \;\;\;\;\frac{y}{x \cdot \left(x \cdot 0.16666666666666666 + \frac{1}{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (sinh y) -1e-11)
   (sinh y)
   (if (<= (sinh y) 2e-43)
     (/ y (* x (+ (* x 0.16666666666666666) (/ 1.0 x))))
     (sinh y))))

C

double code(double x, double y) {
	double tmp;
	if (sinh(y) <= -1e-11) {
		tmp = sinh(y);
	} else if (sinh(y) <= 2e-43) {
		tmp = y / (x * ((x * 0.16666666666666666) + (1.0 / 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) <= (-1d-11)) then
        tmp = sinh(y)
    else if (sinh(y) <= 2d-43) then
        tmp = y / (x * ((x * 0.16666666666666666d0) + (1.0d0 / 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) <= -1e-11) {
		tmp = Math.sinh(y);
	} else if (Math.sinh(y) <= 2e-43) {
		tmp = y / (x * ((x * 0.16666666666666666) + (1.0 / x)));
	} else {
		tmp = Math.sinh(y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if math.sinh(y) <= -1e-11:
		tmp = math.sinh(y)
	elif math.sinh(y) <= 2e-43:
		tmp = y / (x * ((x * 0.16666666666666666) + (1.0 / x)))
	else:
		tmp = math.sinh(y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (sinh(y) <= -1e-11)
		tmp = sinh(y);
	elseif (sinh(y) <= 2e-43)
		tmp = Float64(y / Float64(x * Float64(Float64(x * 0.16666666666666666) + Float64(1.0 / x))));
	else
		tmp = sinh(y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (sinh(y) <= -1e-11)
		tmp = sinh(y);
	elseif (sinh(y) <= 2e-43)
		tmp = y / (x * ((x * 0.16666666666666666) + (1.0 / x)));
	else
		tmp = sinh(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[Sinh[y], $MachinePrecision], -1e-11], N[Sinh[y], $MachinePrecision], If[LessEqual[N[Sinh[y], $MachinePrecision], 2e-43], N[(y / N[(x * N[(N[(x * 0.16666666666666666), $MachinePrecision] + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sinh[y], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (sinh.f64 y) < -9.99999999999999939e-12 or 2.00000000000000015e-43 < (sinh.f64 y)

   1. Initial program 99.5%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 76.2%

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

   if -9.99999999999999939e-12 < (sinh.f64 y) < 2.00000000000000015e-43

   1. Initial program 79.7%

      \[\frac{\sin x \cdot \sinh y}{x} \]
   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 79.7%

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

      1. associate-/l* 99.9%

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

   6. Simplified 99.9%

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

      1. clear-num 99.9%

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

      2. associate-/r/ 99.8%

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

   8. Applied egg-rr 99.8%

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

   9. Taylor expanded in x around 0 77.5%

      \[\leadsto \frac{y}{\color{blue}{\left(0.16666666666666666 \cdot x + \frac{1}{x}\right)} \cdot x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sinh y \leq -1 \cdot 10^{-11}:\\ \;\;\;\;\sinh y\\ \mathbf{elif}\;\sinh y \leq 2 \cdot 10^{-43}:\\ \;\;\;\;\frac{y}{x \cdot \left(x \cdot 0.16666666666666666 + \frac{1}{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \]

Alternative 6: 53.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.7:\\ \;\;\;\;x \cdot \frac{y}{x}\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{+241}:\\ \;\;\;\;\sqrt{y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot \left(x \cdot 0.16666666666666666 + \frac{1}{x}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 2.7)
   (* x (/ y x))
   (if (<= x 9.5e+241)
     (sqrt (* y y))
     (/ y (* x (+ (* x 0.16666666666666666) (/ 1.0 x)))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 2.7) {
		tmp = x * (y / x);
	} else if (x <= 9.5e+241) {
		tmp = sqrt((y * y));
	} else {
		tmp = y / (x * ((x * 0.16666666666666666) + (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 (x <= 2.7d0) then
        tmp = x * (y / x)
    else if (x <= 9.5d+241) then
        tmp = sqrt((y * y))
    else
        tmp = y / (x * ((x * 0.16666666666666666d0) + (1.0d0 / x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 2.7) {
		tmp = x * (y / x);
	} else if (x <= 9.5e+241) {
		tmp = Math.sqrt((y * y));
	} else {
		tmp = y / (x * ((x * 0.16666666666666666) + (1.0 / x)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 2.7:
		tmp = x * (y / x)
	elif x <= 9.5e+241:
		tmp = math.sqrt((y * y))
	else:
		tmp = y / (x * ((x * 0.16666666666666666) + (1.0 / x)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 2.7)
		tmp = Float64(x * Float64(y / x));
	elseif (x <= 9.5e+241)
		tmp = sqrt(Float64(y * y));
	else
		tmp = Float64(y / Float64(x * Float64(Float64(x * 0.16666666666666666) + Float64(1.0 / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 2.7)
		tmp = x * (y / x);
	elseif (x <= 9.5e+241)
		tmp = sqrt((y * y));
	else
		tmp = y / (x * ((x * 0.16666666666666666) + (1.0 / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 2.7], N[(x * N[(y / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 9.5e+241], N[Sqrt[N[(y * y), $MachinePrecision]], $MachinePrecision], N[(y / N[(x * N[(N[(x * 0.16666666666666666), $MachinePrecision] + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 2.7000000000000002

   1. Initial program 86.7%

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

   2. Taylor expanded in y around 0 38.1%

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

   3. Taylor expanded in x around 0 28.5%

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

      1. associate-/l* 36.8%

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

      2. associate-/r/ 60.9%

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

   5. Applied egg-rr 60.9%

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

   if 2.7000000000000002 < x < 9.50000000000000019e241

   1. Initial program 99.7%

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

   2. Taylor expanded in y around 0 56.0%

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

   3. Taylor expanded in x around 0 17.6%

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

      1. associate-/l* 4.8%

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

      2. *-inverses 4.8%

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

      3. /-rgt-identity 4.8%

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

      4. add-sqr-sqrt 2.5%

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

      5. sqrt-unprod 35.5%

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

   5. Applied egg-rr 35.5%

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

   if 9.50000000000000019e241 < x

   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 80.4%

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

      1. associate-/l* 80.4%

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

   6. Simplified 80.4%

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

      1. clear-num 80.4%

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

      2. associate-/r/ 80.4%

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

   8. Applied egg-rr 80.4%

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

   9. Taylor expanded in x around 0 68.5%

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

4. Final simplification 56.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.7:\\ \;\;\;\;x \cdot \frac{y}{x}\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{+241}:\\ \;\;\;\;\sqrt{y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot \left(x \cdot 0.16666666666666666 + \frac{1}{x}\right)}\\ \end{array} \]

Alternative 7: 51.3% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{y} + y \cdot -0.16666666666666666\\ \mathbf{if}\;x \leq 6.8 \cdot 10^{+89}:\\ \;\;\;\;x \cdot \frac{y}{x}\\ \mathbf{elif}\;x \leq 2.35 \cdot 10^{+154}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(x \cdot \left(x \cdot y\right)\right)\\ \mathbf{elif}\;x \leq 9.2 \cdot 10^{+165}:\\ \;\;\;\;\frac{1}{t_0} + -0.16666666666666666 \cdot \frac{x \cdot x}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot \left(x \cdot 0.16666666666666666 + \frac{1}{x}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (/ 1.0 y) (* y -0.16666666666666666))))
   (if (<= x 6.8e+89)
     (* x (/ y x))
     (if (<= x 2.35e+154)
       (* -0.16666666666666666 (* x (* x y)))
       (if (<= x 9.2e+165)
         (+ (/ 1.0 t_0) (* -0.16666666666666666 (/ (* x x) t_0)))
         (/ y (* x (+ (* x 0.16666666666666666) (/ 1.0 x)))))))))

C

double code(double x, double y) {
	double t_0 = (1.0 / y) + (y * -0.16666666666666666);
	double tmp;
	if (x <= 6.8e+89) {
		tmp = x * (y / x);
	} else if (x <= 2.35e+154) {
		tmp = -0.16666666666666666 * (x * (x * y));
	} else if (x <= 9.2e+165) {
		tmp = (1.0 / t_0) + (-0.16666666666666666 * ((x * x) / t_0));
	} else {
		tmp = y / (x * ((x * 0.16666666666666666) + (1.0 / x)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 / y) + (y * (-0.16666666666666666d0))
    if (x <= 6.8d+89) then
        tmp = x * (y / x)
    else if (x <= 2.35d+154) then
        tmp = (-0.16666666666666666d0) * (x * (x * y))
    else if (x <= 9.2d+165) then
        tmp = (1.0d0 / t_0) + ((-0.16666666666666666d0) * ((x * x) / t_0))
    else
        tmp = y / (x * ((x * 0.16666666666666666d0) + (1.0d0 / x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (1.0 / y) + (y * -0.16666666666666666);
	double tmp;
	if (x <= 6.8e+89) {
		tmp = x * (y / x);
	} else if (x <= 2.35e+154) {
		tmp = -0.16666666666666666 * (x * (x * y));
	} else if (x <= 9.2e+165) {
		tmp = (1.0 / t_0) + (-0.16666666666666666 * ((x * x) / t_0));
	} else {
		tmp = y / (x * ((x * 0.16666666666666666) + (1.0 / x)));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (1.0 / y) + (y * -0.16666666666666666)
	tmp = 0
	if x <= 6.8e+89:
		tmp = x * (y / x)
	elif x <= 2.35e+154:
		tmp = -0.16666666666666666 * (x * (x * y))
	elif x <= 9.2e+165:
		tmp = (1.0 / t_0) + (-0.16666666666666666 * ((x * x) / t_0))
	else:
		tmp = y / (x * ((x * 0.16666666666666666) + (1.0 / x)))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(1.0 / y) + Float64(y * -0.16666666666666666))
	tmp = 0.0
	if (x <= 6.8e+89)
		tmp = Float64(x * Float64(y / x));
	elseif (x <= 2.35e+154)
		tmp = Float64(-0.16666666666666666 * Float64(x * Float64(x * y)));
	elseif (x <= 9.2e+165)
		tmp = Float64(Float64(1.0 / t_0) + Float64(-0.16666666666666666 * Float64(Float64(x * x) / t_0)));
	else
		tmp = Float64(y / Float64(x * Float64(Float64(x * 0.16666666666666666) + Float64(1.0 / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (1.0 / y) + (y * -0.16666666666666666);
	tmp = 0.0;
	if (x <= 6.8e+89)
		tmp = x * (y / x);
	elseif (x <= 2.35e+154)
		tmp = -0.16666666666666666 * (x * (x * y));
	elseif (x <= 9.2e+165)
		tmp = (1.0 / t_0) + (-0.16666666666666666 * ((x * x) / t_0));
	else
		tmp = y / (x * ((x * 0.16666666666666666) + (1.0 / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(1.0 / y), $MachinePrecision] + N[(y * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 6.8e+89], N[(x * N[(y / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.35e+154], N[(-0.16666666666666666 * N[(x * N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 9.2e+165], N[(N[(1.0 / t$95$0), $MachinePrecision] + N[(-0.16666666666666666 * N[(N[(x * x), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y / N[(x * N[(N[(x * 0.16666666666666666), $MachinePrecision] + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x < 6.8000000000000004e89

   1. Initial program 87.6%

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

   2. Taylor expanded in y around 0 39.3%

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

   3. Taylor expanded in x around 0 26.7%

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

      1. associate-/l* 34.4%

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

      2. associate-/r/ 57.1%

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

   5. Applied egg-rr 57.1%

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

   if 6.8000000000000004e89 < x < 2.34999999999999992e154

   1. Initial program 99.7%

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

      1. associate-*l/ 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in y around 0 58.6%

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

      1. associate-/l* 58.7%

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

   6. Simplified 58.7%

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

   7. Taylor expanded in x around 0 30.0%

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

      1. +-commutative 30.0%

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

      2. fma-def 30.0%

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

      3. *-commutative 30.0%

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

      4. unpow2 30.0%

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

      5. associate-*l* 30.0%

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

      6. *-commutative 30.0%

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

   9. Simplified 30.0%

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

   10. Taylor expanded in x around inf 30.0%

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

       1. unpow2 30.0%

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

       2. associate-*r* 30.0%

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

   12. Simplified 30.0%

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

   if 2.34999999999999992e154 < x < 9.20000000000000063e165

   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 1.5%

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

   5. Taylor expanded in x around 0 100.0%

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

      1. *-commutative 100.0%

         \[\leadsto \frac{1}{\frac{1}{y} + \color{blue}{y \cdot -0.16666666666666666}} + -0.16666666666666666 \cdot \frac{{x}^{2}}{\frac{1}{y} + -0.16666666666666666 \cdot y} \]

      2. unpow2 100.0%

         \[\leadsto \frac{1}{\frac{1}{y} + y \cdot -0.16666666666666666} + -0.16666666666666666 \cdot \frac{\color{blue}{x \cdot x}}{\frac{1}{y} + -0.16666666666666666 \cdot y} \]

      3. *-commutative 100.0%

         \[\leadsto \frac{1}{\frac{1}{y} + y \cdot -0.16666666666666666} + -0.16666666666666666 \cdot \frac{x \cdot x}{\frac{1}{y} + \color{blue}{y \cdot -0.16666666666666666}} \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{1}{\frac{1}{y} + y \cdot -0.16666666666666666} + -0.16666666666666666 \cdot \frac{x \cdot x}{\frac{1}{y} + y \cdot -0.16666666666666666}} \]

   if 9.20000000000000063e165 < x

   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 73.4%

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

      1. associate-/l* 73.5%

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

   6. Simplified 73.5%

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

      1. clear-num 73.4%

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

      2. associate-/r/ 73.4%

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

   8. Applied egg-rr 73.4%

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

   9. Taylor expanded in x around 0 62.2%

      \[\leadsto \frac{y}{\color{blue}{\left(0.16666666666666666 \cdot x + \frac{1}{x}\right)} \cdot x} \]
3. Recombined 4 regimes into one program.

4. Final simplification 56.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6.8 \cdot 10^{+89}:\\ \;\;\;\;x \cdot \frac{y}{x}\\ \mathbf{elif}\;x \leq 2.35 \cdot 10^{+154}:\\ \;\;\;\;-0.16666666666666666 \cdot \left(x \cdot \left(x \cdot y\right)\right)\\ \mathbf{elif}\;x \leq 9.2 \cdot 10^{+165}:\\ \;\;\;\;\frac{1}{\frac{1}{y} + y \cdot -0.16666666666666666} + -0.16666666666666666 \cdot \frac{x \cdot x}{\frac{1}{y} + y \cdot -0.16666666666666666}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot \left(x \cdot 0.16666666666666666 + \frac{1}{x}\right)}\\ \end{array} \]

Alternative 8: 50.4% accurate, 15.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 8.8 \cdot 10^{+164} \lor \neg \left(y \leq 3.5 \cdot 10^{+223}\right):\\ \;\;\;\;x \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;y + \left(x \cdot x\right) \cdot \left(y \cdot -0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y 8.8e+164) (not (<= y 3.5e+223)))
   (* x (/ y x))
   (+ y (* (* x x) (* y -0.16666666666666666)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= 8.8e+164) || !(y <= 3.5e+223)) {
		tmp = x * (y / x);
	} else {
		tmp = y + ((x * x) * (y * -0.16666666666666666));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= 8.8d+164) .or. (.not. (y <= 3.5d+223))) then
        tmp = x * (y / x)
    else
        tmp = y + ((x * x) * (y * (-0.16666666666666666d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= 8.8e+164) || !(y <= 3.5e+223)) {
		tmp = x * (y / x);
	} else {
		tmp = y + ((x * x) * (y * -0.16666666666666666));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= 8.8e+164) or not (y <= 3.5e+223):
		tmp = x * (y / x)
	else:
		tmp = y + ((x * x) * (y * -0.16666666666666666))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= 8.8e+164) || !(y <= 3.5e+223))
		tmp = Float64(x * Float64(y / x));
	else
		tmp = Float64(y + Float64(Float64(x * x) * Float64(y * -0.16666666666666666)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= 8.8e+164) || ~((y <= 3.5e+223)))
		tmp = x * (y / x);
	else
		tmp = y + ((x * x) * (y * -0.16666666666666666));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, 8.8e+164], N[Not[LessEqual[y, 3.5e+223]], $MachinePrecision]], N[(x * N[(y / x), $MachinePrecision]), $MachinePrecision], N[(y + N[(N[(x * x), $MachinePrecision] * N[(y * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 8.80000000000000022e164 or 3.5000000000000001e223 < y

   1. Initial program 89.7%

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

   2. Taylor expanded in y around 0 45.6%

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

   3. Taylor expanded in x around 0 25.7%

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

      1. associate-/l* 29.5%

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

      2. associate-/r/ 55.5%

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

   5. Applied egg-rr 55.5%

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

   if 8.80000000000000022e164 < y < 3.5000000000000001e223

   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 3.8%

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

      1. associate-/l* 3.8%

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

   6. Simplified 3.8%

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

   7. Taylor expanded in x around 0 68.0%

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

      1. associate-*r* 68.0%

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

      2. *-commutative 68.0%

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

      3. unpow2 68.0%

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

   9. Simplified 68.0%

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

4. Final simplification 55.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 8.8 \cdot 10^{+164} \lor \neg \left(y \leq 3.5 \cdot 10^{+223}\right):\\ \;\;\;\;x \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;y + \left(x \cdot x\right) \cdot \left(y \cdot -0.16666666666666666\right)\\ \end{array} \]

Alternative 9: 51.1% 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 90.1%

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

2. Taylor expanded in y around 0 44.2%

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

3. Taylor expanded in x around 0 25.3%

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

   1. associate-/l* 28.5%

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

   2. associate-/r/ 54.3%

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

5. Applied egg-rr 54.3%

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

6. Final simplification 54.3%

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

Alternative 10: 28.3% 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 90.1%

   \[\frac{\sin x \cdot \sinh y}{x} \]
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 44.2%

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

   1. associate-/l* 54.1%

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

6. Simplified 54.1%

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

7. Taylor expanded in x around 0 28.5%

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

8. Final simplification 28.5%

   \[\leadsto y \]

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 2023230 
(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))